A theorem about coprime action

It is well known that if an elementary abelian p-group P acts on a p'-group Q and Q = vertical bar Q, P vertical bar, then Q . Does a similar statement hold for C-Q(P)? Under further assumptions, the answer is yes. Goldschmidt proves theorems of this flavour in [D.M. Goldschmidt, Weakly embedded 2-local subgroups of finite groups, J. Algebra 21 (1972) 341-351. [1]; D.M. Goldschmidt, Strongly closed 2-subgroups of finite groups, Ann. of Math. (2) 102 (1975) 475-489. [21] and uses them to construct signallizer functors. For the same reason we prove a result of this type, under the assumption that Q is soluble. (C) 2008 Elsevier Inc. All rights reserved.