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Questions about the branch of algebra that deals with groups.

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1 answer
294 views

The center of Sylow subgroups

$\DeclareMathOperator\Syl{Syl}$Let $G$ be a finite group, $\Phi(G)$ is the Frattini subgroup of $G$. And $G/\Phi(G)$ is a simple group. Let $P\in \Syl_{p}(\Phi(G))$, where $p\in \pi(\Phi(G))$ and $\p …
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1 answer
133 views

The intersection of product of Sylow subgroups

Suppose $G$ is solvable, and $\pi(G)= \{2,m,n\}$, $O_{2}(G)=1$. Then can we use the solvability of $G$ to prove that $O_{2^{\prime}}(G) \neq 1$? Let $\bar{G}= G / O_{2^{\prime}}(G)$, what about $O_{ …