Normal subgroup contained in the hypercenter: Difference between revisions
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| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] || if the whole group and the normal subgroup are powered over a prime, so is the quotient group. || [[normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication]] || any finite non-nilpotent group as a subgroup of itself || {{intermediate notions short|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication|normal subgroup contained in the hypercenter}} | | [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] || if the whole group and the normal subgroup are powered over a prime, so is the quotient group. || [[normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication]] || any finite non-nilpotent group as a subgroup of itself || {{intermediate notions short|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication|normal subgroup contained in the hypercenter}} | ||
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Latest revision as of 06:26, 24 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]
Definition
A subgroup of a group is termed a normal subgroup contained in the hypercenter if it is a normal subgroup and it is contained in the hypercenter of the whole group (the hypercenter is the subgroup at which the transfinite upper central series eventually terminates).
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| normal subgroup of nilpotent group | |FULL LIST, MORE INFO | |||
| central subgroup | |FULL LIST, MORE INFO | |||
| subgroup of abelian group | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| amalgam-characteristic subgroup | normal subgroup contained in the hypercenter is amalgam-characteristic | |FULL LIST, MORE INFO | ||
| normal subgroup satisfying the subgroup-to-quotient powering-invariance implication | if the whole group and the normal subgroup are powered over a prime, so is the quotient group. | normal subgroup contained in the hypercenter satisfies the subgroup-to-quotient powering-invariance implication | any finite non-nilpotent group as a subgroup of itself | |FULL LIST, MORE INFO |