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HOMOGENOUS FINITARY SYMMETRIC GROUPS

2015-03-01
Kegel, Otto. H.
Kuzucuoğlu, Mahmut
We characterize strictly diagonal type of embeddings of finitary symmetric groups in terms of cardinality and the characteristic. Namely, we prove the following. Let n be an infinite cardinal. If G = boolean OR(infinity)(i=1) G(i), where G(i), congruent to FSym(kappa(ni)), (H = boolean OR(infinity)(i=1) H-i, where H-i congruent to Alt(kappa(ni))), is a group of strictly diagonal type and xi = (p(1), p(2), ...) 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) ... p(i) .