Profinite groupos with few conjugacy classes of $p$-elements

Abstract

It is proved that a profinite group G has fewer than 2 ℵ 0 conjugacy classes of p-elements for an odd prime p if and only if its p-Sylow subgroups are finite. (Here, by a p-element one understands an element that either has p-power order or topologically generates a group isomorphic to Zp.) A weaker result is proved for p = 2.

Profinite groups with few conjugacy classes of 𝑝-elements

“…Proof. The result is very similar to Proposition 2.2 in [7], which was the special case when N 0 u = G, and the proof is essentially the same. However the printed proof in [7] contains small errors (corrected in the arXiv version) and so we give the proof in its entirety.…”

“…The result is very similar to Proposition 2.2 in [7], which was the special case when N 0 u = G, and the proof is essentially the same. However the printed proof in [7] contains small errors (corrected in the arXiv version) and so we give the proof in its entirety. We construct a descending chain (N k ) k 0 of open normal subgroups and a family (R k ) k 0 of finite subsets of N 0 u ∩ P such that for each k 1 and x ∈ R k there are elements x (1) , x (2) ∈ N k−1 x ∩ P for which (i) N k x (1) and N k x (2) are not conjugate in G/N k and (ii) N k x (1) and N k x (2) have order at least p k in G/N k .…”

“…A well-known theorem of Zelmanov [9] asserts that profinite torsion groups are locally finite; that is, their finite subsets generate finite subgroups. Inspired by these results, we use an extension of Zelmanov's theorem (also due to Zelmanov; see Theorem C below) and some ideas from Wilson [7] to prove the following result:…”

“…(a) First suppose that G is finite. We use results proved in[7].From [7, Lemmas 3.3, 3.7] applied to quotient groups G/K 1 with K 1 ⊳ G and K 1 K, we conclude that K has no perfect composition factors of order divisible by p, and then the conclusion follows from [7, Lemmas 3.3, 4.1]. (It is worth mentioning that [7, Lemma 3.7] depends on the classification of the finite simple groups; in particular this is the reason why the prime 2 is excluded.)…”