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Characteristicity is transitive

6,110 bytes added, 16:10, 19 December 2014
Statement
{{trivial result}}
{{subgroup metaproperty satisfaction|property = characteristic subgroup|metaproperty = transitive subgroup property}}[[difficulty level::1| ]]==Statement== Suppose <math>H \le K \le G</math> are groups such that <math>H</math> is a [[characteristic subgroup]] of <math>K</math>, and <math>K</math> is a [[characteristic subgroup]] of <math>G</math>. Then, <math>H</math> is a [[characteristic subgroup]] of <math>G</math>. ==Related facts== ===Close relation with normality=== A [[normal subgroup]] is a subgroup that is invariant under all [[inner automorphism]]s. Below, we take <math>H \le K \le G</math>, with <math>H</math> the bottom group, <math>K</math> the middle group, and <math>G</math> the top group. {| class="sortable" border="1"! Statement !! Change in assumption !! Change in conclusion|-| [[Normality is not transitive]] || <math>H</math> normal in <math>K</math>, <math>K</math> normal in <math>G</math> || <math>H</math> ''not'' normal in <math>G</math>|-| [[Characteristic of normal implies normal]] || <math>K</math> normal in <math>G</math> || <math>H</math> normal in <math>G</math>|-| [[Left transiter of normal is characteristic]] || <math>H</math> in <math>K</math> such that if <math>K</math> is normal in <math>G</math>, <math>H</math> is normal in <math>G</math> || <math>H</math> is characteristic in <math>K</math>|} ===Generalizations=== [[Balanced implies transitive]]: Any subgroup property that can be expressed as a [[balanced subgroup property (function restriction formalism)|balanced subgroup property]] is transitive. Characteristicity is a special case. Other special cases include: {| class="sortable" border="1"! Property !! Balanced with respect to ... !! Proof|-| [[Fully invariant subgroup]] || [[endomorphism]]s || [[Full invariance is transitive]]|-| [[Central factor]] || [[inner automorphism]]s || [[Central factor is transitive]]|-| [[Transitively normal subgroup]] || [[normal automorphism]]s || [[Transitive normality is transitive]]|-| [[Injective endomorphism-invariant subgroup]] || [[injective endomorphism]]s || [[Injective endomorphism-invariance is transitive]]|} ===Analogues in other algebraic structures=== {| class="sortable" border="1"! Statement !! Analogy correspondence !! Additional comments|-| [[Characteristicity is transitive in Lie rings]] || [[Lie ring]] <math>\leftrightarrow</math> [[group]], [[characteristic subring of a Lie ring]] <math>\leftrightarrow</math> [[characteristic subgroup]] || |-| [[Derivation-invariance is transitive]] || [[Lie ring]] <math>\leftrightarrow</math> [[group]], [[derivation of a Lie ring]] <math>\leftrightarrow</math> [[automorphism of a group]], [[derivation-invariant Lie subring]] <math>\leftrightarrow</math> [[characteristic subgroup]] || |-| [[Characteristicity is transitive for any variety of algebras]] || |||} ===Generalizations in the one-of-its-kind sense of the statement=== {| class="sortable" border="1"! Property !! Meaning !! Proof|-| [[Second-order characteristic subgroup]] || no subgroup equivalent in the second-order theory of groups || [[Second-order characteristicity is transitive]]|-| [[Monadic second-order characteristic subgroup]] || no subgroup equivalent is the monadic second-order theory of groups || [[Monadic second-order characteristicity is transitive]]|-| [[Purely definable subgroup]] || definable in the pure theory of groups || [[Pure definability is transitive]]|} ===Other related facts===
* [[Automorph-conjugacy is transitive]]* [[SQ-dual::Characteristicity is quotient-transitive]] ==StatementDefinitions used== ===Characteristic subgroup=== {{further|[[Characteristic subgroup]]}} A subgroup <math>H</math> of a group <math>G</math> is termed a characteristic subgroup if whenever <math>\sigma</math> is an automorphism of <math>G</math>, <math>\sigma</math> restricts to an automorphism of <math>H</math>.
===Property-theoretic statement===This is written using the [[function restriction expression]]:
The [[subgroup property]] of being [[characteristic subgroup|characteristic]] satisfies the [[subgroup metaproperty]] of being [[transitive subgroup property|transitive]].Automorphism <math>\to</math> Automorphism
===Verbal statement===In other words, every automorphism of the whole group restricts to an automorphism of the subgroup.
A [[characteristic ===Transitive subgroup]] of a [[characteristic subgroup]] is characteristic in the whole group.property===
===Symbolic statement==={{further|[[Transitive subgroup property]]}}
Let A subgroup property <math>p</math> is termed transitive if whenever <math>H\le K \le G</math> be a [[characteristic subgroup]] of are groups such that <math>H</math> satisfies property <math>p</math> in <math>K</math>, and <math>K</math> a characteristic subgroup of satisfies property <math>p</math> in <math>G</math>. Then, <math>H</math> is a characteristic subgroup of also satisfies property <math>p</math> in <math>G</math>.
==Proof==
 {{fillintabular proof format}} ===Hands-on proof=== '''Given''': A group <math>G</math> with a characteristic subgroup <math>K</math>. <math>H</math> is a characteristic subgroup of <math>K</math>. <math>\sigma</math> is an automorphism of <math>G</math>. '''To prove''': <math>\sigma(H) = H</math> and <math>\sigma</math> restricts to an automorphism of <math>H</math>. '''Proof''': {| class="sortable" border="1"! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation|-| 1 || <math>\sigma(K) = K</math>, and <math>\sigma</math> restricts to an automorphism of <math>K</math>, that we call <math>\sigma'</math>. || definition of characteristic subgroup || <math>K</math> is characteristic in <math>G</math>, <math>\sigma</math> is an automorphism of <math>G</math>. || || direct|-| 2 || <math>\sigma'(H) = H</math>, and <math>\sigma'</math> restricts to an automorphism of <math>H</math> || definition of characteristic subgroup || <math>H</math> is characteristic in <math>K</math> || Step (1) || direct|-| 3 || <math>\sigma(H) = H</math> and <math>\sigma</math> restricts to an automorphism of <math>H</math>. || || || Steps (1), (2) || <toggledisplay>By the meaning of restriction, restricting from <math>G</math> to <math>K</math> and then again from <math>K</math> to <math>H</math> is equivalent to directly restricting from <math>G</math> to <math>H</math>. We have that <math>\sigma</math> on <math>G</math> restricts to <math>\sigma'</math> on <math>K</math>, which in turn restricts to an automorphism of <math>H</math>. Thus, the restriction of <math>\sigma</math> ''directly'' to <math>H</math> must also be the same automorphism.</toggledisplay>|} {{frexp metaproperty satisfaction}} {{proof generalizes|[[balanced implies transitive]]}} The idea behind this proof is to observe that characteristicity can be written as the balanced subgroup property: Automorphism <math>\to</math> Automorphism In other words, every automorphism of the big group restricts to an automorphism of the subgroup. Any balanced subgroup property is transitive, and this gives the proof.
==References==
===Textbook references===
* {{booklink-stated|DummitFoote}}, Page 135, Page |137 (|Problem 8(b))}}* {{booklink-stated|AlperinBell}}, Page |17, ''|Lemma 4''}}* {{booklink-stated|RobinsonGT}}, Page |28, | Section 1.5 (''Characteristic and Fully invariant subgroups''), 1.5.6(ii)}}* {{booklink-stated|KhukhroNGA|4|Section 1.1|passing mention}}
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