Central factor is not finite-join-closed
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) not satisfying a subgroup metaproperty (i.e., finite-join-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Join-closed subgroup property (?), .
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Contents
Statement
Statement with symbols
It is possible to have a finite group with subgroups
such that both
and
are central factors of
but the product
(which in this case is also the join of subgroups
) is not a central factor.
Definitions used
Central factor
Further information: Central factor
A subgroup of a group
is termed a central factor if
, or equivalently, every inner automorphism of
restricts to an inner automorphism of
.
Proof
Example of the extraspecial group
For any prime , either of the two isomorphism classes of extraspecial groups of order
gives a counterexample.