Homogenous finitary symmetric groups

Abstract

We characterize strictly diagonal type of embeddings of finitary symmetric groups in terms of cardinality and the ‎characteristic. Namely, we prove the following. Let $kappa$ be an infinite cardinal. If $G=underset{i=1}{stackrel{infty}bigcup} G_i$, where $G_icong FSym(kappa n_i)$, ($H=underset{i=1}{stackrel{infty}bigcup}H_i$, where $H_icong Alt(kappa n_i)$), is a group of strictly diagonal type and $xi=(p_1, p_2, ldots )$ is an infinite sequence of primes, then $G$ is isomorphic to the homogenous finitary symmetric group $FSym(kappa)(xi)$ ($H$ is isomorphic to the homogenous alternating group $Alt(kappa)(xi))$, where $n_0=1$, $n_i=p_1p_2cdots p_i$.