]> 2026-06-23T17:44:04+00:00 Characteristic subgroup 0 en tabular 1 Term invariant under all automorphisms automorphism-invariant strongly normal normal under outer automorphisms 2024-12-02T06:23:10Z 2460646.766088 Characteristic subgroup 0 Characteristic subgroup#Frobenius95 0 Characteristic subgroup#BurnsideFiniteGroups;92 92 0 Characteristic subgroup#DummitFoote;135;Section 4.4 (''Automorphisms'');formal definition 135 Section 4.4 (''Automorphisms'') formal definition 0 Characteristic subgroup#Herstein;70;Problem 7(a);introduced in exercise 70 Problem 7(a) introduced in exercise 0 Characteristic subgroup#Artin;234;Section 8 (''generators and relations''), Exercise 7;introduced in exercise 234 Section 8 (''generators and relations''), Exercise 7 introduced in exercise 0 Characteristic subgroup#Fraleigh;428;Exercises 8.6, Concepts, Point 4;introduced in exercise 428 Exercises 8.6, Concepts, Point 4 introduced in exercise 0 Characteristic subgroup#Gallian;168 168 0 Characteristic subgroup#CohnBasicAlgebra;46;Section 2.6;definition in paragraph 46 Section 2.6 definition in paragraph 0 Characteristic subgroup#Hungerford;103;Chapter II, Section 7;definition in paragraph 103 Chapter II, Section 7 definition in paragraph 0 Characteristic subgroup#IsaacsGT;11;definition in paragraph 11 definition in paragraph 0 Characteristic subgroup#AlperinBell;17;formal definition in paragraph 17 formal definition in paragraph 0 Characteristic subgroup#KhukhroNGA;4;Section 1.1;definition in paragraph 4 Section 1.1 definition in paragraph 0 Characteristic subgroup#FGTAsch;25 25 0 Characteristic subgroup#RobinsonGT;28;Section 1.5;definition in paragraph 28 Section 1.5 definition in paragraph 0 Characteristic subgroup#RotmanGT;104;formal definition 104 formal definition 0 Characteristic subgroup#Frobenius95 0 1 plainlist 2 [[Definition style::tabular]] Characteristic subgroup# QUERYa06e7f16fa3dc8cb712b7a1b798724eb 0 1 table 2 [[Defining ingredient::Characteristic subgroup]] Characteristic subgroup# QUERY55395050ea172d0d6c3f628a0a46b1b8 0 2 table 5 [[Fact about:: <q>[[Page::Characteristic subgroup]] [[Number::1]]</q> ]] Characteristic subgroup# QUERY6ac54b11840a557f9c295d89c5e47962 0 2 table 3 [[Fact about.Page::Characteristic subgroup]] Characteristic subgroup# QUERY2ff4463348e01ea87a0c9241fbb7966c 0 1 table 2 [[Survey article about::Characteristic subgroup]] Characteristic subgroup# QUERY95a4ba54f66285f3a26d650fde6b7f11 0 2 plainlist 3 [[Survey article about.Defining ingredient::Characteristic subgroup]] Characteristic subgroup# QUERYb11caf297924194e58e78e7366aa526f 0 1 plainlist 2 [[Analogue of::Characteristic subgroup]] Characteristic subgroup# QUERY7fc565309111a9c9e1d29eb101b286d3 0 1 plainlist 2 [[Variation of::Characteristic subgroup]] Characteristic subgroup# QUERY198fc0a4636c1b745c4fe2bf3c1ebf9a 0 1 plainlist 2 [[Opposite of::Characteristic subgroup]] Characteristic subgroup# QUERY154f057d9dc2b7aa69f911bae4a9cf1f 0 2 plainlist 4 [[Fact about.Page::Characteristic subgroup]] [[Category:Subgroup property implications]] Characteristic subgroup# QUERYa1f7e61d8b1b48bd6ed9f18a212fc062 0 2 plainlist 4 [[Fact about.Page::Characteristic subgroup]] [[Category:Subgroup property non-implications]] Characteristic subgroup# QUERY1f3141e0f114f7c5cf30753e39cddf94 0 2 plainlist 4 [[Fact about.Page::Characteristic subgroup]] [[Category:Subgroup metaproperty satisfactions]] Characteristic subgroup# QUERY43ed068db99e49ffd0eb6decb98a1f04 0 2 plainlist 4 [[Fact about.Page::Characteristic subgroup]] [[Category:Subgroup metaproperty dissatisfactions]] Characteristic subgroup# QUERY2cf3ae37f355dbcf847dba5637ce95de 0 2 plainlist 4 [[Fact about.Page::Characteristic subgroup]] [[Category:Subgroup property satisfactions]] [[Category:Subgroup property satisfactions]] [[Fact about.Page::Characteristic subgroup]] Characteristic subgroup# QUERYf8128bb60541907c61e48c4626880971 0 2 plainlist 4 [[Fact about.Page::Characteristic subgroup]] [[Category:Subgroup property dissatisfactions]] [[Category:Subgroup property dissatisfactions]] [[Fact about.Page::Characteristic subgroup]] Characteristic subgroup# QUERY33525c29530957d864b95d3b42e7e35e 0 1 plainlist 2 [[Uses property satisfaction of::Characteristic subgroup]] Characteristic subgroup# QUERY7c5592e855c53ffa9aa708de671f3ab4 0 1 plainlist 2 [[Proves property satisfaction of::Characteristic subgroup]] Characteristic subgroup# QUERYe5a59a67fe83053408b3e4ec128c2d46 0 1 table 2 [[Variation of::Normality]] Characteristic subgroup# QUERYc59ca6f7b6c7879085cb4a7c4dccd769 0 2 table 6 <q> <q>[[Satisfies property::Characteristic subgroup]]</q> OR [[Satisfies property.Stronger than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERYe565c9b7bcdbdfe0820af432788f561c 0 2 table 6 <q> <q>[[Dissatisfies property::Characteristic subgroup]]</q> OR [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY19258d880aff1bcdd33cd59710c409a5 0 0 plainlist 1 [[:Characteristicity is transitive]] Characteristic subgroup# QUERY7bf750b01d97f613f8288d39f7e032fe 0 0 plainlist 1 [[:Characteristicity is strongly