JOURNAL OF COMBINATORIAL THEORY (A) 16, 209-214 (1974) Proof of the Fundamental Lemma of Complexity (Strong Version) for Arbitrary Finite Semigroups* JOHN RHODES University of California, Berkeley, Cal$x-nia 94720 Communicated by the Managing Editors Received May 6, 1971 DEDICATED TO ALEXANDER DONIPHAN WALLACE 1. STATEMENT OF THE THEOREM All semigroups considered are of finite order. All undefined notation is explained in [4, Chapters 1, 5-91. The results proved here together with some corollaries were announced in [5]. We assume the reader is familiar with the results of [4, Chapters 1, 5-91. See also [2], [3], and [7], The purpose of this paper is to prove the following theorem, Theorem A, the so-called Fundamental Lemma of Complexity (strong form). We recall that 0: S -++Y T means that 0 is an epimorphism (equal onto homomorphism) such that 0 restricted to any subgroup of S is one-to-one. (1.1) THEOREM A. S --Hi T implies (1.2) C(S) < CC,1) 0 C(T). And thus, in particular, #G(S) = #G(T). 2. REDUCTION TO THE WEAK VERSION (2.1) NOTATION. We recall (see [4] or [l]) that for s, 1 E S, we write s&‘t iff Ss = Bt and sS = tS. J? is an equivalence relation on S. We write 0: s vwP)‘+ T * This research was sponsored in part by N.I.H. USDHEW-SROl-GM 14211-03. 209 Copyright 0 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.