# Is a section of a minimal non $P_1$-group a $P_1$-group?

I'm having difficulty with Theorem 3.b from this paper.

I'll begin with some definitions, for a finite $p$-group $G$, define $\mho_n(G) = \langle g^{p^n} \mid g \in G \rangle$. Call $G$ a $P_1$-group if $\mho_n(S) = \{ s^{p^n} \mid s \in S \}$ for every section $S$ of $G$, including $S=G$.

Now suppose $G$ is a minimal non $P_1$-group of exponent $p^2$ and suppose $\Phi(G)$ has exponent $p^2$ too. Thus there exists $z \in \mho_1(\Phi(G)) \cap Z(G)$ with $z^p =1$.

Why is $G/\langle z \rangle$ a $P_1$ group?

I don't see how this follows, I've tried to show $G/\langle z \rangle$ is isomorphic to a subgroup of $G$ but with no success.

Previously in the paper it's been shown that $G$ is a $P_1$-group if every section of exponent $p^2$ is a $P_1$-group.

Reference: Mann, A. (1976). The power structure of p-groups. I. Journal of Algebra, 42(1), pp.121-135.

## 1 Answer

This seems to follow just from the definition of a minimal non-$P_1$-group.

The definition given in the paper is that $G$ is a minimal non-$P_1$-group, if $G$ is not a $P_1$-group and all proper sections of $G$ are $P_1$-groups.

So being a proper section, the quotient $G/\langle z \rangle$ is a $P_1$-group.

• Thanks. I missed that line. I had assumed that minimal meant only proper subgroups enjoyed the property. – Bysshed Aug 5 '17 at 14:49