intersection-closed]] Characteristic subgroup# QUERYb246e87d7703c03fc39400c00ed09fc6 0 0 plainlist 1 [[:Characteristicity is strongly join-closed]] Characteristic subgroup# QUERY0eec044d3c73c725f937ded1e7e6a4b3 0 0 plainlist 1 [[:Characteristicity is quotient-transitive]] Characteristic subgroup# QUERYa71760b4a2caa50922333d640ef073a2 0 0 plainlist 1 [[:Characteristicity does not satisfy intermediate subgroup condition]] Characteristic subgroup# QUERY54145834bd5c7448b5656d41e5b11efd 0 0 plainlist 1 [[:Characteristicity is not upper join-closed]] Characteristic subgroup# QUERY9ded037d559e0d3ca7bb064559c26f49 0 0 plainlist 1 [[:Characteristicity is commutator-closed]] Characteristic subgroup# QUERYbf0141e136c85653672139c09f410196 0 0 plainlist 1 [[:Characteristicity is centralizer-closed]] Characteristic subgroup# QUERYaa1ae581fdf28638a6719b7397333868 0 0 plainlist 1 [[:Characteristicity does not satisfy lower central series]] Characteristic subgroup# QUERY1e6fe3282153ad2e3b068da2568253da 0 0 plainlist 1 [[:Characteristicity is not finite direct power-closed]] Characteristic subgroup# QUERY53d9feae764698b132b1298cf7054f2e 0 0 plainlist 1 [[:Characteristicity is not finite-relative-intersection-closed]] Characteristic subgroup# QUERYdd7201ef2fc5a3f97fb5ed71cbd13cf3 0 0 plainlist 1 [[:Characteristicity satisfies partition difference condition]] Characteristic subgroup# QUERYc81d22bc3e89b700b9ec1953e567e847 0 0 plainlist 1 [[:No subgroup property between normal Sylow and subnormal or between Sylow retract and retract is conditionally lattice-determined]] Characteristic subgroup# QUERYee358842c83a611407fde6b13b8d909e 0 2 table 12 <q>[[Satisfies property::Characteristic subgroup]] [[Dissatisfies property::Fully invariant subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Characteristic subgroup]] [[Dissatisfies property.Weaker than::Fully invariant subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY289079cacd555bbe0c2c7ddb759e58f2 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Fully invariant subgroup]] Characteristic subgroup# QUERY706af4aeca70c5b52f9d6f213b4c761b 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Fully invariant subgroup]] Characteristic subgroup# QUERY98d1861686ad2da1a3e9803e73541e0b 0 2 table 12 <q>[[Satisfies property::Characteristic subgroup]] [[Dissatisfies property::Isomorph-containing subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Characteristic subgroup]] [[Dissatisfies property.Weaker than::Isomorph-containing subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERYfc4ac0899159a3aefa107a001dfc40b5 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Isomorph-containing subgroup]] Characteristic subgroup# QUERY91be1adcc1c68c76ccb96c72835e637a 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Isomorph-containing subgroup]] Characteristic subgroup# QUERY589e93175e8ba79b07f4cd8f6ef8a49b 0 2 table 12 <q>[[Satisfies property::Characteristic subgroup]] [[Dissatisfies property::Isomorph-free subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Characteristic subgroup]] [[Dissatisfies property.Weaker than::Isomorph-free subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY79708ae4ac1f4d3dc2633f350656ac04 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Isomorph-free subgroup]] Characteristic subgroup# QUERYc9ddc45eb1d0c8bf95d7c935153ecce2 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Isomorph-free subgroup]] Characteristic subgroup# QUERYf7daedf972e9452cd31819105bcddd06 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Injective endomorphism-invariant subgroup]] Characteristic subgroup# QUERY1a9f84122651b2cdcefd4750097c0d85 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Injective endomorphism-invariant subgroup]] Characteristic subgroup# QUERY6646a500388cb4adcfa5a3a68d3ff610 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Strictly characteristic subgroup]] Characteristic subgroup# QUERY5806094ca073dc22430df0bb0a4c5839 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Strictly characteristic subgroup]] Characteristic subgroup# QUERY3379c0a64324624be1874110185ea295 0 2 table 12 <q>[[Satisfies property::Characteristic subgroup]] [[Dissatisfies property::Homomorph-containing subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Characteristic subgroup]] [[Dissatisfies property.Weaker than::Homomorph-containing subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY02e2389cfd5d75ed20583a00a6f5c16c 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Homomorph-containing subgroup]] Characteristic subgroup# QUERYb78d7e953d9150cb36dc3960e703441d 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Homomorph-containing subgroup]] Characteristic subgroup# QUERYec394b9310f12ef8cff563849f835bb1 0 2 table 12 <q>[[Satisfies property::Characteristic subgroup]] [[Dissatisfies property::Subhomomorph-containing subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Characteristic subgroup]] [[Dissatisfies property.Weaker than::Subhomomorph-containing subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY449ab975e0ee6114dc45b878914633be 