Note on a problem concerning the Frattini subgroups

A Note on Deaconescu’s Result concerning Lehmer’s Problem

Let φ(n) be the Euler function of n. We prove that there are at most finitely many composite integers n such that φ(n) | n − 1 and P (φ(n)) ≡ 0 (mod n), where P (X) ∈ Z[X] is any monic non-constant polynomial.

Abelian groups have/are near Frattini subgroups

The notions of nearly-maximal and near Frattini subgroups considered by J.B. Riles in [20] and the natural related notions are characterized for abelian groups.

Frattini and related subgroups of Mapping Class Groups

Let Γg,b denote the orientation-preserving Mapping Class Group of a closed orientable surface of genus g with b punctures. For a group G let Φf (G) denote the intersection of all maximal subgroups of finite index in G. Motivated by a question of Ivanov as to whether Φf (G) is nilpotent when G is a finitely generated subgroup of Γg,b, in this paper we compute Φf (G) for certain subgroups of Γg,b...

Near Frattini Subgroups of Certain Generalized Free Products of Groups

Let G = A ∗H B be the generalized free product of the groups A and B with the amalgamated subgroup H. Also, let λ(G) and ψ(G) represent the lower near Frattini subgroup of G and the near Frattini subgroup of G respectively. We show that G is ψ−free provided: (i) G is any ordinary free product of groups; (ii) G = A ∗H B and there exists an element c in G\H such thatH ∩H = 1; (iii) G = A ∗H B and...

A Note on Counting Isotropy Subgroups