Prime localizations of Burger–Mozes-type groups

Abstract

This article concerns Burger-Mozes universal groups acting on regular trees locally like a given permutation group of finite degree. We also consider locally isomorphic generalizations of the former due to Le Boudec and Lederle. For a large class of such permutation groups and primes p we determine their local p-Sylow subgroups as well as subgroups of their p-localization, which is identified as a group of the same type in certain cases.Date: March 23, 2020.

Orbits of Sylow subgroups of finite permutation groups

“…We note the following classic result which, together with Tornier's result [17] for the alternating and symmetric groups, deals completely with the case where n p = p. We need the following three preliminary results. The first concerns primitive prime divisors and can be found in [10, Lemma 2.5] (for p = 2) and [14, Lemma 4.1(iii)].…”

“…The almost simple 2-transitive permutation groups are listed in [5,Table 7.4]. If soc(G) = A n then by Tornier [17] we have that pn p > n and again case (a) holds. Also, if n p = p then one of (b), (c) or (d) holds by Theorem 5.1, so we also assume that n p p 2 .…”

“…Given G Sym(Ω) with |Ω| = n, the Burger-Mozes group U (G) [3] is the group of all automorphisms of the n-regular tree T n such that, for all vertices v, the stabiliser of v induces the group G on the set of all neighbours of v. Tornier [17] introduced the following property when studying local Sylow p-subgroups of Burger-Mozes groups. We use Syl p (G) to denote the set of all Sylow p-subgroups of a group G and for a group G acting on a set Ω we use G ω to denote the stabiliser in G of the point ω ∈ Ω.…”

“…Tornier's motivation for studying Property ( * ) p was as follows: Let T be a finite subtree of T n and for any H Aut(T n ) let H (T ) denote the pointwise stabiliser of T in H. Then Tornier [17, Proposition 10] showed that for G Sym(Ω) and P ∈ Syl p (G), the group U (P ) (T ) is a local Sylow p-subgroup of U (G) (T ) if and only if G has Property ( * ) p . Tornier [17, Proposition 11] showed that if G is the alternating group A n or symmetric group S n acting on n points then G has Property ( * ) p , with p dividing |G|, if and only if n p p > n, where n p is the highest power of p that divides n. Moreover, if either P has the same orbits as G or |Ω| = p n , then ( * ) p holds [17,Propositions 12 and 13].…”