A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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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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Inverse (finite group) isomorphism of a certain form exists

I have been working something in group theory for a long time and I have everything worked out but this one problem. I have reduced that problem to a conjecture. It takes some work to set it up, but 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 = ...