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**Universal locally finite central extensions of groups.**
*(English)*
Zbl 0582.20022

An interesting new class of locally finite simple groups is studied; they generalize the well-known locally finite groups introduced by P. Hall [J. Lond. Math. Soc. 34, 305-319 (1959; Zbl 0088.023)]. The starting point is a periodic abelian group A. The group G is called a universal locally finite central extension of A if (i) A lies in the centre of G; (ii) G is locally finite; (iii) if \(A\leq B\leq D\) are groups such that A is in the centre of D and of finite index in D, then every monomorphism of B into G that fixes the elements of A can be extended to a monomorphism of D into G.

All the universal locally finite general extensions of A form the class ULF(A). Theorem 3 says (much more than) that this class is non-empty; if A is countable, then ULF(A) contains a countable member which is unique up to isomorphism; if A is finite, then there are \(2^{\aleph_ 0}\) mutually non-embeddable members of ULF(A) of cardinality \(\aleph_ 1\), and their central quotients are also mutually non-embeddable. Theorem 1 says, inter alia, that if \(G\in ULF(A)\), then A is the centre of G and G/A is simple; moreover G/A uniquely determines A. If A contains the direct cube of a cyclic group of prime order, then G/A is NOT a direct limit of finite simple groups. There are many more interesting results in the paper. The method relies on embedding lemmas for amalgams of locally finite groups that exploit the permutational product of groups.

All the universal locally finite general extensions of A form the class ULF(A). Theorem 3 says (much more than) that this class is non-empty; if A is countable, then ULF(A) contains a countable member which is unique up to isomorphism; if A is finite, then there are \(2^{\aleph_ 0}\) mutually non-embeddable members of ULF(A) of cardinality \(\aleph_ 1\), and their central quotients are also mutually non-embeddable. Theorem 1 says, inter alia, that if \(G\in ULF(A)\), then A is the centre of G and G/A is simple; moreover G/A uniquely determines A. If A contains the direct cube of a cyclic group of prime order, then G/A is NOT a direct limit of finite simple groups. There are many more interesting results in the paper. The method relies on embedding lemmas for amalgams of locally finite groups that exploit the permutational product of groups.

Reviewer: B.H.Neumann

### MSC:

20E25 | Local properties of groups |

20E32 | Simple groups |

20F50 | Periodic groups; locally finite groups |

20E22 | Extensions, wreath products, and other compositions of groups |