This submission contains theories that lead to a formalization of the proof of the Jordan-Hölder theorem about composition series of finite groups. The theories formalize the notions of isomorphism classes of groups, simple groups, normal series, composition series, maximal normal subgroups. Furthermore, they provide proofs of the second isomorphism theorem for groups, the characterization theo...

If P is a p-group for some prime p we shall write M (P ) to denote the set of all maximal subgroups of P and Md(P ) = {P1, ..., Pd} to denote any set of maximal subgroups of P such that ∩d i=1 Pi = Φ(P ) and d is as small as possible. In this paper, the structure of a finite group G under some assumptions on the c-normal or s-quasinormally embedded subgroups in Md(P ), for each prime p, and Syl...