Construction of some simple locally finite groups

We construct a proper class of simple locally finite groups. Namely for each infinite cardinal $\kappa$, we construct uncountably many pairwise non-isomorphic simple locally finite groups of cardinality $\kappa$, as a direct limit of finitary symmetric groups. The construction of the groups of similar kind for countably infinite order has been common knowledge as indicated in [2]. The countable ones are classified using the lattice of Steinitz numbers by Kroshko–Sushchansky in [3]. We give the classification of the uncountable ones by the pair, the cardinality of the group and the characteristic which corresponds to a Steinitz number. We study the structure of the centralizers of arbitrary elements in this new class of groups and correct some of the errors in the section about the centralizers of elements in $S(\xi)$ in [3].