ADVANCES IN MATHEMATICS 45, 319--330 (1982) Commutators and Commutator Subgroups ROBERT M. GURALNICK Department of Mathematics, University of Southern California, Los Angeles, California 90007 1. INTRODUCTION If G is a group and x, y ~ G, then [x,y] =x ly-lxy is the commutator of x and y. Set FG= {[x,y][x, yC G} and 2(G)=n, where n is the smallest integer such that every element of G' is a product of n commutators. The problem of determining when G' = FG (i.e., L(G)= 1) is of particular interest. Fire [2] constructs a group G of order 256 with G' elementary abelian of order 16 and [/'l = 15. Here it is shown that the smallest groups G with G' 4: FG are of order 96. If G=SL,(F), then Shoda [13] (for F algebraically closed) and Thompson [15] (with some exceptions if F has characteristic 0) show G'= FG. Similar results are known for A n, the Mathieu groups, the Suzuki groups, and semisimple algebraic groups over algebraically closed fields (cf. [9, 11, 16, 17]). It is still an open question whether G' =FG for all finite simple groups. Isaacs [8] gives examples of finite perfect groups with G' 4:FG. For examples of finite groups with 2(G) arbitrarily large see [5, 10]. In [6], all pairs (m, n) are determined such that there exists a group G with G' ~- C(n) (the cyclic group of order n) and 2(G) > m. In particular if G' is cyclic and either IGI < 240 or [G')< 60, then G' =FG. Dark and Newell [1] investigate similar questions for the other terms in the descending central series. Let Sylo(G) denote the set of Sylow p-subgroups of G and d(G) the minimal number of elements needed to generate G. In this paper, commutator subgroups with rank 2 Sylow p-subgroups are considered, and the following is proved. THEOREM A. Let P ~ Syl,(G) with P* = P (~ G' abelian and d(P*) ~<2. Then P* ~ FG. Rodney [12] proves this in the case P=P* _~ C(p) × C(p). Examples are given to show that the result is false for d(P*) = 3. However, we do obtain: 319 0001 8708/82/090319-12$05,00/0 Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.