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Register a new userOn paratopological gyrogroups
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The concept of gyrogroups is a generalization of groups which do not explicitly have associativity. Recently, Atiponrat extended the idea of topological (paratopological) groups to topological (paratopological) gyrogroups. In this paper, we prove that every regular (Hausdorff) locally gyroscopic invariant paratopological gyrogroup $G$ is completely regular (function Hausdorff). These results improve theorems of Banakh and Ravsky for paratopological groups. Also, we extend the Pontrjagin conditions of (para)topological groups to (para)topological gyrogroups.
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Topological gyrogroups, with a weaker algebraic structure than groups, have been investigated recently. In this paper, we prove that every feathered strongly topological gyrogroup is paracompact, which implies that every feathered strongly topological gyrogroup is a $D$-space and gives partial answers to two questions posed by A.V.Arhangel skivi ~(2010) in cite{AA1}. Moreover, we prove that every locally compact $NSS$-gyrogroup is first-countable. Finally, we prove that each Lindel{o}f $P$-gyrogroup is Ra$check{imath}$kov complete.
Given a $T_0$ paratopological group $G$ and a class $mathcal C$ of continuous homomorphisms of paratopological groups, we define the $mathcal C$-$semicompletion$ $mathcal C[G)$ and $mathcal C$-$completion$ $mathcal C[G]$ of the group $G$ that contain $G$ as a dense subgroup, satisfy the $T_0$-separation axiom and have certain universality properties. For special classes $mathcal C$, we present some necessary and sufficient conditions on $G$ in order that the (semi)completions $mathcal C[G)$ and $mathcal C[G]$ be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group $G$ whose $mathcal C$-semicompletion $mathcal C[G)$ fails to be a $T_1$-space, where $mathcal C$ is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group $G$ contains an $omega$-bounded sequentially compact subgroup $H$ such that $H$ is a topological group but its closure in $G$ fails to be a subgroup.
A discrete subset $S$ of a topological gyrogroup $G$ with the identity $0$ is said to be a {it suitable set} for $G$ if it generates a dense subgyrogroup of $G$ and $Scup {0}$ is closed in $G$. In this paper, it was proved that each countable Hausdorff topological gyrogroup has a suitable set; moreover, it is shown that each separable metrizable strongly topological gyrogroup has a suitable set.
Consider a group word w in n letters. For a compact group G, w induces a map G^n rightarrow G$ and thus a pushforward measure {mu}_w on G from the Haar measure on G^n. We associate to each word w a 2-dimensional cell complex X(w) and prove in Theorem 2.5 that {mu}_w is determined by the topology of X(w). The proof makes use of non-abelian cohomology and Nielsens classification of automorphisms of free groups [Nie24]. Focusing on the case when X(w) is a surface, we rediscover representation-theoretic formulas for {mu}_w that were derived by Witten in the context of quantum gauge theory [Wit91]. These formulas generalize a result of ErdH{o}s and Turan on the probability that two random elements of a finite group commute [ET68]. As another corollary, we give an elementary proof that the dimension of an irreducible complex representation of a finite group divides the order of the group; the only ingredients are Schurs lemma, basic counting, and a divisibility argument.
Suppose that $X=G/K$ is the quotient of a locally compact group by a closed subgroup. If $X$ is locally contractible and connected, we prove that $X$ is a manifold. If the $G$-action is faithful, then $G$ is a Lie group.
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