turkmath.org

Let ξ = (p_1, p_2, . . . ,) be a sequence of not necessarily distinct primes. The direct limit group S(χ(ξ)) = SU^∞_{i=1} S(n_i) associated with ξ is called homogeneous symmetric group. The automorphisms of these class was studied by Lavreniuk and Suschansky. Let N = (n_1, n_2, . . .) where n_i = p_1p_2 . . . p_i for all i > 1. An automorphism α of the homogeneous symmetric group S(χ(ξ)) is called M-level preserving automorphism if for some subsequence M = (m_1, m_2, . . .) of N we have α(S(m_i)) = S(mi) for all i > 0. It is proved that there are uncountably many level preserving automorphisms of the homogeneous symmetric group. In this talk, we will answer the following natural question: Are there any automorphism of S(χ(ξ)) which does not preserve any level n_i? We construct uncountably many non-level preserving automorphisms of the group. This is a joint work with Prof. Dr. Mahmut Kuzucuoğlu.