Every finite group has a finite composition series
P-Frattini-realizable implies not non-abelian cyclic-center
Order has only two prime factors implies solvable
Supersolvable implies nilpotent derived subgroup
Abelian implies every element is automorphic to its inverse
Finite minimal simple implies 2-generated
Finitely generated and residually finite implies Hopfian
Finitely generated implies finitely many homomorphisms to any finite group
Linearly primitive implies cyclic-center
Ascending chain condition on normal subgroups implies Hopfian
Ascending chain condition on subnormal subgroups implies subnormal join property
Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic
Strongly p-solvable implies Glauberman type for odd p
Maximal among abelian normal implies self-centralizing in nilpotent
Finitely many homomorphisms to any finite group implies every subgroup of finite index has finitely many automorphic subgroups
Automorphism group is transitive on non-identity elements implies characteristically simple
Glauberman type implies ZJ-functor controls fusion
Finite normal implies amalgam-characteristic
Conjugacy-separable implies residually finite
Cyclic quotient of automorphism group by class-preserving automorphism group implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group
Nilpotent implies every subgroup is subnormal
Every subgroup is subnormal implies normalizer condition
Schur multiplier of cyclic group is trivial
Class-inverting automorphism implies every element is automorphic to its inverse
Nilpotent derived subgroup implies subnormal join property
Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup
Nilpotent implies center is normality-large
Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup
Metabelian implies subnormal join property
Finitely generated abelian is subgroup-closed
Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini
Prime power order implies subgroups of all orders dividing the group order
Nilpotent implies every maximal subgroup is normal
Normalizer condition implies every maximal subgroup is normal
Multiplicative group of a field implies every finite subgroup is cyclic
At most n elements of order dividing n implies every finite subgroup is cyclic
FZ implies generalized subnormal join property
Locally cyclic implies abelian automorphism group
Algebra group implies power degree group for field size
SQ-universal implies no nontrivial identity
Additive group of a field implies characteristic in holomorph
Class two implies generated by abelian normal subgroups
Abelian implies every subgroup is normal
Finite supersolvable implies subgroups of all orders dividing the group order
Odd-order cyclic group is characteristic in holomorph
Finitely generated inner automorphism group implies every locally inner automorphism is inner
4-Engel implies locally nilpotent for groups
Normalizer condition implies locally nilpotent
Nilpotent automorphism group implies nilpotent of class at most one more
Characteristically simple implies CSCFN-realizable
Solvable automorphism group implies solvable of derived length at most one more
Finitely presented implies all homomorphisms to any finite group can be listed in finite time
Finitely presented and residually finite implies solvable word problem
Schur multiplier of free group is trivial
Finitely generated and free implies Hopfian
Abelian automorphism group implies class two
3-Engel implies locally nilpotent for groups
Abelian implies self-centralizing in holomorph
Conjugacy-separable implies every quotient-pullbackable automorphism is class-preserving
Cyclic implies abelian automorphism group
Centerless and maximal in automorphism group implies every automorphism is normal-extensible
Conjugacy-separable and aperiodic implies every extensible automorphism is class-preserving
Finitely presented and conjugacy-separable implies solvable conjugacy problem
Proper and normal in quasisimple implies central
Finite abelian implies same orbit sizes of conjugacy classes and irreducible representations under automorphism group
Free implies every subgroup is descendant
Finitely generated abelian implies Hopfian
Locally finite Artinian p-group implies hypercentral
Identity functor controls strong fusion for saturated fusion system on abelian group
Finitely generated and nilpotent implies Hopfian
Finitely generated abelian implies residually finite
Noetherian implies subnormal join property
Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup
Finitely generated implies every subgroup of finite index has finitely many automorphic subgroups