Questions tagged [gr.group-theory]

Questions about the branch of abstract algebra that deals with groups.

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Number of commuting pairs in p-group

Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z} )$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. The set $U$ is a Sylow $p$-...
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Does Levi operator always map one-word varieties to one-word varieties?

Suppose $\mathfrak{U}$ is a group variety. Now let’s define $L(\mathfrak{U})$ as the class of all such groups $G$, such that $\forall g \in G$ $\langle \langle g \rangle \rangle \in \mathfrak{U}$ (...
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Efficiently computable graph conductance measures

Treating electrical networks from a graph theory point of view, do there exist measures that characterize the overall electrical conductance of the graph whilst being efficiently computable? There ...
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Does asymmetric fraction of finite groups tend to $0$?

Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
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Amalgamated subgroup of an HNN extension finitely generated

Baumslag proved that if $G= A \ast_{C} B$ is an amalgamated free product where $A$ and $B$ are finitely presented, $G$ is finitely presented if and only if $C$ is finitely generated. Similarly, by ...
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Wreath product $S_k\wr S_n$ inside $S_{kn}$

I want to understand wreath products a little better. Currently, my intuition about them is as follows. Take $nk$ disks, pile them in order forming $n$ piles of $k$ disks (one pile contains disks {1,...
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Algebraic varieties associated to finite groups

Have the following equations been studied in the literature? Let $G$ be a finite group. Then I am looking for functions $f : G \rightarrow \mathbb{C}~ \backslash \left\lbrace 0 \right\rbrace$ such ...
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Is $S_\omega/F_\omega$ embeddable to $S_\omega$?

Let $S_\omega$ be the group of bijections $f:\omega\to\omega$, and let $F_\omega = \{\pi\in S_\omega: \exists N\in \omega(\pi(k) = k \text{ for all } k\geq N)\}$. It is easy to see that $F_\omega$ is ...
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Can the set of endomorphisms of $(\mathbb R,+)$ have cardinality strictly between $\frak c$ and $\frak{c^c}$?

Let $\frak c$ be the cardinality of the reals. I know that in ZF the set of endomorphisms of $(\mathbb R,+)$ can have at least two different cardinalites: If we allow the axiom of choice, you can ...
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In what sense is SL(2,q) “very far from abelian”?

I am far from an expert in this area. I'd be grateful if someone could explain in what sense $\mathop{SL}(2,q)$ is "very far from abelian," to quote Emanuele Viola? Why does Theorem 1 (below) justify ...
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When are extensions of algebraically good groups algebraically good?

Let $G$ be a discrete group. The pro-algebraic completion of $G$ is a pro-algebraic group $G^{\mathrm{alg}}$ together with a morphism $s:G\to G^{\mathrm{alg}}$ which is initial among all morphisms ...
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Spacetime symmetries

We know some nice space-time have a lot of symmetries. It is said that Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$ de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and anti-de Sitter spacetime ...
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Anti-holomorphic involutions of a complex linear algebraic group

Let $G$ be a connected linear algebraic group over the field of complex number ${\Bbb C}$. Let $G({\Bbb C})$ denote the complex Lie group of ${\Bbb C}$-points of $G$. Let $\sigma$ be an anti-...
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Commensurator of a subgroup of matrices

Let $k$ be a totally real number field and let $\mathcal{O}_k$ denote its ring of integers. If $H$ is a subgroup of $\text{GL}(n, \mathbb{R})$ let denote with $H(k)$ and $H(\mathcal{O}_k)$ the ...
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Is there an equivariant simplicial deformation retract of Teichmüller space?

Let $S_g$ be a surface of genus $g \ge 2$. By analogy with Teichmüller space for $S_g$, Culler and Vogtmann studied Outer Space $CV_n$, with points projective classes of marked metric graphs with ...
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Generating the monoid of injections of the free group into itself

Let $F$ be the free group of rank $2$ (or any finite rank if this does not matter). The set of injections $F\to F$ forms a monoid $M$ by compositions. Is there a simple looking set of generators of $M$...
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Diagonal automorphisms for twisted Chevalley groups

Let $G$ be a Chevalley group over a field $k$ of characteristic $0$. We know that a diagonal automorphism $\phi_h$ of $G$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ ...
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Let $G$ be a group acts trivially on an abelian group $A$. Let $\varepsilon$ be a normalized 2-cocycle in $Z^{2}(G,A)$. Assume that $G=H_{1} \times H_{2}$ and let $\varepsilon_{1}=res_{H_{1}\times ... 2answers 132 views Existence of n-axial elements in groups with at least 2 ends Let$G$be a finitely generated group. Fix some symmetric finite generating set$S$for$G$, and write$\Gamma$for the Cayley graph of$G$with respect to$S$. Given finite subsets$X,S,Y$of$G$, ... 2answers 295 views Is random walk drift rational? (Question mildly edited for clarity by request of Matt F.) If$G$is a finitely presented group, let$|\cdot|$denote the word metric with respect to a finite set of generators. Suppose$\nu$is a ... 0answers 164 views Diameter of finite rational matrix groups Suppose$G$is a finite subgroup of$\mathrm{GL}(n,\mathbb{Q})$. For a set$\mathcal{M} \subseteq G$that generates$G$, define the$\mathcal{M}$-diameter$\mathit{diam}(G, \mathcal{M})$of$G$to be ... 1answer 142 views Cohomology of linear algebraic groups Let$R$be a commutative ring. Let$G\subset \mathrm{GL}_m$be a linear algebraic subgroup. Has the group cohomology$H^i(G(R),R^m)$been studied in the literature? For example, do we know (1)$H^...
I say in advance that I am really new to Group Theory, so if my question is trivial I apologize in advance. Let $A$ and $H$ be groups and $\Omega$ be a $H$-set. In this set-up, we can define the ...