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Subhomomorph-containing subgroup]] Characteristic subgroup# QUERYc2337bffec85c3bcf50d29bea912601b 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Subhomomorph-containing subgroup]] Characteristic subgroup# QUERY1b5d110484086573c7b9a95431e213bc 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Variety-containing subgroup]] Characteristic subgroup# QUERY00abdfb4fec0108d4b6f1e11c7d75da9 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Variety-containing subgroup]] Characteristic subgroup# QUERY5d7f8a3300028b63e88bdab520854742 0 2 table 12 <q>[[Satisfies property::Characteristic subgroup]] [[Dissatisfies property::Quotient-isomorph-containing subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Characteristic subgroup]] [[Dissatisfies property.Weaker than::Quotient-isomorph-containing subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY940d0dc142b53e2f0d925c192c74cacc 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Quotient-isomorph-containing subgroup]] Characteristic subgroup# QUERYc215b46ba2fe2265775f1cab7165a7d0 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Quotient-isomorph-containing subgroup]] Characteristic subgroup# QUERY719471de9aa3b27c38dccd0bb6b26cf1 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Quotient-subisomorph-containing subgroup]] Characteristic subgroup# QUERYb09cf91adc084706a468e64e3783bc04 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Quotient-subisomorph-containing subgroup]] Characteristic subgroup# QUERY1497b8611b61782a0e9adfd10e9f3cc8 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Elementarily characteristic subgroup]] Characteristic subgroup# QUERY959ec0f2db6c82958aa297fbfb82ab9e 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Elementarily characteristic subgroup]] Characteristic subgroup# QUERY28e918a371d0bb6698a8d7b8c8a406ff 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Purely definable subgroup]] Characteristic subgroup# QUERYabf8ce238ae193adf988017b113bb7e1 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Purely definable subgroup]] Characteristic subgroup# QUERY248024949ff3986a43529c5f1cbcf1d5 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::MSO-definable subgroup]] Characteristic subgroup# QUERY5d52d64caf08f439bd43708bac55ce37 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::MSO-definable subgroup]] Characteristic subgroup# QUERY6d9d5561f9ba123436a95e5dc06cfcc8 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Intermediately characteristic subgroup]] Characteristic subgroup# QUERY704222ed62ed5715db8b90c84e87880d 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Intermediately characteristic subgroup]] Characteristic subgroup# QUERY2544fc0257f93512aae6e7fbdfe06f4d 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Transfer-closed characteristic subgroup]] Characteristic subgroup# QUERY7c4c04c7484732e264814ef81763db8d 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Transfer-closed characteristic subgroup]] Characteristic subgroup# QUERY190f4887be65077a7b0e65b607dd3565 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Verbal subgroup]] Characteristic subgroup# QUERY68fb79d427b314b546899f21c3a7835b 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Verbal subgroup]] Characteristic subgroup# QUERY2a9edd6312f251e9120e0a5ef6256271 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Existentially bound-word subgroup]] Characteristic subgroup# QUERYc2b64d608f47364e65eaf71a9fc34e9d 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Existentially bound-word subgroup]] Characteristic subgroup# QUERY16ac3f6154b6c3bc4512010ad0b7f2a8 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Bound-word subgroup]] Characteristic subgroup# QUERY5b90c2002a8057fe5e0687b4c39efc13 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Bound-word subgroup]] Characteristic subgroup# QUERYb1bbdc5ca69bedb882120e69631151a6 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Finite direct power-closed characteristic subgroup]] Characteristic subgroup# QUERY158f5d6f2f92ef06f7f39729a75d9b66 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Finite direct power-closed characteristic subgroup]] Characteristic subgroup# QUERY6490e55919a3a5e51447badcd4d730dd 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Direct power-closed characteristic subgroup]] Characteristic subgroup# QUERY42e6f8d23ef0b22cd5da94f0ba3070b3 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Direct power-closed characteristic subgroup]] Characteristic subgroup# QUERY413b0913d35d12a006616299a28d36dc 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Quasiautomorphism-invariant subgroup]] Characteristic subgroup# QUERY74d52ce674776d5bdc562099d5f34469 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Quasiautomorphism-invariant subgroup]] Characteristic subgroup# QUERY63ef07d616842183d149ee17d77070c6 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::1-automorphism-invariant subgroup]] Characteristic subgroup# QUERY6c4aa736d696dbff27836eb80ee290f3 