Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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How many conjugacy classes of cyclic subgroups of order $p^2$ does $\operatorname{GL}_{n}(\Bbb Z / p\Bbb Z)$ have?
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Let $f\in \Hom((\Bbb Z/p^2\Bbb Z),\GL_{n}(\Bbb Z / p\Bbb Z))$ be an injective homomorphism. What is the number ...
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When is the product of regular matrices regular
$\DeclareMathOperator{\GL}{\operatorname{GL}}$We say that a matrix $g \in \GL_n(F)$ is regular if it has a centraliser of minimal dimension, or equivalently, if the minimal and characteristic ...
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Example of family of Cayley graphs with Ramanujan behaviour on finite $p$-groups
This is a very general question: are there known examples of Ramanujan behaviour of Cayley graphs obtained from family of finite p-groups?
${\mathrm{\bf Adjacency~matrix:}}$ Given a graph ${\mathcal{G}...
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What are the irreps in this canonical action of $\operatorname{PGL}_2(F_q)$?
Consider the permutation action of $\operatorname{PGL}_2(\mathbb F_q)$ on $\mathbb P^1(\mathbb F_q)$ by fractional linear transformations. We can consider the associated (complex) representation of ...
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Neighbor count in sphere packing in N dimensions
So I'm really interested in building a mathematical model for how powerful computer chips could be given extra spatial dimensions. Obviously this is a squishy problem, since "computer chips" ...
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Reference request concerning splitting fields for groups that are related to special symmetric groups
Denote the symmetric group of order $n!$ by $S_n$. Let $H:=S_p$ for an odd prime $p$.
Every finite field $k$ is a splitting field $(^*)$ for $kH$, in particular $k:=\mathbb{F}_p$.
Questions:
Is $k:=\...
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Embedding of wreath product
Consider the wreath product $G=\mathbb{Z}_2\wr O_n(\mathbb{R}),$ where $O_n(\mathbb{R})$ is the set of orthogonal groups over reals. Can we show $G$ embeds in a nice enough group (for example, some ...
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surjective algebra homomorphism $k\widetilde{G} \to k_\alpha G$
I am having trouble understanding the following statement:
Let $Z$ be an abelian group written multiplicatively and let $1 \to Z \to \widetilde{G} \xrightarrow{\pi} G \to 1$ be a central extension of ...
- 41
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Structure/description of a finitely presented group
I am unable to see the structure of the following finitely presented group.
$$\langle a,b,c,d : [a,b]=c=a^p,\ [c,b]=c^p=d^p,\ b^{p^2}=c^{p^2}=1 \rangle$$
I have tried in GAP, but it is not showing any ...
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On isoclinism classes of finite p-groups
With reference to
James, Rodney, The groups of order (p^6) ((p) an odd prime)., Math. Comput. 34, 613-637 (1980). ZBL0428.20013., My question is can we get isoclinism class $\phi_2$ for a finite p-...
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Determining the irreducible invariant subspaces of a permutation action by computing eigenspaces of a matrix
Let $\Sigma\subseteq\mathrm{Sym}(n)$ be a permutation group on $N:=\{1,...,n\}$.
My goal is to determine the irreducible invariant subspaces of the permutation action of $\Sigma$ on $\Bbb R^n$, and I ...
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Wonderful compactification of $\mathrm{SL}(2)/\mathrm{SO}(2)$
Let $\mathbb{P}^2 = \mathbb{P}(\operatorname{Sym}^2\mathbb{C}^2)$ be the projective space of $2\times 2$ symmetric matrices over $\mathbb{C}$ modulo scalar.
Define an $\mathrm{SL}(2)$-action on $\...
user125056
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How dense can a transitive sets of points be?
How dense can a finite set of points on the $d$-dimensional unit sphere be if I require that the symmetry group of that arrangement is still transitive on the points?
As a measure for density I use ...
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Restricted wreath product as fundamental group of a space with coinciding Reidemeister and Nielsen numbers
I am studying a group $\mathbb{Z}_n \wr \mathbb{Z}^k$, where $\wr$ denotes the restricted wreath product:
$$
\mathbb{Z}_n \wr \mathbb{Z}^k = \bigoplus_{x\in\mathbb{Z}^k}(\mathbb{Z_n})_x\rtimes\mathbb{...
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Question on models for $EG$ for a $G$-CW complex
I am having trouble finding information on a definition in P. Hanham's PhD thesis paper. recall that given a discrete group $G$ a $G$-CW-complex $X$ is a CW-complex equipped with a topological $G$ ...
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Orthogonal projections of permutation matrices are symmetry group of regular simplex?
Each permutaion $\pi\in S_n$ corresponds to a permutation matrix $P_\pi$ with $(0,1)$ entries. To be more specific, $P_\pi = [e_{\pi(1)},\ldots,e_{\pi(n)}]$, where $e_1,\ldots,e_n$ are standard basis ...
