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Register a new userEmbedding topological semigroups into the hyperspaces over topological groups
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We study algebraic and topological properties of subsemigroups of the hyperspace exp(G) of non-empty compact subsets of a topological group G endowed with the Vietoris topology and the natural semigroup operation. On this base we prove that a compact Clifford topological semigroup S is topologically isomorphic to a subsemigroup of exp(G) for a suitable topological group G if and only if S is a topological inverse semigroup with zero-dimensional idempotent semilattice.
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We study algebraic and topological properties of topological semigroups containing a copy of the bicyclic semigroup C(p,q). We prove that each topological semigroup S with pseudocompact square contains no dense copy of C(p,q). On the other hand, we construct a (consistent) example of a pseudocompact (countably compact) Tychonov semigroup containing a copy of C(p,q).
In this paper, we investigate algebraic and topological properties of the Riordan groups over finite fields. These groups provide a new class of topologically finitely generated profinite groups with finite width. We also introduce, characterize index-subgroups of our Riordan groups, and finally we show exactly the range of Hausdorff dimensions of these groups. The latter results are analogous to the work of Barnea and Klopsch for the Nottingham groups.
Let $G$, $R$ and $A$ be topological groups. Suppose that $G$ and $R$ act continuously on $A$, and $G$ acts continuously on $R$. In this paper, we define a partially crossed topological $G-R$-bimodule $(A,mu)$, where $mu:Arightarrow R$ is a continuous homomorphism. Let $Der_{c}(G,(A,mu))$ be the set of all $(alpha,r)$ such that $alpha:Grightarrow A$ is a continuous crossed homomorphism and $mualpha(g)=r^{g}r^{-1}$. We introduce a topology on $Der_{c}(G,(A,mu))$. We show that $Der_{c}(G,(A,mu))$ is a topological group, wherever $G$ and $R$ are locally compact. We define the first cohomology, $H^{1}(G,(A,mu))$, of $G$ with coefficients in $(A,mu)$ as a quotient space of $Der_{c}(G,(A,mu))$. Also, we state conditions under which $H^{1}(G,(A,mu))$ is a topological group. Finally, we show that under what conditions $H^{1}(G,(A,mu))$ is one of the following: $k$-space, discrete, locally compact and compact.
We consider the lattice of subsemigroups of the general linear group over an Artinian ring containing the group of diagonal matrices and show that every such semigroup is actually a group.
Let $G$ be a simple algebraic group over an algebraically closed field $k$ and let $C_1, ldots, C_t$ be non-central conjugacy classes in $G$. In this paper, we consider the problem of determining whether there exist $g_i in C_i$ such that $langle g_1, ldots, g_t rangle$ is Zariski dense in $G$. First we establish a general result, which shows that if $Omega$ is an irreducible subvariety of $G^t$, then the set of tuples in $Omega$ generating a dense subgroup of $G$ is either empty or dense in $Omega$. In the special case $Omega = C_1 times cdots times C_t$, by considering the dimensions of fixed point spaces, we prove that this set is dense when $G$ is an exceptional algebraic group and $t geqslant 5$, assuming $k$ is not algebraic over a finite field. In fact, for $G=G_2$ we only need $t geqslant 4$ and both of these bounds are best possible. As an application, we show that many faithful representations of exceptional algebraic groups are generically free. We also establish new results on the topological generation of exceptional groups in the special case $t=2$, which have applications to random generation of finite exceptional groups of Lie type. In particular, we prove a conjecture of Liebeck and Shalev on the random $(r,s)$-generation of exceptional groups.
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