154,644 research outputs found
Normal Subgroup of Product of Groups
In [6] it was formalized that the direct product of a family of groups gives a new group. In this article, we formalize that for all j ∈ I, the group G = Πi∈IGi has a normal subgroup isomorphic to Gj. Moreover, we show some relations between a family of groups and its direct product.Okazaki Hiroyuki - Shinshu University, Nagano, JapanArai Kenichi - Shinshu University, Nagano, JapanShidama Yasunari - Shinshu University, Nagano, JapanGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.Grzegorz Bancerek and Andrzej Trybulec. Miscellaneous facts about functions. Formalized Mathematics, 5(4):485-492, 1996.Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Artur Korniłowicz. The product of the families of the groups. Formalized Mathematics, 7(1):127-134, 1998.Wojciech A. Trybulec. Classes of conjugation. Normal subgroups. Formalized Mathematics, 1(5):955-962, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.Wojciech A. Trybulec. Subgroup and cosets of subgroups. Formalized Mathematics, 1(5):855-864, 1990.Wojciech A. Trybulec. Lattice of subgroups of a group. Frattini subgroup. Formalized Mathematics, 2(1):41-47, 1991.Wojciech A. Trybulec and Michał J. Trybulec. Homomorphisms and isomorphisms of groups. Quotient group. Formalized Mathematics, 2(4):573-578, 1991.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990
Hilbert space compression for free products and HNN-extensions
Given the Hilbert space compression of two groups, we find bounds on the
Hilbert space compression of their free product. We also investigate the
Hilbert space compression of an HNN-extension of a group relative to a finite
normal subgroup or a finite index subgroup.Comment: 18 page
W*-superrigidity of mixing Gaussian actions of rigid groups
We generalize W*-superrigidity results about Bernoulli actions of rigid
groups to general mixing Gaussian actions. We thus obtain the following: If
\Gamma\ is any ICC group which is w-rigid (i.e. it contains an infinite normal
subgroup with the relative property (T)) then any mixing Gaussian action
\sigma\ of \Gamma\ is W*-superrigid. More precisely, if \rho\ is another free
ergodic action of a group \Lambda\ such that the crossed-product von Neumann
algebras associated with \rho\ and \sigma\ are isomorphic, then \Lambda\ and
\Gamma\ are isomorphic, and the actions \rho\ and \sigma\ are conjugate. We
prove a similar statement whenever \Gamma\ is a non-amenable ICC product of two
infinite groups
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