# Changes

## Characteristicity is transitive

, 16:10, 19 December 2014
Statement
{{trivial result}}
{{subgroup metaproperty satisfaction|property = characteristic subgroup|metaproperty = transitive subgroup property}}[[difficulty level::1| ]]==Statement== Suppose $H \le K \le G$ are groups such that $H$ is a [[characteristic subgroup]] of $K$, and $K$ is a [[characteristic subgroup]] of $G$. Then, $H$ is a [[characteristic subgroup]] of $G$. ==Related facts== ===Close relation with normality=== A [[normal subgroup]] is a subgroup that is invariant under all [[inner automorphism]]s. Below, we take $H \le K \le G$, with $H$ the bottom group, $K$ the middle group, and $G$ the top group. {| class="sortable" border="1"! Statement !! Change in assumption !! Change in conclusion|-| [[Normality is not transitive]] || $H$ normal in $K$, $K$ normal in $G$ || $H$ ''not'' normal in $G$|-| [[Characteristic of normal implies normal]] || $K$ normal in $G$ || $H$ normal in $G$|-| [[Left transiter of normal is characteristic]] || $H$ in $K$ such that if $K$ is normal in $G$, $H$ is normal in $G$ || $H$ is characteristic in $K$|} ===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]] $\leftrightarrow$ [[group]], [[characteristic subring of a Lie ring]] $\leftrightarrow$ [[characteristic subgroup]] || |-| [[Derivation-invariance is transitive]] || [[Lie ring]] $\leftrightarrow$ [[group]], [[derivation of a Lie ring]] $\leftrightarrow$ [[automorphism of a group]], [[derivation-invariant Lie subring]] $\leftrightarrow$ [[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 $H$ of a group $G$ is termed a characteristic subgroup if whenever $\sigma$ is an automorphism of $G$, $\sigma$ restricts to an automorphism of $H$.
===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 $\to$ 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 $p$ is termed transitive if whenever $H\le K \le G$ be a [[characteristic subgroup]] of are groups such that $H$ satisfies property $p$ in $K$, and $K$ a characteristic subgroup of satisfies property $p$ in $G$. Then, $H$ is a characteristic subgroup of also satisfies property $p$ in $G$.
==Proof==
{{fillintabular proof format}} ===Hands-on proof=== '''Given''': A group $G$ with a characteristic subgroup $K$. $H$ is a characteristic subgroup of $K$. $\sigma$ is an automorphism of $G$. '''To prove''': $\sigma(H) = H$ and $\sigma$ restricts to an automorphism of $H$. '''Proof''': {| class="sortable" border="1"! Step no. !! Assertion/construction !! Facts used !! Given data used !! Previous steps used !! Explanation|-| 1 || $\sigma(K) = K$, and $\sigma$ restricts to an automorphism of $K$, that we call $\sigma'$. || definition of characteristic subgroup || $K$ is characteristic in $G$, $\sigma$ is an automorphism of $G$. || || direct|-| 2 || $\sigma'(H) = H$, and $\sigma'$ restricts to an automorphism of $H$ || definition of characteristic subgroup || $H$ is characteristic in $K$ || Step (1) || direct|-| 3 || $\sigma(H) = H$ and $\sigma$ restricts to an automorphism of $H$. || || || Steps (1), (2) || <toggledisplay>By the meaning of restriction, restricting from $G$ to $K$ and then again from $K$ to $H$ is equivalent to directly restricting from $G$ to $H$. We have that $\sigma$ on $G$ restricts to $\sigma'$ on $K$, which in turn restricts to an automorphism of $H$. Thus, the restriction of $\sigma$ ''directly'' to $H$ 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 $\to$ 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}}