The \(Z^*\)-theorem for compact Lie groups. (English) Zbl 0758.22006

A subgroup \(H\) of a finite group \(G\) is said to control \(p\)-fusion in \(G\) if the inclusion \(H\to G\) induces an equivalence of the categories of finite \(p\)-subgroups of \(H\) and \(G\). One way of stating the classical \(Z^*\)-theorem (due to Glauberman for \(p=2\) and to the classification for odd \(p\)) is the following: if the centralizer \(C_ G(A)\) of a \(p\)- subgroup \(A\) of \(G\) controls \(p\)-fusion in \(G\), then \(G=C_ G(A)\cdot O_{p'}(G)\), where \(O_{p'}(G)\) denotes the maximal normal subgroup of \(G\) of order prime to \(p\). The main result of the paper asserts that the same result holds for a (not necessarily connected) compact Lie group \(G\) and a subgroup \(A\) which is either a (not necessarily finite) \(p\)- subgroup or a \(p\)-toral subgroup. The notion of control of fusion can be generalized using the category of all finite \(p\)-subgroups, or all \(p\)- subgroups, or all \(p\)-toral subgroups; it turns out that one gets equivalent notions. This work is motivated by a recent theorem of Mislin which asserts that \(H\) controls \(p\)-fusion in a compact Lie group \(G\) if and only if \(BH\to BG\) induces an isomorphism in \(\text{mod-}p\) cohomology.
Reviewer: G.Mislin (Zürich)


22E20 General properties and structure of other Lie groups
22C05 Compact groups
57T10 Homology and cohomology of Lie groups
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
20E07 Subgroup theorems; subgroup growth
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