# Fully invariant not implies verbal in finite abelian group

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite abelian group. That is, it states that in a finite abelian group, every subgroup satisfying the first subgroup property (i.e., fully invariant subgroup) need not satisfy the second subgroup property (i.e., verbal subgroup)
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## Statement

A fully characteristic subgroup of a finite abelian group need not be a verbal subgroup.

## Proof

### Example of a group of prime-cube order

Let $C_p$ and $C_{p^2}$ denote the cyclic groups of order $p$ and $p^2$ respectively. Let $G = C_p \times C_{p^2}$. Consider the omega-1 subgroup:

$\Omega_1(G) := \{ x \in G \mid px = 0 \}$.

In other words, it is the subgroup of $G$ comprising the elements of order dividing $p$. Then, $\Omega_1(G)$ is fully characteristic: any endomorphism of $G$ preserves the condition. On the other hand, $\Omega_1(G)$ is not verbal: the only possible verbal subgroups of $G$ are the agemo subgroups: the whole group $G$, the set of multiples of $p$, and the trivial subgroup. None of these equals $\Omega_1(G)$.