Subgroups such as Hall systems, system normalizers, F-normalizers and F-covering subgroups arise naturally in the study of finite solvable groups. Here we present practical algorithms to compute such subgroups. Besides being of interest in the theory of formations, the algorithms also yield fast methods for constructing head complements. These complements are closely related to the maximal subg...

We prove that each ω-categorical, generically stable group is solvable-byfinite.

We show that for any finite group G and for any d there exists a word w ∈ Fd such that a d-tuple in G satisfies w if and only if it generates a solvable subgroup. In particular, if G itself is not solvable, then it cannot be obtained as a quotient of the one relator group Fd/ 〈w〉. As a corollary, the probability that a word is satisfied in a fixed nonsolvable group can be made arbitrarily small...

Let $G$ be a finite group. The graph $D(G)$ is a divisibility graph of $G$. Its vertex set is the non-central conjugacy class sizes of $G$ and there is an edge between vertices $a$ and $b$ if and only if $a|b$ or $b|a$. In this paper, we investigate the structure of the divisibility graph $D(G)$ for a non-solvable group with $sigma^{ast}(G)=2$, a finite simple group $G$ that satisfies the one-p...