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.