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Existentially closed L\({\mathfrak X}\)-groups. (English) Zbl 0598.20036
If \({\mathfrak Y}\) is a class of groups, then a \({\mathfrak Y}\)-group G is existentially closed if every system of finitely many equations and inequations with coefficients from G, which has a solution in some \({\mathfrak Y}\)-supergroup of G, can already be solved in G. Properties of such groups when \({\mathfrak Y}\) is the class of all locally finite groups follow from the work of P. Hall [J. Lond. Math. Soc. 34, 305-319 (1959; Zbl 0088.023)] and when \({\mathfrak Y}\) is the class of all locally finite p-groups from the work of B. Maier [Arch. Math. 37, 113-128 (1981; Zbl 0474.20015)]. In this paper, the author considers classes \({\mathfrak Y}\) of the form L\({\mathfrak X}\) (the class of all locally \({\mathfrak X}\)-groups) when \({\mathfrak X}\) is closed under the taking of subgroups, extensions, quotient groups and cartesian powers of finitely generated subgroups. (Not all these properties are required in every case however.)
A group is said to be verbally complete if every element of the group is an instance of every non-trivial word (taken from some fixed countable free group). It is shown that e.c. L\({\mathfrak X}\)-groups are verbally complete in many cases; for example when the group is periodic or when the class \({\mathfrak X}\) contains a non-trivial p-group for each prime p. The general question, however, remains open. More particularly, if \({\mathfrak X}\) is closed under taking subgroups, extensions, quotient groups and cartesian powers of finitely generated subgroups, is an e.c. L\({\mathfrak X}\)-group verbally complete ?
The next section considers characteristic subgroups of e.c. L\({\mathfrak X}\)- groups. Again, it is shown that in many cases these must be proper or trivial but the general case is not fully understood. In particular, the author asks under what conditions an e.c. L\({\mathfrak X}\)-group (\({\mathfrak X}\) locally finite) is characteristically simple.
The normal subgroup structure of countable e.c. L\({\mathfrak X}\)-groups is then considered and it is shown that such groups have a unique chief series (i.e. the normal subgroups are totally ordered by inclusion). This does not require all the restrictions on \({\mathfrak X}\) but using all these restrictions, it can be shown that the order type of the set of normal subgroups is that of the rationals.
The automorphisms of e.c. L\({\mathfrak X}\)-groups are considered next. Finally it is shown that there are \(2^{\aleph_ 0}\) isomorphism types of e.c. locally-(soluble with torsion elements all \(\pi\)-elements) groups whenever \(\pi\) is an infinite class of primes. This is in contrast to the case for L\({\mathfrak X}\) the class of locally finite groups or locally finite p-groups where there is a unique countable e.c. L\({\mathfrak X}\)- group.
Reviewer: J.R.J.Groves

MSC:
20E25 Local properties of groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F50 Periodic groups; locally finite groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20F14 Derived series, central series, and generalizations for groups
20E07 Subgroup theorems; subgroup growth
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
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