Elements of maximal order in finite $p$-groups

Abstract

Let G be a finite p-group such that every 3-generator subgroup has class at most p. If K(G) denotes the subgroup of G generated by the elements of maximal order in G, then K(G) has index at most p in G. G. Zappa proved [5] that if G is a finite p-group of class at most p, then H,(G) has index at most p in G, unless H,(G)= 1. Here H,(G) denotes the subgroup of G generated by those elements of G having order different from p. If we let K(G) be the subgroup of G generated by the elements of maximal order in G, then, of course, K(G) C H,(G), whenever H,(G) ? 1. So we describe a construction which yields a family of examples, for each odd prime p and arbitrary exponent greater than p2, of finite p-groups of class p with K(G)?H,(G). Thus, our result is a strengthening of Zappa's, although the techniques are essentially the same. Recently, I. D. Macdonald [3], [4] extended Zappa's results in a different direction. Our main goal is the proof of the following THEOREM. Let G be afinite p-group such that every 3-generator subgroup has class at most p. Then [G: K(G)] <p. The notation will be the same as in [1]. The computational aspect of the proof will be facilitated greatly by the following lemma. LEMMA. Let G be a finite p-group of exponent pm+l, with the exponent of G2 a divisor of pm. If yeK(G), and xeG\K(G), then _pm = [y, lx]P'm(mod G,+,). PROOF. Since we argue modulo G,+,, we may as well assume that Gp+l= 1, that is, G has class at most p. From yeK(G), and xeG\K(G), it follows that xyEG\K(G). By the Hall-Petrescu Identity [2, Chapter III, Satz 9.4, Hilfsatz 9.5], m m mm jm P (P (1) xP yP = (xy)P I Ck k=2 Presented to the Society, March 27, 1970; received by the editors May 7, 1971. AMS 1970 subject classifications. Primary 20D15, 20D25.