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Let G be a group and let x ∈ G be a left 3-Engel element of odd order. We show that x is in the locally nilpotent radical of G. We establish this by proving that any finitely generated sandwich group, generated by elements of odd orders, is nilpotent. This can be seen as a group theoretic analog of a well-known theorem on sandwich algebras by Kostrikin and Zel'manov. We also give some applications of our main result. In particular, for any given word w = w(x1, . . . , xn) in n variables, we show that if the variety of groups satisfying the law w³ = 1 is a locally finite variety of groups of exponent 9, then the same is true for the variety of groups satisfying the law (xn+1³w³)³ = 1. [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]

A finite group is called a CH-group if for every x, y ϵ G\Z(G), xy = yx implies that ∥CG(x)∥ = ∥CG(y)∥. Applying results of Schmidt ['Zentralisatorverbände endlicher Gruppen', Rend. Sem. Mat. Univ. Padova 44 (1970), 97-131] and Rebmann ['F-Gruppen', Arch. Math. 22 (1971), 225-230] concerning CA-groups and F-groups, the structure of CH-groups is determined, up to that of CH-groups of prime-power order. Upper bounds are found for the derived length of nilpotent and solvable CH-groups. [ABSTRACT FROM AUTHOR]

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]

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]

Let G and A be finite groups with A acting on G by automorphisms. In this paper we introduce the concept of "good action"; namely we say the action of A on G is good, if H = H , B C H (B) for every subgroup B of A and every B -invariant subgroup H of G. This definition allows us to prove a new noncoprime Hall-Higman type theorem. If A is a nilpotent group acting on the finite solvable group G with C G (A) = 1 , a long standing conjecture states that h (G) ⩽ ℓ (A) where h (G) is the Fitting height of G and ℓ (A) is the number of primes dividing the order of A counted with multiplicities. As an application of our result we prove the main theorem of this paper which states that the above conjecture is true if A and G have odd order, the action of A on G is good and some other fairly general conditions are satisfied. [ABSTRACT FROM AUTHOR]