Omega-1 of center not is minimal characteristic

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This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., Omega-1 of center) does not always satisfy a particular subgroup property (i.e., minimal characteristic subgroup)
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Let p be a prime number and G be a nilpotent p-group. Then, \Omega_1(Z(G)) is not necessarily a minimal characteristic subgroup.


Example of an Abelian group

Consider G = \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}. Then, \Omega_1(Z(G)) = \Omega_1(G) = 2\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}. On the other hand, the subgroup \operatorname{Agemo}^1(G) = 2\mathbb{Z}/4\mathbb{Z} is a strictly smaller nontrivial characteristic subgroup of G.