# Verbal not implies iterated agemo

From Groupprops

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., iterated agemo subgroup of group of prime power order) neednotsatisfy the second subgroup property (i.e., verbal subgroup)

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## Statement

It is possible to have a group of prime power order and a verbal subgroup (hence, a Verbal subgroup of group of prime power order (?)) of that is not an iterated agemo subgroup of .

## Proof

`Further information: prime-cube order group:U(3,p)`

Suppose is an odd prime. Consider the group , which is the unique (up to isomorphism) non-abelian group of order and exponent .

We define , so is the commutator subgroup of . Clearly, is a verbal subgroup of . On the other hand, is *not* an iterated agemo subgroup of , because is trivial, so the only agemo subgroups (and hence the only iterated agemo subgroups) are the whole group and the trivial subgroup.