Central factor is not finite-join-closed

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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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Statement with symbols

It is possible to have a finite group G with subgroups H, K such that both H and K are central factors of G but the product HK (which in this case is also the join of subgroups \langle H, K \rangle) is not a central factor.

Definitions used

Central factor

Further information: Central factor

A subgroup H of a group G is termed a central factor if HC_G(H) = G, or equivalently, every inner automorphism of G restricts to an inner automorphism of H.


Example of the extraspecial group

For any prime p, either of the two isomorphism classes of extraspecial groups of order p^5 gives a counterexample.