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We prove that a group with spectrum {1, 2, 3, 4, 5, 8} is locally finite and is isomorphic to M10, where M10 = A6 ⋅ 2 is the 3-transitive Mathieu group of degree 10, i.e. the corresponding maximal subgroup of the sporadic simple group M11. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we prove that if G is a group which is the union of the conjugates of a finite subgroup H, then we have if , or H is a generalized dihedral group. This result extends Theorem 3 of Cutolo et al. (2005) proved only in the case . [Copyright &y& Elsevier]

We consider finite groups G for which any two cyclic subgroups of the same order are conjugate in G. We prove various structure results and, in particular, that any such group has at most one non-abelian composition factor, and this is isomorphic to PSL(2, pm), with m odd if p is odd, or to Sz(22m+1), or to one of the sporadic groups M11, M23, or J1. [ABSTRACT FROM AUTHOR]

We give a structure theorem for the finite groups with three conjugacy class sizes. In particular, they are solvable groups with derived length at most 3 or nilpotent groups. [ABSTRACT FROM PUBLISHER]

We show that every group of exponent 2 m · 3 n ( m, n ∈ ℕ, n ≤ 2) that acts freely on some abelian group is finite. [ABSTRACT FROM AUTHOR]

Abstract: Let G be a residually finite group. If G admits an automorphism of prime order p with finite Reidemeister number, then G is virtually nilpotent (of class bounded by a function of p). [Copyright &y& Elsevier]

Abstract: We prove that if G is a solvable group with abelian Sylow 2-subgroups, the index of the Fitting subgroup of G is at most the square of the largest irreducible character degree of G. [Copyright &y& Elsevier]

The article discusses integral theorems on the actions of abelian groups on groups of automorphisms. It illustrates the notation and preliminary results of these theorems with the use of various algebraic formulas to prove that it is possible to omit the hypothesis that abelian groups on groups of automorphisms is fixed-point-free (FPF).