Abstract :
We generalize Lemmas 7.3–7.7 from the Odd Order paper. Those lemmas we extend fromp-solvable groups to π-separable groups, where π is an arbitrary set of primes. Note that even in the case π={p} some of our results present proper generalizations. In some proofs we use the solvability of groups of odd order (in the case π<3 our results do not depend on the solvability of groups of odd order). We give also very short (and, sometimes, new) proofs of some related known results, for example, ThompsonʹsA×B-Lemma. Corollary 3 extends Lemma 8.2 of W. Feit and J. G. Thompson (Pacific J. Math.13, 1963, 775–1029) fromp-groups to solvable groups.
