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# Questions tagged [gr.group-theory]

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

5,572 questions
69 views

### 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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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 $|S| ... 1answer 96 views ### A closed formula for$\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 $$I(\sigma_1,\cdots,\sigma_p)=(-1)^\sigma\left(\left(\det\left(\frac{\partial}{\partial U_{ij}}\right)_{i,j=1}^n\right)^p \prod_{... 0answers 177 views ### Is SL_n(\mathbb{Q}_p) virtually torsion free? Recall that a group is virtually torsion free if it admits a finite index subgroup which is torsion free. Question. Is it known whether SL_n(\mathbb{Q}_p) is virtually torsion free for n > 1? ... 0answers 471 views ### How is this group theoretic construct called? Let G be a finite group, S\subset G a generating set, |g| = |g|_S = word length with respect to S. Define the "defect" of g,h to be$$\psi(g,h) = |g|+|h|-|gh|$$Then \psi:G\times G \... 0answers 54 views ### Lattices are not solvable in non-compact semisimple Lie groups I'm trying to prove the following result. If G is a non compact semisimple Lie group with no compact factors (lying in some SL(l,\mathbb{R})), and \Gamma is a lattice in G, then \Gamma is ... 1answer 155 views ### Word length zeta function Let G be a group with a finite symmetric set S of generators. Let \ell_S(x) denote the word-length of a given x\in G. For s\in\mathbb C set$$ Z(s)=\sum_{x\in G^*}\ell_S(x)^{-s},$$where$G^...
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### 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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### 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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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 $A\subset ... 0answers 81 views ### “Brunnian” words in solvable groups Let$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$as soon ... 1answer 116 views ### Greatest Common Divisor of two specified sequences of numbers (search for equality) I consider two sequences of numbers$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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