View source for Characteristicity is transitive

{{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]]

==Definitions 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>.

This is written using the [[function restriction expression]]:

Automorphism <math>\to</math> Automorphism

In other words, every automorphism of the whole group restricts to an automorphism of the subgroup.

===Transitive subgroup property===

{{further|[[Transitive subgroup property]]}}

A subgroup property <math>p</math> is termed transitive if whenever <math>H \le K \le G</math> are groups such that <math>H</math> satisfies property <math>p</math> in <math>K</math> and <math>K</math> satisfies property <math>p</math> in <math>G</math>, <math>H</math> also satisfies property <math>p</math> in <math>G</math>.

==Proof==

{{tabular 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|137|Problem 8(b)}}
* {{booklink-stated|AlperinBell|17|Lemma 4}}
* {{booklink-stated|RobinsonGT|28| Section 1.5 (''Characteristic and Fully invariant subgroups''), 1.5.6(ii)}}
* {{booklink-stated|KhukhroNGA|4|Section 1.1|passing mention}}