ON PI-SEPARABLE GROUPS

We generlize Lemmas 7.3-7.7 from the Odd Order paper. Those lemmas we extend from p-solvable groups to pi-separable groups, where pi is an a rbitrary set of primes. Note that even in the ease pi = {p} some of ou r results present proper generalizations. In some proofs we use the so lvability of groups of odd order (in the case /pi/ < 3 our results do not depend on the solvability of groups of odd order). We give also ve ry short (and, sometimes, new) proofs of some related known results, f or example, Thompson's A x B-Lemma. Corollary 3 extends Lemma 8.2 of W . Felt and J. G. Thompson (Pacific J. Math. 13, 1963, 775-1029) from p -groups to solvable groups. (C) 1996 Academic Press, Inc.