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Register a new userGyrogroup through its Grothendieck Group Completion and Right gyrogroup action
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In this article, we discuss the Grothendieck group completion (GGC) of a gyrogroup. Consequently, we show that there is a one to one correspondence between actions and representations of a gyrogroup, and actions and representations of its Grothendieck group completion. We also introduce the concept of an action of a right gyrogroup.
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Using computational methods, we complete the determination of the $3$-modular character table of the Chevalley group $F_4(2)$ and its covering group.
We study the algebraic $K$-theory and Grothendieck-Witt theory of proto-exact categories, with a particular focus on classes of examples of $mathbb{F}_1$-linear nature. Our main results are analogues of theorems of Quillen and Schlichting, relating the $K$-theory or Grothendieck-Witt theories of proto-exact categories defined using the (hermitian) $Q$-construction and group completion.
Let $(mathcal{G},Gamma)$ be an abstract graph of finite groups. If $Gamma$ is finite, we can construct a profinite graph of groups in a natural way $(hat{mathcal{G}},Gamma)$, where $hat{mathcal{G}}(m)$ is the profinite completion of $mathcal{G}(m)$ for all $m in Gamma$. The main reason for this is that $Gamma$ is finite, so it is already profinite. In this paper we deal with the infinite case, by constructing a profinite graph $overline{Gamma}$ where $Gamma$ is densely embedded and then defining a profinite graph of groups $(widehat{mathcal{G}},overline{Gamma})$. We also prove that the fundamental group $Pi_1(widehat{mathcal{G}},overline{Gamma})$ is the profinite completion of $Pi_1^{abs}(mathcal{G},Gamma)$. This answers Open Question 6.7.1 of the book Profinite Graphs and Groups, published by Luis Ribes in 2017. Later we generalise the main theorem of a paper by Luis Ribes and the second author, proving that if $R$ is a virtually free abstract group and $H$ is a finitely generated subgroup of $R$, then $overline{N_{R}(H)}=N_{hat{R}}(overline{H})$ answering Open Question 15.11.10 of the book of Ribes. Finally, we generalise the main theorem of a paper by Sheila Chagas and the second author, showing that every virtually free group is subgroup conjugacy separable. This answers Open Question 15.11.11 of the same book of Ribes.
It is well known that $G=langle x,y:x^2=y^3=1rangle$ represents the modular group $PSL(2,Z)$, where $x:zrightarrowfrac{-1}{z}, y:zrightarrowfrac{z-1}{z}$ are linear fractional transformations. Let $n=k^2m$, where $k$ is any non zero integer and $m$ is square free positive integer. Then the set $$Q^*(sqrt{n}):={frac{a+sqrt{n}}{c}:a,c,b=frac{a^2-n}{c}in Z~textmd{and}~(a,b,c)=1}$$ is a $G$-subset of the real quadratic field $Q(sqrt{m})$ cite{R9}. We denote $alpha=frac{a+sqrt{n}}{c}$ in $ Q^*(sqrt{n})$ by $alpha(a,b,c)$. For a fixed integer $s>1$, we say that two elements $alpha(a,b,c)$, $alpha(a,b,c)$ of $Q^*(sqrt{n})$ are $s$-equivalent if and only if $aequiv a(mod~s)$, $bequiv b(mod~s)$ and $cequiv c(mod~s)$. The class $[a,b,c](mod~s)$ contains all $s$-equivalent elements of $Q^*(sqrt{n})$ and $E^n_s$ denotes the set consisting of all such classes of the form $[a,b,c](mod~s)$. In this paper we investigate proper $G$-subsets and $G$-orbits of the set $Q^*(sqrt{n})$ under the action of Modular Group $G$
Two groups are said to have the same nilpotent genus if they have the same nilpotent quotients. We answer four questions of Baumslag concerning nilpotent completions. (i) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one is finitely presented and the other is not. (ii) There exists a pair of finitely presented, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has a solvable conjugacy problem and the other does not. (iii) There exists a pair of finitely generated, residually torsion-free-nilpotent groups of the same nilpotent genus such that one has finitely generated second homology $H_2(-,Z)$ and the other does not. (iv) A non-trivial normal subgroup of infinite index in a finitely generated parafree group cannot be finitely generated. In proving this last result, we establish that the first $L^2$ betti number of a finitely generated parafree group of rank $r$ is $r-1$. It follows that the reduced $C^*$-algebra of the group is simple if $rge 2$, and that a version of the Freiheitssatz holds for parafree groups.
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