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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Complete universal locally finite groups

Author: Ken Hickin
Journal: Trans. Amer. Math. Soc. 239 (1978), 213-227
MSC: Primary 20E25; Secondary 20F50
MathSciNet review: 0480750
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Abstract: This paper will partly strengthen a recent application of model theory to the construction of sets of pairwise nonembeddable universal locally finite groups [8]. Our result is

Theorem. There is a set $ \mathcal{U}$ of $ {2^{{\aleph _1}}}$ universal locally finite groups of order $ {\aleph _1}$ with the following properties:

0.1. If $ U \ne V \in \mathcal{U}$ and A and B are uncountable sugroups of U and V, then A and B are not isomorphic. Let A be an uncountable subgroup of $ U \in \mathcal{U}$.

0.2. A does not belong to any proper variety of groups, and

0.3. A is not isomorphic to any of its proper subgroups.

0.4. Every $ U \in \mathcal{U}$ is a complete group (every automorphism of U is inner).

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Keywords: Locally finite groups, universal homogeneous groups, complete groups, subgroup-incomparability, regular representation, group amalgams
Article copyright: © Copyright 1978 American Mathematical Society