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::1-automorphism-invariant subgroup]] Characteristic subgroup# QUERYd6918b8dd475e9c6cc73716db4902168 0 1 plainlist 2 [[Stronger than::Characteristic subgroup]] Characteristic subgroup# QUERY078105b3e0b1eaea76e88857fe6aa3c1 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Transitive subgroup property]] Characteristic subgroup# QUERY44526c56ff7ed747ba487959721bcc04 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Intermediate subgroup condition]] Characteristic subgroup# QUERY33724b432faf50b158473dc204dacd97 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Transfer condition]] Characteristic subgroup# QUERY9561e9e56f1330d820cdd0512b444b0b 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Quotient-transitive subgroup property]] Characteristic subgroup# QUERYfa5b8b9e552f34e4e02144d82bdbbb60 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Intersection-closed subgroup property]] Characteristic subgroup# QUERY3576f92402fcdccc4a97973cdb75789b 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Join-closed subgroup property]] Characteristic subgroup# QUERYb31f110fba57fc68a0e1b32b1d59585b 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Trim subgroup property]] Characteristic subgroup# QUERYcd68c2296c274fb9fa798d72ab366cbd 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Inverse image condition]] Characteristic subgroup# QUERY1b456ec4996b72e708093db3b324149a 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Image condition]] Characteristic subgroup# QUERY5a2c3563f66829388369d6befe10fc5b 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Satisfies metaproperty::Centralizer-closed subgroup property]] Characteristic subgroup# QUERYa7e402d12f33041c6c552054247d3668 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Transitive subgroup property]] Characteristic subgroup# QUERYcb0866fccdc93b34c9a9e8ea4b10f068 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Intermediate subgroup condition]] Characteristic subgroup# QUERY77907eac27a7d83138e52568cc2edcbc 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Transfer condition]] Characteristic subgroup# QUERYbe7bc32d19636a5a94c49792ea6aa8f2 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Quotient-transitive subgroup property]] Characteristic subgroup# QUERY1b9ad510a4fff1d4a4c02ec1d3b1b3a1 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Intersection-closed subgroup property]] Characteristic subgroup# QUERYbb3b2a447559dd7124d69f55a34dbf71 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Join-closed subgroup property]] Characteristic subgroup# QUERY603b894b9bdc4a69dbfc9d68d1d0fe9a 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Trim subgroup property]] Characteristic subgroup# QUERYaf9506c9f05870bd49466e9cc1dbcaa6 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Inverse image condition]] Characteristic subgroup# QUERY970074efe53a8f46068be0aa19a6cb47 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Image condition]] Characteristic subgroup# QUERYe7053561254024021bd21575a2157689 0 1 plainlist 4 [[Stronger than::Characteristic subgroup]] [[Dissatisfies metaproperty::Centralizer-closed subgroup property]] Characteristic subgroup# QUERY842b14971a0d68b8173876ab44b3ada9 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic central factor]] Characteristic subgroup# QUERYbfe95b507d82b01833739b1701f1c428 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic central factor]] Characteristic subgroup# QUERYcd0bef60c610c7b2465feee939d26aac 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Conjugacy-closed characteristic subgroup]] Characteristic subgroup# QUERYcad13c2465dba94d713d233ea65bebe9 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Conjugacy-closed characteristic subgroup]] Characteristic subgroup# QUERY66ffbd41f8709a69431490bf56ae4ccc 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic transitively normal subgroup]] Characteristic subgroup# QUERY25963084d0acf267f30708750bd46952 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic transitively normal subgroup]] Characteristic subgroup# QUERY776a7372d20037dca5f04451399eeca3 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of finite index]] Characteristic subgroup# QUERYf2dfb75695d3c79ab5d8fc0ce2e27a54 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of finite index]] Characteristic subgroup# QUERY2a1279dee5e74e8f8298b86f216bef69 0 1 plainlist 3 [[Category:Subgroup property conjunctions]] [[Conjunction involving::Characteristic subgroup]] Characteristic subgroup# QUERYcece0a4506b79982e792ae7e58496766 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Abelian characteristic subgroup]] Characteristic subgroup# QUERYb46e142a35139698a20574a7376410f1 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Abelian characteristic subgroup]] Characteristic subgroup# QUERYce07e1f4c451b036dcbaf5bb7b050b26 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Nilpotent characteristic subgroup]] Characteristic subgroup# QUERYe989e9326fa47f0ae6aa82062c6280ab 