The Z*-theorem for compact Lie groups

Glauberman’s classical Z∗-theorem is a theorem about involutions of finite groups (i.e. elements of order 2). It is one of the important ingredients for the classification of finite simple groups, which in turn allows to prove the corresponding theorem for elements of arbitrary prime order p. Let us recall the statement: if G is a finite group with a Sylow p-subgroup P , and if x is an element of P of order p such that no other G-conjugate of x lies in P , then the image of x in G/Op′(G) is central, where Op′(G) denotes the maximal normal subgroup of G of order prime to p. The symbol Z∗(G) is the classical notation for the inverse image in G of the centre of G/Op′(G) and this explains the name of the theorem. One can restate the assumption on x in terms of control of fusion. For an arbitrary group G and a prime p, we say that a subgroup H of G controls finite p-fusion in G if the following two conditions are satisfied: (a) every finite p-subgroup of G is conjugate to a subgroup of H, (b) if A is a finite p-subgroup of H and if A is also a subgroup of H for some g ∈ G, then g = ch for some h ∈ H and c in the centralizer CG(A) of A in G. This notion is equivalent to the requirement that the inclusion H → G induces an equivalence between the categories of finite p-subgroups (in a suitable sense, see Section 1). The assumption on x in the Z∗-theorem is then equivalent (at least for finite groups and more generally for compact Lie groups) to the condition that the centralizer CG(x) controls finite p-fusion in G (see Proposition 1.8 below). Also the conclusion of the Z∗-theorem is readily seen to be equivalent to the equation G = CG(x) ·Op′(G) (see Lemma 2.3 below).