We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}S$ is a finite subset of an ordered group that generates a nonabelian ordered group, then $|S^2|\geq 3|S|-2$. This generalizes a classical result from the theory of set addition.

A finite group is called a CH-group if for every x,y∈G∖Z(G), xy=yx implies that . Applying results of Schmidt [‘Zentralisatorverbände endlicher Gruppen’, Rend. Sem. Mat. Univ. Padova44 (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.

Let G be a finitely generated group. We investigate the graph ΓM(G), whose vertices are the maximal subgroups of G and where two vertices M1 and M2 are joined by an edge whenever M1∩M2≠1. We show that if G is a finite simple group then the graph ΓM(G) is connected and its diameter is 62 at most. We also show that if G is a finite group, then ΓM(G) either is connected or has at least two vertices and no edges. Finite groups G with a nonconnected graph ΓM(G) are classified. They are all solvable groups, and if G is a finite solvable group with a connected graph ΓM(G), then the diameter of ΓM(G) is at most 2. In the infinite case, we determine the structure of finitely generated infinite nonsimple groups G with a nonconnected graph ΓM(G). In particular, we show that if G is a finitely generated locally graded group with a nonconnected graph ΓM(G), then G must be finite.

The main result of this paper is an upper bound for the number of maximal subgroups in finite solvable groups. Our result improves an earlier one of Cook, Wiegold and Williamson [1]. At the end, we use our bound to deduce an estimation for the total number of subgroups in finite solvable groups.

It is known that a finite non-abelian group G has a proper centralizer of order if, for example, |G| is even and |Z(G)| is odd, or whenever G is solvable. Often the exponent can be improved to , for example when G is supersolvable, or metabelian, or |G = pαqβ. Here we show more generally that this improvement is possible in many situations where G is factorizable into the product of two subgroups. In particular, much more evidence is presented to support the conjecture that some proper centralizer has order whenever G is a finite non-abelian solvable group.

It is shown that the simple groups G2(q), q = 3f, are characterized by their character table. This result completes characterization of the simple groups G2(q), q odd, by their character table.

Let k be a non-zero complex number and let u and v be elements of a finite group G. Suppose that at most one of u and v belongs to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(v)–X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G, if and only if G/O(G) is isomorphic to one of the following groups: C2, PSL(2, 2n) or pΣL(2, 52a+1), where n≥2 and a ≥ 1.

Subject classification (Amer. Math. Soc. (MOS) 1970): 20 C 20

The proof of Theorem 5 in a paper with the same title is incorrect. In this note weaker versions of that theorem are proved.

Let k be a complex number and let u be an element of a finite group G. Suppose that u does not belong to O(G), the maximal normal subgroup of G of odd order. It is shown that G satisfies X(1) – X(u) = k for every complex nonprincipal irreducible character X in the principal 2-block of G if and only if G/O(G) is isomorphic either to C2, a cyclic group of order 2, or to PSL (2, 2n), n ≧ 2.

Let u be a nonidentity element of a finite group G and let c be a complex number. Suppose that every nonprincipal irreducible character X of G satisfies either X(l) − X(u) = c or X(u) = 0. It is shown that c is an even positive integer and all such groups with c ≤ 8 are described.

The paper establishes a method for bounding the 2-rank of a simple group with one conjugacy class of involutions, by means of its character table. For many groups of 2-rank ≦ 4, this bound is shown to be exact. The main result is that the simple groups G2(q),(q,6) = 1, are characterized bv their character table.

Let G be a finite group. A nontrivial proper subgroup M of G is called a CC-subgpoup if M contains the centralizer in G of each of its nonidentity elements. In this paper groups containing a CC-subgroup of order divisible by 3 are completely determined.

We study the problem of representing a permutation C as a product of a minimum number, fk(C), of cycles of length k. Upper and lower bounds on fk(C) are obtained and exact results are derived for k = 2, 3, 4.