Use with the (group-theory) tag. The tag "finite-groups" refers to questions asked in the field of Group Theory which, in particular, focus on the groups of finite order.

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How are simple groups the building blocks?

I know a bit about simple groups. A finite Abelian group is a (direct) product of finite cyclic groups. The simple finite Abelian groups are exactly $\mathbb{Z}_p$ for $p$ a prime. And so, I ...
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If $M < M < G$ with certain conditions and a special subgroup $U < G$, then we can choose $|G / M| = p$ and $p \nmid |M|$.

Let $G$ be a finite group and $U \le G$ a subgroup of odd order. Assume that $N_G(U) = TU$ with $T = \langle t \rangle$ for some involution $t \notin U$. Also assume $U^g \ne U$ implies $U^g \cap U = ...