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        Questions tagged [gr.group-theory]

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

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        Are almost all permutation configurations from $S_n$ covered by small subsets subgroups of $S_n$?

        Given integer $m\in[1,n]$ fix a set $\mathcal T$ of permutations in $S_n$. Then there are subgroups $G_1,\dots,G_m$ of $S_n$ so that $\mathcal T$ is covered by cosets of $G_1,\dots,G_m$. Do we ...
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        $G^F$ conjugacy class of $F$-stable maximal tori, in an algebraic group $G$ defined over $\mathbb{F}_{q}$

        Let $G$ be an affine algebraic group over $k=\bar{\mathbb{F}_{p}}$. Let $q$ be a power of $p$, and assume that $G$ is defined over $\mathbb{F}_q$. Let $\mathcal{T}$, be the collection of all maximal ...
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        Could we assume without loss of generality that all coefficients are positive?

        Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...
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        Game on groups (generalization of spinning switches puzzle)

        Alice and Bob are playing a game as follows: Initially There're two subgroups $A,B$ of $S_n$ known to both Alice and Bob There're $n$ slots $S_1, \cdots, S_n$ and $n$ boxes $B_1, \cdots, B_n$. ...
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        Shifts-induced group of a toroidal cube

        Consider $[n]^d$ -- a $d$-dimensional toroidal cube with side length $n$ divided into $n^d$ unit cubes. Define $k$-shift as a following permutation type on unit cubes: choose $S \subset [n]$ with $|S| ...
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        A closed formula for $\det(\partial/\partial U)^p\prod_{i=1}^n\prod_{j=1}^p U_{i\sigma_j(i)}$

        This is a continuation of this question. Is there a simple formula for $$I(\sigma_1,\cdots,\sigma_p)=(-1)^\sigma\left(\left(\det\left(\frac{\partial}{\partial U_{ij}}\right)_{i,j=1}^n\right)^p \prod_{...
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        Is $SL_n(\mathbb{Q}_p)$ virtually torsion free?

        Recall that a group is virtually torsion free if it admits a finite index subgroup which is torsion free. Question. Is it known whether $SL_n(\mathbb{Q}_p)$ is virtually torsion free for n > 1? ...
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        How is this group theoretic construct called?

        Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be $$\psi(g,h) = |g|+|h|-|gh|$$ Then $\psi:G\times G \...
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        Lattices are not solvable in non-compact semisimple Lie groups

        I'm trying to prove the following result. If $G$ is a non compact semisimple Lie group with no compact factors (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is ...
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        Word length zeta function

        Let $G$ be a group with a finite symmetric set $S$ of generators. Let $\ell_S(x)$ denote the word-length of a given $x\in G$. For $s\in\mathbb C$ set $$ Z(s)=\sum_{x\in G^*}\ell_S(x)^{-s}, $$ where $G^...
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        1answer
        120 views

        Multiplication in $Z(\mathbb{C}S_n)$ [duplicate]

        I am trying to multiply two generators of center $Z(\mathbb{C}[S_n])$ of ring algebra of symmetric group of $n$ elements. We know that these generators are given by sums of conjugacy classes in $S_n,$ ...
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        2answers
        130 views

        Generators for permutation groups

        Consider (e.g.) the full permutation group $G=S_6$. A valid set of generators and equations for $G$ is $r^6=m^2=(rm)^5=1$. I say this system has width $3$ (because there are $3$ equations), length $10$...
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        Amenable groups with special presentations

        Is there a group with a presentation $\left< X \mid r_i, i \in \mathbb{N} \right>$ (where $X$ is finite) with $\left< X \mid r_i, i \in A \right>$ is amenable if and only if $A\subset ...
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        “Brunnian” words in solvable groups

        Let $G$ be a group, and call a word $W(x_1,\dots,x_n)$ in letters $x_i$ and $x_i^{-1}$ "$G$-Brunnian" if there exist $g_1,\dots,g_n\in G$ with $W(g_1,\dots,g_n)\neq1$, but $W(h_1,\dots,h_n)=1$ as soon ...
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        116 views

        Greatest Common Divisor of two specified sequences of numbers (search for equality)

        I consider two sequences of numbers $A=\{a_1,...,a_n\}$ and $B=\{k-a_1,...,k-a_n\}$, where $a_1 \le a_2 \le ... \le a_n \le k$. I am looking for such conditions under which: $gcd(a_1,...,a_n) = gcd(k-...

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