×

Groups that are locally embeddable in the class of finite groups. (English. Russian original) Zbl 0898.20016

St. Petersbg. Math. J. 9, No. 1, 49-67 (1998); translation from Algebra Anal. 9, No. 1, 71-97 (1997).
A group \(G\) is called locally embeddable in the class of finite groups (or shortly, is a LEF-group), if for every finite set \(H\subseteq G\) there exists a finite set \(K\) with \(H\subseteq K\) and a binary operation \(\odot\) on \(K\) such that \((K,\odot)\) is a group satisfying the condition: if \(h_1,h_2\in H\) then \(h_1\cdot h_2=h_1\odot h_2\).
Note some properties of LEF-groups. 1. A group \(G\) is a LEF-group if and only if for any finite set \(H=\{h_1,\dots,h_t\}\) there exists a group \(F\), a finite group \(L\) and surjective homomorphisms \(\psi\colon F\to G(H)\) and \(\varphi\colon F\to L\) such that for some \(f_1,\dots,f_t\in F\) satisfying \(\psi(f_i)=h_i\), \(1\leq i\leq t\), the homomorphism \(\varphi\) is injective on \(\{f_1,\dots,f_t\}\). (Here \(G(H)\) is the subgroup of \(G\) generated by \(H\).) 2. Every locally residually finite group is a LEF-group. In particular, every free group is a LEF-group.
3. The following classes of groups are contained in the class of LEF-groups: (a) abelian groups; (b) locally finite groups; (c) nilpotent groups; (d) matrix groups; (e) metabelian groups. 4. A finitely presented LEF-group is residually finite. 5. A finitely presented infinite simple group is not a LEF-group. 6. (a) The Cartesian product of an arbitrary family of LEF-groups is a LEF-group. (b) If \(A\) and \(B\) are LEF-groups then the restricted wreath product \(A\text{ wr }B\) is a LEF-group. (c) If \(A\) is a LEF-group and \(B\) is a locally finite group, then the unrestricted wreath product \(A\text{ Wr }B\) is a LEF-group.

MSC:

20E25 Local properties of groups
20E34 General structure theorems for groups
20F22 Other classes of groups defined by subgroup chains
20F50 Periodic groups; locally finite groups
20E07 Subgroup theorems; subgroup growth
20E26 Residual properties and generalizations; residually finite groups
PDFBibTeX XMLCite