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Nilpotent characteristic subgroup]] Characteristic subgroup# QUERYa11bc73b56cd5c1600f6a8e1f2853154 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Cyclic characteristic subgroup]] Characteristic subgroup# QUERY6618f2fa7b8c221e14f252bb7bf351a0 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Cyclic characteristic subgroup]] Characteristic subgroup# QUERY02ca209bc957d753f84b80cff82d3857 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Solvable characteristic subgroup]] Characteristic subgroup# QUERYb7a6bca567ff20f89bf3d4474aae0f00 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Solvable characteristic subgroup]] Characteristic subgroup# QUERY8e58908649c322fa359c70cc60515cb5 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Finite characteristic subgroup]] Characteristic subgroup# QUERY633f25086b6a0603aab8b3121ced4363 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Finite characteristic subgroup]] Characteristic subgroup# QUERY3225b7fb4b80cd4ce5655c173371934d 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Perfect characteristic subgroup]] Characteristic subgroup# QUERY8949f4bac1dd08acea88cc9de987e9f4 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Perfect characteristic subgroup]] Characteristic subgroup# QUERYe0f2f68748c31a6404b50c756eba2e62 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Simple characteristic subgroup]] Characteristic subgroup# QUERYc01e369c34bb80a30b248d56d3a6d338 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Simple characteristic subgroup]] Characteristic subgroup# QUERYcb704b90f80d548b16df5f2c1beacec3 0 1 plainlist 3 [[Category:Conjunctions of group property with subgroup property]] [[Conjunction involving::Characteristic subgroup]] Characteristic subgroup# QUERY4f3000bd3e72cd9442ec78b53216178e 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of finite group]] Characteristic subgroup# QUERY5b6aaf0c8426fd26d474c735410c6a28 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of finite group]] Characteristic subgroup# QUERYec4dcca50357e8e25082fffe2a4c12c8 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of abelian group]] Characteristic subgroup# QUERY729d4e409f16571607a748f05b907053 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of abelian group]] Characteristic subgroup# QUERY5215ed0fb4152858a385daa19ed3bbad 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Subgroup of cyclic group]] Characteristic subgroup# QUERY6c4eae9ec06fe36631bd2d1f20c9fccd 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Subgroup of cyclic group]] Characteristic subgroup# QUERY8e3d68a8d2ff1e23a95613aadbd8df83 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of finite abelian group]] Characteristic subgroup# QUERYd66310f6e7f4f65616fb20392a12e24f 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of finite abelian group]] Characteristic subgroup# QUERY93bccd707c78262e32c16383c6e569f8 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of nilpotent group]] Characteristic subgroup# QUERY918ded01e4682596880ef7d0a193ea15 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of nilpotent group]] Characteristic subgroup# QUERYd7ed4d04965aab8cb99d8eaf8f793f78 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of group of prime power order]] Characteristic subgroup# QUERY5098661c23f5b2b419f572f9c905f6a4 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of group of prime power order]] Characteristic subgroup# QUERY86d3afc1c74a0abc04485b3c28e29389 0 1 list 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of solvable group]] Characteristic subgroup# QUERY0002b18e48460ec8ce54ea30778aa5c7 0 1 broadtable 4 [[Stronger than::Characteristic subgroup]] [[Weaker than::Characteristic subgroup of solvable group]] Characteristic subgroup# QUERY489c02c66a5471fb28b477cd0f268f6b 0 2 table 12 <q>[[Satisfies property::Normal subgroup]] [[Dissatisfies property::Characteristic subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Normal subgroup]] [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY5e2655e13ad05870c73678d116507e6e 0 1 list 4 [[Stronger than::Normal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY9c20fd8454bcc1441e1ba02d01c45f2f 0 1 broadtable plainlist 4 [[Stronger than::Normal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY4e47f44a59a6362fe976eceed7a72aad 0 2 table 12 <q>[[Satisfies property::Subnormal subgroup]] [[Dissatisfies property::Characteristic subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Subnormal subgroup]] [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERYffea599966136912409925611f23a7ac 0 1 list 4 [[Stronger than::Subnormal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY71ff367159e37f6299e2dadd7cc2da42 0 1 broadtable 4 [[Stronger than::Subnormal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERYbd6092293b9f4f205574c979a8a7e266 0 2 table 12 <q>[[Satisfies property::Cofactorial automorphism-invariant subgroup]] [[Dissatisfies property::Characteristic subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Cofactorial automorphism-invariant