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Let n be a positive integer. We say that a group G is an (n + ½)-Engel group if it satisfies the law [x, ny, x ] = 1. The variety of (n + ½)-Engel groups lies between the varieties of n-Engel groups and (n + 1)-Engel groups. In this paper, we study these groups, and in particular, we prove that all (4 + ½)-Engel {2,3}-groups are locally nilpotent. We also show that if G is a (4 + ½)-Engel p-group, where p ≥ 5 is a prime, then Gp is locally nilpotent. [ABSTRACT FROM AUTHOR]

This note provides an affirmative answer to Problem 2.6 of Praeger and Schneider ['Group factorisations, uniform automorphisms, and permutation groups of simple diagonal type', Israel J. Math. 228 (2) (2018), 1001–1023]. We will build groups $G$ (abelian, nonabelian and simple) for which there are two automorphisms $\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D6FD}$ of $G$ such that the map $$\begin{eqnarray}T=T_{\unicode[STIX]{x1D6FC}}\times T_{\unicode[STIX]{x1D6FD}}:G\longrightarrow G\times G,\quad g\mapsto (g^{-1}g^{\unicode[STIX]{x1D6FC}},g^{-1}g^{\,\unicode[STIX]{x1D6FD}})\end{eqnarray}$$ is surjective. [ABSTRACT FROM AUTHOR]

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]

Let G be a finite soluble group, and let h ( G ) be the Fitting length of G . If φ is a fixed-point-free automorphism of G , that is C G ( φ ) = { 1 } , we denote by W ( φ ) the composition length of 〈 φ 〉 . A long-standing conjecture is that h ( G ) ≤ W ( φ ) , and it is known that this bound is always true if the order of G is coprime to the order of φ . In this paper we find some bounds to h ( G ) in function of W ( φ ) without assuming that ( | G | , | φ | ) = 1 . In particular we prove the validity of the “universal” bound h ( G ) < 7 W ( φ ) 2 . This improves the exponential bound known earlier from a special case of a theorem of Dade. [ABSTRACT FROM AUTHOR]

Let G be a group generated by the set Γ = {g ∊ G ∣ g2 = 1 ≠ g} of its involutions. We prove that if (ab)4 = 1 for every a; b ∊ Γ, then G is a locally finite 2-group. This answers in the affirmative Question 18.58 in the Kourovka notebook ([3]). [ABSTRACT FROM AUTHOR]

Local finiteness is proved for groups of exponent dividing 72 with no elements of order 6. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with finite soluble groups G admitting a factorisation , with A and B proper subgroups having coprime order. We are interested in bounding the Fitting height of G in terms of some group-invariants of A and B, including the Fitting heights and the derived lengths. [ABSTRACT FROM AUTHOR]

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]

Abstract: Let G be a group and a group of automorphisms of G. An -orbit of G is a set of the form , where g is an element of G. The aim of this paper is to prove that if is abelian and G is a union of a finite number of -orbits then G admits a normal abelian subgroup of finite index. This result answers affirmatively a question raised by Neumann and Rowley (1998) in . [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).

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]

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]