Variations on the Baer–Suzuki theorem

Abstract

The Baer-Suzuki theorem says that if p is a prime, x is a p-element in a finite group G and x, x g is a p-group for all g ∈ G, then the normal closure of x in G is a p-group. We consider the case where x g is replaced by y g for some other p-element y. While the analog of Baer-Suzuki is not true, we show that some variation is. We also answer a closely related question of Pavel Shumyatsky on commutators of conjugacy classes of p-elements.

Order of products of elements in finite groups

“…As we will show below, Theorem A is an easy consequence of a well-known theorem of Guralnick and Robinson on extensions of the Baer-Suzuki theorem [12]. (Some modifications of this result are also obtained by Guralnick and G. Malle in [10]. ) We mention now that certain generalizations of Theorem A are not possible.…”

On varieties of groups satisfying an Engel type identity

“…Given a prime p and a finite group G, we denote by O p (G) the unique maximal normal p-subgroup of G. In the proposition that follows we will require a recent result of Guralnick and Malle [3]: if G is a finite group generated by a normal commutator-closed set of p-elements, then either G is a p-group or p = 5 and G/O 5 (G) is a direct product of copies of A 5 , the alternating group of degree 5. Proposition 3.3.…”

A generalisation of a theorem of Wielandt

“…We show that in general these generalisations fail to guarantee the subnormality of odd p-subgroups. For other variations on the Baer-Suzuki Theorem the interested reader may consult [17], [10], [6], [7], [8] and [9]. Lemma 2.2.…”