The normalized positive definite class functions CG −→ C are determined for all those direct limits G of finite alternating groups Alt(Ω i ) for which the embeddings Alt(Ω i ) −→ Alt(Ω j ) are natural in the sense that every non-trivial Alt(Ω i )-orbit in Ω j is natural.

1. IntroductionA class [Xscr ] of groups is said to be countably recognizable, if every group all of whose countable subgroups are contained in countable [Xscr ]-subgroups is itself an [Xscr ]-group. Many examples of such classes are discussed in section 8·3 of [20]. In the present work we are concerned with the question of how far countable recognizability can be obtained for classes of finitary linear groups. Recall that a group is said to be finitary [ ]-linear if it is isomorphic to a subgroup of FGL[ ](V), the group of all invertible [ ]-linear transformations α of the [ ]-vector space V with the property that the image of the endomorphism α−idV has finite [ ]-dimension. This generalizes the notion of linearity. A survey about features of finitary linear groups is given in [18].

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Every finite group G has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the nonsolvable length of the group and denoted by λ(G). For every integer n, we define a particular class of groups of nonsolvable length n, called n-rarefied, and we show that every finite group of nonsolvable length n contains an n-rarefied subgroup. As applications of this result, we improve the known upper bounds on λ(G) and determine the maximum possible nonsolvable length for permutation groups and linear groups of fixed degree resp. dimension.

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