Some generalizations of Shao and Beltrán’s theorem

Let [Formula: see text] and [Formula: see text] be finite groups of relative coprime orders and [Formula: see text] act on [Formula: see text] via automorphisms. In this paper, we prove that when every maximal [Formula: see text]-invariant subgroup of [Formula: see text] that contains the normalizer of some Sylow subgroup has prime index, then [Formula: see text] is supersolvable; if every non-nilpotent maximal [Formula: see text]-invariant subgroup of [Formula: see text] has prime index or is normal in [Formula: see text], then [Formula: see text] is a Sylow tower group.

Finite Groups with Minimal CSS-subgroups

Let G be a finite group. A subgroup H of G is called SS-quasinormal in G if there is a supplement B of H to G such that H permutes with every Sylow subgroup of B. A subgroup H of G is called CSS-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩K is SS-quasinormal in G. In this paper, we investigate the influence of minimal CSS-subgroups of G on its structure. Our results improve and generalize several recent results in the literature.

Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s∗-semipermutable in G. Some recent results are generalized and unified.

On finite state automaton actions of HNN extensions of free abelian groups

HNN extensions of free abelian groups are considered. For arbitrary prime $p$ it is introduced a class of such extensions that act by finite automaton permutations over an alphabet $ \mathsf{X} $ of cardinality $p$ and belong to $p$-Sylow subgroup of the group of automaton permutations over such $ \mathsf{X} $. As a corollary it implies that all corresponding HNN extensions are residually $p$-finite.

Rigid automorphisms of linking systems

A rigid automorphism of a linking system is an automorphism that restricts to the identity on the Sylow subgroup. A rigid inner automorphism is conjugation by an element in the center of the Sylow subgroup. At odd primes, it is known that each rigid automorphism of a centric linking system is inner. We prove that the group of rigid outer automorphisms of a linking system at the prime $2$ is elementary abelian and that it splits over the subgroup of rigid inner automorphisms. In a second result, we show that if an automorphism of a finite group G restricts to the identity on the centric linking system for G, then it is of $p'$ -order modulo the group of inner automorphisms, provided G has no nontrivial normal $p'$ -subgroups. We present two applications of this last result, one to tame fusion systems.

THE CYCLIC GRAPH OF A Z-GROUP

For a group G, we define a graph $\Delta (G)$ by letting $G^{\scriptsize\#}=G{\setminus} \lbrace 1\rbrace $ be the set of vertices and by drawing an edge between distinct elements $x,y\in G^{\scriptsize\#}$ if and only if the subgroup $\langle x,y\rangle $ is cyclic. Recall that a Z-group is a group where every Sylow subgroup is cyclic. In this short note, we investigate $\Delta (G)$ for a Z-group G.

Finite Groups with Certain S-Permutable and GS-Maximal Subgroups

Let G be a finite group and H a subgroup of G. We say that H is S-permutable in G if H permutes with every Sylow subgroup of G. A group G is called a generalized smooth group (GS-group) if [G/L] is totally smooth for every subgroup L of G of prime order. In this paper, we investigate the structure of G under the assumption that each subgroup of prime order is S-permutable if the maximal subgroups of G are GS-groups.

New characterizations of p-nilpotency of finite groups

Suppose that [Formula: see text] is a finite group and [Formula: see text] is a subgroup of [Formula: see text]. [Formula: see text] is said to be an [Formula: see text]-quasinormal subgroup of [Formula: see text] if there is a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] permutes with every Sylow subgroup of [Formula: see text]. In this note, we fix in every non-cyclic Sylow subgroup [Formula: see text] of [Formula: see text] some subgroup [Formula: see text] satisfying [Formula: see text] and study the [Formula: see text]-nilpotency of [Formula: see text] under the assumption that every subgroup [Formula: see text] of [Formula: see text] with [Formula: see text] is [Formula: see text]-quasinormal in [Formula: see text]. The Frobenius theorem is generalized.