subgroup]] [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERYe748665390d832079effb6ab7450063f 0 1 list 4 [[Stronger than::Cofactorial automorphism-invariant subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY7225d4b66f4e2b66c2f1bdb123842f9c 0 1 broadtable 4 [[Stronger than::Cofactorial automorphism-invariant subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERYa2ea570295eeb524e8edfb2f3a668634 0 1 list 4 [[Stronger than::Coprime automorphism-invariant subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY703d00c10221cfcb8c11d6672030baf9 0 1 broadtable 4 [[Stronger than::Coprime automorphism-invariant subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY7cc3370edb5c51054c42b271e6e8be64 0 1 list 4 [[Stronger than::Coprime automorphism-invariant normal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY84b9e1c95ef03188517b5246060f85b6 0 1 broadtable 4 [[Stronger than::Coprime automorphism-invariant normal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY25f5deb71227dcc0170af8e79a3ab886 0 2 table 12 <q>[[Satisfies property::Left-transitively 2-subnormal subgroup]] [[Dissatisfies property::Characteristic subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Left-transitively 2-subnormal subgroup]] [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY26eae43ad995034c1acb1a2854853467 0 1 list 4 [[Stronger than::Left-transitively 2-subnormal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY063fecc27a5354e3e9a2d600c82a81f6 0 1 broadtable 4 [[Stronger than::Left-transitively 2-subnormal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY257c6a81e1748688da1e35e996e2f604 0 2 table 12 <q>[[Satisfies property::Left-transitively fixed-depth subnormal subgroup]] [[Dissatisfies property::Characteristic subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Left-transitively fixed-depth subnormal subgroup]] [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY28774feb4f1c5d5d409f7cfd2e05e794 0 1 list 4 [[Stronger than::Left-transitively fixed-depth subnormal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERYcb0535d20cc4d7733d2d9c9bf1751f87 0 1 broadtable 4 [[Stronger than::Left-transitively fixed-depth subnormal subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY5c2f552f62ec346663e78d9a7d08d1f9 0 2 table 12 <q>[[Satisfies property::Automorph-conjugate subgroup]] [[Dissatisfies property::Characteristic subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Automorph-conjugate subgroup]] [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY427c649892c9858ab4b0970939ed27c6 0 1 list 4 [[Stronger than::Automorph-conjugate subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY8ce6e5cf2ebf6060595d26df6194f69f 0 1 broadtable 4 [[Stronger than::Automorph-conjugate subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY3e3142cba27ed1cbe1c8350dc6a7ebbb 0 2 table 12 <q>[[Satisfies property::Core-characteristic subgroup]] [[Dissatisfies property::Characteristic subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Core-characteristic subgroup]] [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERYbaa48534184cd4a665c1489b56047baa 0 1 list 4 [[Stronger than::Core-characteristic subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY82ef2f51c5f2eb6ecde2f95f9ae0edcc 0 1 broadtable 4 [[Stronger than::Core-characteristic subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY4e01c1c6442473b95d77ab6ab481923b 0 2 table 12 <q>[[Satisfies property::Closure-characteristic subgroup]] [[Dissatisfies property::Characteristic subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Closure-characteristic subgroup]] [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERYf26a6397720d223d939b20e12501f088 0 1 list 4 [[Stronger than::Closure-characteristic subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY44ab7d2361b862e17ca9f742275e61aa 0 1 broadtable 4 [[Stronger than::Closure-characteristic subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERYbbab77a5ed1817f5b2138ef7dd890706 0 2 table 12 <q>[[Satisfies property::Procharacteristic subgroup]] [[Dissatisfies property::Characteristic subgroup]] [[Category:Particular subgroups]] OR [[Satisfies property.Stronger than::Procharacteristic subgroup]] [[Dissatisfies property.Weaker than::Characteristic subgroup]] [[Category:Particular subgroups]]</q> Characteristic subgroup# QUERY01863e78f77ee70bb9e8f7d37dad0966 0 1 list 4 [[Stronger than::Procharacteristic subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERYd3a18f2db21b816d91e1206a5af42d58 0 1 broadtable 4 [[Stronger than::Procharacteristic subgroup]] [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERYb6a72c49f3505848890d75ad4b10dd06 0 1 plainlist 2 [[Weaker than::Characteristic subgroup]] Characteristic subgroup# QUERY9813ececc8534fee4e5555b3e40ad625 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Transitive subgroup property]] Characteristic subgroup# QUERYef14d2b6f9e2830eb882b3d8d43b8828 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Intermediate subgroup condition]] Characteristic subgroup# QUERY53881db0e5fdaa9c368e031e2773fb27 