The study of symmetry: groups, subgroups, homomorphisms, group actions.

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Non simplicity of a group of order $p^{100}q$ given some conditions.

Let $G$ be a group $p$, $q$ primes $p\neq q$, $|G|=p^{100}q$. 1.- Assume that $G$ has two different $p$-Sylow subgroups and intersection for each pair of $p$-Sylow subgroups is trivial. Show that ...