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=underseti=1stackrelinftybigcupG i , where G i =FSym(kappan i ) , (H=underseti=1stackrelinftybigcupH i , where H i =Alt(kappan 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 1 p 2 ldotsp i .

Keywords

• Finitary symmetric groups
• Centralizer
• Locally finite simple groups