Conjugacy-closedness is not join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., conjugacy-closed subgroup) not satisfying a subgroup metaproperty (i.e., join-closed subgroup property).
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We can have a situation where are conjuacy-closed subgroups of but the join is not conjugacy-closed in .
Example of the dihedral group
Further information: dihedral group:D8
Let be the dihedral group of order eight:
Suppose are subgroups given as follows:
- Since and are both subgroups of order two, they are both conjugacy-closed in .
- The join is an Abelian subgroup of order four. It is clearly not conjugacy-closed, because the elements and in this subgroup are conjugate in .