1-closed transversal not implies permutably complemented
This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., subgroup having a 1-closed transversal) need not satisfy the second subgroup property (i.e., permutably complemented subgroup)
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Example of a non-abelian group of prime exponent
Further information: prime-cube order group:U(3,p)
Let be a prime number. Let be a non-abelian group of order and exponent (hence is isomorphic to prime-cube order group:U(3,p)). Let be the center of . is a normal subgroup of and it has no permutable complement because nilpotent and non-abelian implies center is not complemented (which in turn follows from the fact that nilpotent implies center is normality-large). On the other hand, does have a 1-closed transversal.
More generally, any subgroup of a group of prime exponent has a 1-closed transversal but is not necessarily permutably complemented.