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Complete existentially closed locally finite groups. (English) Zbl 0563.20037
A group G is existentially closed (e.c.) in a class \({\mathfrak X}\) of groups, if \(G\in {\mathfrak X}\), and each finite system of equations and inequations, with coefficients in G, having a solution in some \({\mathfrak X}\)-group containing G, already has a solution in G. It turns out that when \({\mathfrak X}\) is the class of locally finite groups, one obtains in this way the universal locally finite groups, which are usually defined somewhat differently. This paper deals with groups that are e.c. in \({\mathfrak X}\), where \({\mathfrak X}\) is either the class of all locally finite \(\pi\)-groups (for a set \(\pi\) of primes), or the class of locally finite- soluble \(\pi\)-groups. The main result of this impressive paper is Theorem 1: There is a set \({\mathfrak A}\) of \(2^{\omega_ 1}\) non-isomorphic e.c. \({\mathfrak X}\)-groups of cardinality \(\omega_ 1\) with the following properties: (i) if U and V are distinct members of \({\mathfrak A}\) and A and B are uncountable subgroups of U and V respectively, then A and B are not isomorphic; (ii) if A is an uncountable subgroup of \(U\in {\mathfrak A}\), then A is not isomorphic to any of its proper subgroups; (iii) (\(\diamond)\) every \(U\in {\mathfrak A}\) is a complete group. Here, \(\diamond\) denotes that Jensen’s set theoretic principle \(\diamond\), which cannot be proved from ZFC but is consistent with it, is used in the proof of property (iii) of the groups constructed.
For universal locally finite groups, that is, the case when \({\mathfrak X}\) is the class of all locally finite groups, the corresponding result was proved by K. Hickin [Trans. Am. Math. Soc. 239, 213-227 (1978; Zbl 0386.20014)], without using \(\diamond\). The present construction, which is very intricate, is an amalgam of Hickin’s methods with model theoretic techniques. The paper is carefully written, and the relevant set theory and model theory is clearly described for the benefit of group theorists unaccustomed to this kind of thing.
Now universal locally finite groups are well known to be simple. On the other hand, it is shown to follow from the classification of finite simple groups that if \({\mathfrak X}\) (as described in the first paragraph) is not the class of all locally finite groups, then an e.c. \({\mathfrak X}\)- group cannot be simple. In particular, for such \({\mathfrak X}\), the groups in the set \({\mathfrak A}\) are not simple. By (iii) they are not characteristically simple either, disproving a conjecture of Kegel that e.c. \({\mathfrak X}\)-groups are characteristically simple. Actually a more careful study of the construction shows that if \(U\in {\mathfrak A}\), then every non-trivial normal subgroup of U is uncountable. Thus, since it is also known that the set of normal subgroups is totally ordered by inclusion, distinct normal subgroups of U are non-isomorphic and hence one sees that U is not characteristically simple without using \(\diamond.\)
It is interesting to note that while one obtains uncountable complete locally finite p-groups in this way, a very simple argument shows that a countable locally finite p-group cannot be complete. In fact no example appears to be known of a complete countably infinite locally finite group.
Reviewer: B.Hartley

MSC:
20F50 Periodic groups; locally finite groups
20E25 Local properties of groups
20E07 Subgroup theorems; subgroup growth
03C60 Model-theoretic algebra
20E15 Chains and lattices of subgroups, subnormal subgroups
20F28 Automorphism groups of groups
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