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Let G and A be groups where A acts on G by automorphisms. We say "the action of  A on  G is good" if the equality H = [ H , B ] C H (B) holds for any subgroup B of A and for any B -invariant subgroup H of G. It is straightforward that every coprime action is a good action. In this work, we extend some results due to Ward, Gross, Shumyatsky, Jabara and Meng and Guo under coprime action to good action. [ABSTRACT FROM AUTHOR]

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]

We describe sharply 2-transitive groups whose point stabilizer is a nilpotent {2,3}-group without elements of order 9 and, more generally, in which the third power of each element belongs to the FC-center. In particular, we will prove that these groups are finite. [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]

Let G be a finite soluble group and h( G) its Fitting length. The aim of this paper is to give certain upper bounds for h( G) as functions of the Fitting length of at least three Hall subgroups of G which factorize G in a particular way. [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]

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]

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]

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).