homogenous finitary symmetric groups

We characterize strictly diagonal type of embeddings of finitary symmetric groups in terms of cardinality and the characteristic. namely, we prove thefollowing. let $kappa$ be an infinite cardinal. if$g=underset{i=1}{stackrel{infty}bigcup} g_i,$ where $ g_i=fsym(kappa n_i)$,$(h=underset{i=1}{stackrel{infty}bigcup}h_i, $ where $ h_i=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 homogenousalternating group $alt(kappa)(xi))$, where $n_0=1, n_i=p_1p_2ldots p_i$.