Epabelian group
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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Contents
Definition
A group is termed epabelian if it satisfies the following equivalent conditions:
- For any group
with a central subgroup
such that the quotient group
is isomorphic to
,
must be an abelian group.
- The epicenter of
equals
.
-
is an abelian group and its Schur multiplier (which is necessarily equal to its exterior square) is the trivial group.
- (Certainly necessary, not sure it is sufficient): For any elements
, either
is cyclic or there exists a positive integer
and an element
such that
and
.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
cyclic group | ||||
locally cyclic group | ||||
periodic divisible abelian group |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
abelian group | ||||
epinilpotent group |
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Any finitely generated epabelian group is cyclic. This follows from the more general fact that finitely generated and epinilpotent implies cyclic.