Let $G$ be a finite $p$ group such that $G=A*B$ central product of two subgroup $A$ and $B$. What will be the frattini subgroup of $G$, i.e $\Phi(G)=?$.
Let $G$ be a finite $p$ group such that $G=A*B$ central product of two subgroup $A$ and $B$. What will be the frattini subgroup of $G$.
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group-theory
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0Means relation between $\Phi(G)$ and $\Phi(A)$ and $\Phi(B)$? – 2017-02-22
1 Answers
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Firstly, note that $\Phi(G)=G^pG^{\prime}$ in $p$-groups. As $A$ and $B$ commute, because $G$ is a central extension, we have $G^{\prime}=A^{\prime}B^{\prime}$ (you should verify this yourself). Similarly, $G^p=A^pB^p$. Therefore, noting that $B^p$ and $A^{\prime}$ commute, we have: $$\begin{align*} \Phi(G)&=G^pG^{\prime}\\&=A^pB^pA^{\prime}B^{\prime}\\&=A^pA^{\prime}B^pB^{\prime}\\&=\Phi(A)\Phi(B).\end{align*}$$