Quotient-powering-invariant subgroup
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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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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and normal subgroup satisfying the subgroup-to-quotient powering-invariance implication
View other subgroup property conjunctions | view all subgroup properties
Definition
A normal subgroup of a group
is termed a quotient-powering-invariant subgroup if, for any prime number
such that
is a powered for
, the quotient group
is also powered for
.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
quotient-transitive subgroup property | Yes | quotient-powering-invariance is quotient-transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
union-closed subgroup property | Yes | quotient-powering-invariance is union-closed | If ![]() ![]() ![]() ![]() ![]() ![]() |
Relation with other properties
Stronger properties
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
powering-invariant normal subgroup | quotient-powering-invariant implies powering-invariant | powering-invariant and normal not implies quotient-powering-invariant | |FULL LIST, MORE INFO | |
powering-invariant subgroup | quotient-powering-invariant implies powering-invariant | (via powering-invariant normal) | Powering-invariant normal subgroup|FULL LIST, MORE INFO | |
normal subgroup satisfying the subgroup-to-quotient powering-invariance implication | |FULL LIST, MORE INFO | |||
normal subgroup | (by definition) | (via powering-invariant normal) | Powering-invariant normal subgroup|FULL LIST, MORE INFO |
Properties whose conjunction with powering-invariance implies quotient-powering-invariance
The relevant subgroup property is normal subgroup satisfying the subgroup-to-quotient powering-invariance implication. In fact, the conjunction of this with powering-invariant subgroup precisely gives quotient-powering-invariant subgroup.
This property is implied both by being a central subgroup and by being a normal subgroup contained in the hypercenter.