Quotient-powering-invariant subgroup: Difference between revisions

Revision as of 23:42, 11 February 2013

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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 $p$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Intermediate notions
normal subgroup of finite group
normal subgroup of periodic group
normal subgroup of finite index		normal of finite index implies quotient-powering-invariant
finite normal subgroup		finite normal implies quotient-powering-invariant
direct factor	normal subgroup with normal complement		\|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

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

@@ Line 19: / Line 19: @@
 |-
 | [[Weaker than::finite normal subgroup]] || || [[finite normal implies quotient-powering-invariant]] || ||
+|-
+| [[Weaker than::direct factor]] || normal subgroup with normal complement || || || {{intermediate notions short|direct factor|quotient-powering-invariant subgroup}}
+|-
+| [[Weaker than::complemented normal subgroup]] || normal subgroup with a (possibly non-normal) complement) || [[complemented normal implies quotient-powering-invariant]] || || {{intermediate notions short|quotient-powering-invariant subgroup|complemented normal subgroup}}
 |}
@@ Line 27: / Line 31: @@
 |-
 | [[Stronger than::powering-invariant subgroup]] || || [[quotient-powering-invariant implies powering-invariant]] || [[powering-invariant not implies quotient-powering-invariant]] || {{intermediate notions short|powering-invariant subgroup|quotient-powering-invariant subgroup}}
+|}
+===Properties whose conjunction with powering-invariance implies quotient-powering-invariance===
+{| class="sortable" border="1"
+! 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]]
 |}