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Transfer condition]] Characteristic subgroup# QUERY90d30e1ff7d1f4a840d8433759908d5d 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Quotient-transitive subgroup property]] Characteristic subgroup# QUERYff802c4276cdd032e5a87c54a10da58b 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Intersection-closed subgroup property]] Characteristic subgroup# QUERYded17ca13b4b9837edd560a91448f99c 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Join-closed subgroup property]] Characteristic subgroup# QUERYc4c5592a53a24840968ea58d28779d3f 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Trim subgroup property]] Characteristic subgroup# QUERY124342af7182112386db3c517db89bb7 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Inverse image condition]] Characteristic subgroup# QUERY7f395da0b6f4dda84742b6a412b1679e 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Image condition]] Characteristic subgroup# QUERY0ac6cf28d427b2c6a9627f7dfa2a85a0 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Satisfies metaproperty::Centralizer-closed subgroup property]] Characteristic subgroup# QUERY9903ebc8159cc1e28fce0bacbf9c81e9 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Transitive subgroup property]] Characteristic subgroup# QUERYa96e14564a1990d3e0f41ac8784d47a5 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Intermediate subgroup condition]] Characteristic subgroup# QUERYb5349faa8747d9f49fb202c2fb987062 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Transfer condition]] Characteristic subgroup# QUERYd5b75fa1b9c1b95515fa6ea834c724b9 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Quotient-transitive subgroup property]] Characteristic subgroup# QUERYf72f508fa24d8fc19165ad073425d30a 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Intersection-closed subgroup property]] Characteristic subgroup# QUERY6fb2d77428634aa87e68666c0d2c8cbe 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Join-closed subgroup property]] Characteristic subgroup# QUERY79ca6f0bc9bf997fe8531c37565abac9 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Trim subgroup property]] Characteristic subgroup# QUERYfd35f0f126e726de98792b1b2052676e 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Inverse image condition]] Characteristic subgroup# QUERYc9fb04dc9481bdac413a5d6da9bf8332 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Image condition]] Characteristic subgroup# QUERY0a2239614da0505b7dd0f77716fe0cb8 0 1 plainlist 4 [[Weaker than::Characteristic subgroup]] [[Dissatisfies metaproperty::Centralizer-closed subgroup property]] Characteristic subgroup# QUERYc3eff2e839f04fd2a62d38bff9c0e748 0 1 plainlist 2 [[Satisfies metaproperty::Second-order subgroup property]] Characteristic subgroup# QUERYe7955396f5bf66153df1ba4f6eab9191 0 1 plainlist 2 [[Satisfies metaproperty::Function restriction-expressible subgroup property]] Characteristic subgroup# QUERY6aa24c02f8e67de24223fa849790e0e6 0 1 plainlist 2 [[Satisfies metaproperty::Relation implication-expressible subgroup property]] Characteristic subgroup# QUERYda878a4b9f5ff2713c910190c3a8d6ad 0 1 plainlist 2 [[Satisfies metaproperty::GAP-testable subgroup property]] Characteristic subgroup# QUERYab3deb33606903472662717bd3ca61c2 0 1 plainlist 2 [[Satisfies metaproperty::GAP-listable subgroup property]] Characteristic subgroup# QUERYe6c7e410aa0a9124a787ec94e3720a50 0 1 plainlist 2 [[Satisfies metaproperty::GAP-codable subgroup property]] Characteristic subgroup# QUERY0db1d57349eaa4fa10fe190d719c5331 Derived subgroup 0 en Derived subgroup Center 0 en Center Center of dihedral group:D8 0 en Center of dihedral group:D8 Characteristicity is transitive 0 en Characteristicity is transitive Fully invariant implies characteristic 0 en Fully invariant implies characteristic Left-transitively complemented normal implies characteristic 0 en Left-transitively complemented normal implies characteristic Isomorph-containing implies characteristic 0 en Isomorph-containing implies characteristic Left-transitively permutable implies characteristic 0 en Left-transitively permutable implies characteristic Self-centralizing and minimal normal implies characteristic 0 en Self-centralizing and minimal normal implies characteristic Normality-preserving endomorphism-invariant implies characteristic 0 en Normality-preserving endomorphism-invariant implies characteristic Derivation-invariant Lie subring 0 en Derivation-invariant Lie subring Isomorph-normal characteristic subgroup 0 en Isomorph-normal characteristic subgroup Characteristic implies normal 0 en Characteristic implies normal Characteristicity is transitive 0 en Characteristicity is transitive Characteristic of CDIN implies CDIN 0 en Characteristic of CDIN implies CDIN Characteristic implies automorph-conjugate 0 en Characteristic implies automorph-conjugate Critical subgroup 0 en Critical subgroup Direct factor of characteristic subgroup 0 en Direct factor of characteristic subgroup Isomorph-normal characteristic subgroup 0 en Isomorph-normal characteristic