The Number of Pairwise Non-commuting Elements and the Index of the Centre in a Finite Group

If the group G contains at most n pairwise non-commuting elements, then \G:Z(G)\ ^ c for some constant c. This answers a question of B. H. Neumann and also solves a problem of P. Erdos. Introduction Let G be a group and associate with G a graph T = T(G) as follows: the vertices of F are the elements of G and two vertices g, h of T are joined by an edge if and only if g and h commute as elements of G. In 1975, P. Erdos [2] raised the following problem. If the size of the maximal empty subgraph of V(G) is n — n(G) < oo, then what is the maximum of cc(T), the minimal number of complete subgraphs covering Gl A paper of B. H. Neumann [5] contains implicitly the estimate cc(T) ̂ a* for some constant a. Later I. M. Isaacs [3] gave an elegant proof for the upper bound What B. H. Neumann actually proves is the following. If the group G contains at most n pairwise non-commuting elements then for the index of the centre we have \G:Z{G)\ ^ a for some constant a. He also asks for a better estimate. Our main result is the following. THEOREM. \G\Z{G)\ ^ c for some constant c. As the cosets of the centre are abelian subsets of G, we also obtain an answer to the question of Erdos. COROLLARY. CC(T) < c for some constant c. Both our results are optimal in a sense as shown by an example due to Isaacs [1]. Received 10 March 1986; revised 20 June 1986. 1980 Mathematics Subject Classification 20D60. J. London Math. Soc. (2) 35 (1987) 287-295 288 L. PYBER EXAMPLE. Let S be an extra special group of order 2, then (i) n(S) = 2m+\, (ii) |S:Z(S)| = 2 2 , (iii) cc{T{S))>2+\. 1. Preliminaries All the groups to be considered will be finite. This is not a real restriction for it is easy to see that if n{G) < oo, then there exists a finite group Go with G0/Z(G0) s G/Z(G) and n(G) = n(G0) (see for example [4, §2]). If A' is a subset of G consisting of pairwise non-commuting elements we call it an independent subset of G. The maximum size of such subsets is «(G). The conjugacy class of geG is denoted by ClG(g). The maximum size of a conjugacy class in G will be denoted by k(G) and CG(g) denotes the centralizer of g inG. Instead of k(G), n(G), CG(g) and C\G(g) we simply write k, n, C{g) and Cl (g) if it does not lead to confusion. As an elementary /?-group P isomorphic to Cp is equivalent to an /--dimensional vector space over the /^-element field, we call some elements of P linearly independent if and only if no non-trivial product of these elements is equal to 1. We shall also use the following well-known facts. (a) If A and B are subgroups of G then \G:A(]B\^\G:A\\G:B\. (b) For all gl, g2eG, C\G(gl)ClG(g2) =2 C\G(Slg2). (c) \G\ = \G:CG(g)\\ClG(g)\. (d) If H c G then there is a set R with \R\ ^ log \G:H| such that R and H together generate G. (Here log denotes logarithm to the base 2.) (e) If 7 is a subset of a finite group G with \Y\ > ^|G| then YY = G [3, p. 57]. 3. Reduction Our most important observation is the following.