Homogenous finitary symmetric groups

Document Type: Ischia Group Theory 2014

Authors

1 Mathematisches Institut Albert Ludwigs Universitat Eckerstr

2 Middle East Technical University

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 $G_i\cong FSym(\kappa n_i)$, ($H=\underset{i=1}{\stackrel{\infty}\bigcup}H_i$, where $H_i\cong Alt(\kappa n_i)$), 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 $n_0=1$, $n_i=p_1p_2\cdots p_i$.

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