Central factor is not finite-join-closed

From Groupprops
Jump to: navigation, search
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 (?), .
View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about central factor|Get more facts about finite-join-closed subgroup propertyGet more facts about join-closed subgroup property|

Statement

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.

Proof

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.