Otto H. Kegel

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

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.

Zur Einfachheit der multiplikativen Gruppe eines existentiell abgeschlossenen Schiefkörpers

Let E be an existentially closed skew field in the class of all skew fields with given central subfield Z. For the multiplicative group E* it is shown that the group G ≔ E*/Z* is simple.