Homogenous finitary symmetric groups

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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 $n_0=1, n_i=p_1p_2ldots p_i$.