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Carter subgroups and injectors in a class of locally finite groups. (English) Zbl 0656.20033

Let \({\mathcal U}\) be the largest subgroup closed class of locally finite groups that satisfies the condition: if G belongs to \({\mathcal U}\) and \(\pi\) is any set of primes, then the Sylow (that is maximal) \(\pi\)-subgroups of G are conjugate in G. Suppose that \({\mathcal K}\) is a class of groups belonging to \({\mathcal U}\) such that (i) \({\mathcal K}\) is closed under taking subgroups and (ii) if \(G\in {\mathcal K}\) and \(C_ p\) is a cyclic group of prime order p, then \(G\times C_ p\in {\mathcal K}\). The main result of the present paper is: Let \({\mathcal X}\) be a Fitting class of \({\mathcal K}\)-groups. Then every \({\mathcal K}\)-group G has \({\mathcal X}\)-injectors, and any two \({\mathcal X}\)-injectors of G are conjugate.
Reviewer: H.Lausch

MSC:

20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
20F50 Periodic groups; locally finite groups
20F17 Formations of groups, Fitting classes
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References:

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