Centralizers of Subgroups in Direct Limits of Symmetric Groups with Strictly Diagonal Embedding

Abstract

Let ξ = (p ₁, p ₂,…) be a given infinite sequence of not necessarily distinct primes. In 1976, the structure of locally finite groups S(ξ) (respectively A(ξ) ) which are obtained as a direct limit of finite symmetric (finite alternating) groups are investigated in [7]. The countable locally finite groups A(ξ) gives an important class in the theory of infinite simple locally finite groups. The classification of these groups using the lattice of Steinitz numbers is completed by Kroshko and Sushchansky in 1998 see [8]. Here we extend the results on the structure of centralizers of elements to centralizers of arbitrary finite subgroups and correct some of the errors in the section of centralizers of elements in [8]. We construct for each infinite cardinal κ, a new class of uncountably many simple locally finite groups of cardinality κ as a direct limit of finitary symmetric groups. We investigate the centralizers of elements and finite subgroups in this new class of simple locally finite groups, and finally, we characterize this class by the lattice isomorphism with the cardinality of the group and the Steinitz numbers.

Key Words:

• Centralizer
• Nonlinear group
• Simple locally finite

2010 Mathematics Subject Classification:

• 20F50
• 20E32

ACKNOWLEDGMENTS

The third author would like to express his sincere thanks to the colleagues and the staff of FRIAS for warm hospitality during his visit to Freiburg.

Notes

Communicated by A. Olshanskii.