Quotient-powering-invariant subgroup: Difference between revisions

Revision as of 02:03, 12 February 2013

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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Definition

A normal subgroup $H$ of a group $G$ is termed a quotient-powering-invariant subgroup if, for any prime number $p$ such that $G$ is a powered for $p$ , the quotient group $G / H$ is also powered for $p$ .

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
quotient-transitive subgroup property	Yes	quotient-powering-invariance is quotient-transitive	If $H \leq K \leq G$ are such that $H$ is quotient-powering-invariant in $G$ and $K / H$ is quotient-powering-invariant in $G / H$ , then $K$ is quotient-powering-invariant in $G$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup of finite group	the whole group is finite
normal subgroup of periodic group	every element in the whole group has finite order
normal subgroup of finite index	the quotient group is finite	normal of finite index implies quotient-powering-invariant
finite normal subgroup	the normal subgroup is finite	finite normal implies quotient-powering-invariant
direct factor	normal subgroup with normal complement	(via complemented normal)	(via complemented normal)	\|FULL LIST, MORE INFO
complemented normal subgroup	normal subgroup with a (possibly non-normal) complement	complemented normal implies quotient-powering-invariant		\|FULL LIST, MORE INFO
characteristic subgroup of abelian group	characteristic subgroup and the whole group is an abelian group	characteristic subgroup of abelian group is quotient-powering-invariant		\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
powering-invariant subgroup		quotient-powering-invariant implies powering-invariant	powering-invariant not implies quotient-powering-invariant	\|FULL LIST, MORE INFO

Properties whose conjunction with powering-invariance implies quotient-powering-invariance

Property	Proof of conjunction statement
central subgroup	powering-invariant and central implies quotient-powering-invariant
normal subgroup contained in the hypercenter	normal subgroup contained in the hypercenter that is powering-invariant is quotient-powering-invariant

Incomparable properties

Property	Meaning	Proof that it does not imply being powering-invariant	Properties for the ambient group for which it does imply being powering-invariant	Proof that being powering-invariant does not imply the property
normal subgroup	invariant under all inner automorphisms	see examples for characteristic or central subgroup below.	finite group or periodic group	any finite non-normal subgroup
characteristic subgroup	invariant under all automorphisms	characteristic not implies powering-invariant	abelian group (plus above cases for normal subgroup)	any finite non-characteristic subgroup
central subgroup	contained in the center	follows from subgroup of abelian group not implies powering-invariant	(all cases for normal subgroup)	any non-abelian group as a subgroup of itself

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 |-
 | [[normal subgroup contained in the hypercenter]] || [[normal subgroup contained in the hypercenter that is powering-invariant is quotient-powering-invariant]]
+|}
+===Incomparable properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof that it does not imply being powering-invariant !! Properties for the ambient group for which it does imply being powering-invariant !! Proof that being powering-invariant does not imply the property
+|-
+| [[normal subgroup]] || invariant under all [[inner automorphism]]s || see examples for characteristic or central subgroup below. || [[finite group]] or [[periodic group]] || any finite non-normal subgroup
+|-
+| [[characteristic subgroup]] || invariant under all [[automorphism]]s || [[characteristic not implies powering-invariant]] || [[abelian group]] (plus above cases for normal subgroup) || any finite non-characteristic subgroup
+|-
+| [[central subgroup]] || contained in the [[center]] || follows from [[subgroup of abelian group not implies powering-invariant]] || (all cases for normal subgroup) || any non-abelian group as a subgroup of itself ||
 |}