subgroup Fully invariant subgroup 0 en Fully invariant subgroup Isomorph-normal characteristic subgroup 0 en Isomorph-normal characteristic subgroup Normal subgroup 0 en Normal subgroup Direct factor of characteristic subgroup 0 en Direct factor of characteristic subgroup Coprime automorphism-invariant subgroup 0 en Coprime automorphism-invariant subgroup 0 Characteristicity is transitive#Characteristic subgroup;1# 379a7510701925b6767a584c6105c9a1 0 Characteristic implies normal#Characteristic subgroup;1# 379a7510701925b6767a584c6105c9a1 0 Normal not implies characteristic#Characteristic subgroup;1# 379a7510701925b6767a584c6105c9a1 0 Characteristic of normal implies normal#Characteristic subgroup# 9a34df27c99679c88be5eda450e18c0d 0 Characteristicity is not upper join-closed#Characteristic subgroup;1# 379a7510701925b6767a584c6105c9a1 0 Characteristicity is not finite direct power-closed#Characteristic subgroup;1# 379a7510701925b6767a584c6105c9a1 0 Characteristic of normal implies normal#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Left transiter of normal is characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristicity is quotient-transitive#Characteristic subgroup;1# 379a7510701925b6767a584c6105c9a1 0 Characteristic upper-hook AEP implies characteristic#Characteristic subgroup# 9a34df27c99679c88be5eda450e18c0d 0 Fully invariant implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Left-transitively complemented normal implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic central factor of WNSCDIN implies WNSCDIN#Characteristic subgroup# 9a34df27c99679c88be5eda450e18c0d 0 Cyclic-quotient characteristic implies upward-closed characteristic#Characteristic subgroup# 9a34df27c99679c88be5eda450e18c0d 0 Isomorph-containing implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Left-transitively permutable implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic of CDIN implies CDIN#Characteristic subgroup# 9a34df27c99679c88be5eda450e18c0d 0 Characteristic of CDIN implies CDIN#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Additive group of a field implies characteristic in holomorph#Characteristic subgroup# 9a34df27c99679c88be5eda450e18c0d 0 Odd-order cyclic group is characteristic in holomorph#Characteristic subgroup# 9a34df27c99679c88be5eda450e18c0d 0 Nilpotent not implies ACIC#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Cyclic characteristic implies hereditarily characteristic#Characteristic subgroup# 9a34df27c99679c88be5eda450e18c0d 0 Self-centralizing and minimal normal implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic not implies strictly characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic and self-centralizing implies coprime automorphism-faithful#Characteristic subgroup# 9a34df27c99679c88be5eda450e18c0d 0 Characteristic not implies potentially fully invariant#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic not implies amalgam-characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic not implies fully invariant#Characteristic subgroup;1# 379a7510701925b6767a584c6105c9a1 0 Characteristic not implies quasiautomorphism-invariant#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic not implies injective endomorphism-invariant#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Normal-potentially characteristic not implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic implies automorph-conjugate#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic subgroup of abelian group not implies local powering-invariant#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Normality-preserving endomorphism-invariant implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic subgroup of abelian group not implies divisibility-closed#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Direct factor not implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Index two not implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 P-automorphism-invariant not implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic not implies characteristic-isomorph-free in finite#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic not implies elementarily characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 No nontrivial homomorphism from quotient group not implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Left-transitively WNSCDIN not implies characteristic#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Series-equivalent characteristic subgroups may be distinct#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 0 Characteristic not implies direct factor#Characteristic subgroup;2# 6fbfe18f13df0d45a525c3bad2c16b88 Satisfies property 132 en Satisfies property Proves property satisfaction of 132 en Proves property satisfaction of Analogue of 132 en Analogue of Conjunction involving 132 en Conjunction involving Uses property satisfaction of 132 en Uses property satisfaction of Defining ingredient 132 en Defining ingredient Stronger than 132 en Stronger than Weaker than 132 en Weaker than Page 132 en Page