Relationship between agemo subgroup and Frattini subgroup in $2$-groups

Question

Let $G$ be a $2$-group. Call $\Phi(G)$ the Frattini subgroup of $G$. Define with $A_1(G)=\langle \{g^2\mid g \in G\}\rangle$ the first agemo subgroup of $G$. Is it true that $A_1(G)=\Phi(G)$?

Answer 1

This is true. The quotient by $A_1(G)$ has $x^2=1$ for all elements $x$ and thus is abelian, so $\Phi(G)\le A_1(G)$. Vice versa, by Burnside's basis theorem, the quotient $G/\Phi(G)$ is elementary abelian of exponent $2$, so $A_1(G)\le\Phi(G)$.