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Reference request for finite simple exceptional group of lie type $E_7(q)$ and its Schur covering group $2.E_7(q)$?
Does anyone have the paper named 'Génerateurs, relations et revêtements de groupes algébriques' written by Robert Steinberg in 1962, or any other reference for simple groups of Lie type $E_7(q)$ and ...
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Abelian subgroup of maximal order
Let $\mathbf{F}_q$ be a finite field of order $q$ ($q$, an odd prime, or a power of the same). I know that for the matrix algebra $M(n,\mathbf{F}_q)$ (or $gl(n,q)$), the maximal dimension for a ...
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Orbit calculation for normaliser when orbits under centraliser action is known
I was solving a problem and in the middle, I came across this. I will be really grateful for any help here. In the following, we fix the integer $s\ge 2$.
Let $U\le \operatorname{GL}(2^s,\mathbb Z)$...
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Relationship between non-zero values of characters and normality in finite groups
Note: Let $G$ be a finite group and $H \leq G$. Then it is clear that if $H \unlhd G$, then $\chi(h) \neq 0$ for irreducible constituents $\chi$ of the permutation character $(1_H)^G$ and $h\in H$. ...
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Which quotients of surface groups are linear?
Let $S$ be a compact connected Riemann surface, and let $\pi = \pi_1(S)$ be its fundamental group. Let $\pi \to G$ be a surjective homomorphism.
Is $G$ linear? (That is, does $G$ admit a ...
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Quotient of Euclidean space with maximal volume growth
Let $\Gamma$ be a discrete subgroup of the isometry group of $\mathbb R^n$ and $O=\mathbb R^n/\Gamma$ is the orbifold.
If there exists a point $p \in O$ such that
$$
\lim_{r \to \infty}\frac{\text{...
- 2,535
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Embedding into a restricted direct product of finite groups
Residually finite groups are those groups embeddable in a direct product of a family of finite groups. What happens if we consider only restricted direct product?
- 2,233
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Algebraic structures on graphs
There are many algebraic structures linked to graphs.
For example one can find zero divisor graphs $[1]$, $[2]$ and many other graphs.
Does there exist any survey paper which characterizes all the ...
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How to relate toric data and group character?
If I collect the Cartan generators of su(3) or sl(3,C), then the diagonal entries can be arranged as
$$
\begin{pmatrix}
1 &1& -2 \\\
0& 1 &-1
\end{pmatrix}
$$
after a row operation ...
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Sidon sets in finite groups
Suppose $G$ is a group, $S \subset G$. Let’s call $S$ a Sidon subset iff $\forall$ quadruples $(a, b, c, d)$ of distinct elements of $S$ we have $ab \neq cd$ (named after Simon Sidon who studied such ...
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Nilpotency of topological groups
A group $G$ is said to be nilpotent if $G$ has a central series of finite length, that is, a series of normal subgroups
$$
\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G
$$
...
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When is the following preorder on the set of central elements of order 2 a total preorder?
Let $G$ be a finite 2-group. Denote by $S$ the set of central elements of $G$ of order exactly $2$. The relation $a\leq b$ iff there is an endomorphism of $G$ sending $b$ to $a$ defines a preorder on $...
user145520
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How a profinite group can be obtained from its normal open subgroups?
Let $\Delta$ be a set, each element of which is a profinite group (2 distinct elements of $\Delta$ may be isomorphic). Under what conditions on $\Delta$, there exists a profinite group $G$ which has $\...
- 2,118
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Nomenclature: does this coset space have a name?
in my work I tripped on a specific coset space and before starting thinking about it by myself, I wanted to check the literature. However, I do not know if the object has a name (which makes ...
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Descending FC series
In analogy to the central series one can define a FC series as a sequence $A_i$ of normal subgroups such that
$$
\{1\} = A_0 \lhd A_1 \lhd A_2 \lhd \cdots \lhd A_n = G
$$
such that $A_{i+1}/ A_i$ is ...
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Matrix factorizations over $GL_2$ of a real quadratic ring of integers
tl;dr: The groups $GL_2(K)$, or $SL_2(K)$, where $K = \mathbb{C,R}$ admits several factorizations (the polar decomposition,
the KAN decomposition, the Schur triangular form, etc). Those
...
- 1,090
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Generating graphs of groups
Suppose $G$ is a group and $S \subset G$ is its finite subset. Let’s define the generating graph of $G$ in respect to $S$ as $Gen(G, S)$ - a graph $\Gamma(V, E)$, where $V = S$ and $E = \{(a, b) \in S ...
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Finding the minimum dimension of $\operatorname{SL}_n(\mathbb{F}_q)$'s nontrivial real representations
Denote by $\mathbb{F}_q$ the finite field with $q$ elements, and $\operatorname{SL}_n(\mathbb{F}_q)$ the special linear group in $n$ variables.
What is the minimum dimension of nontrivial real ...
user148117
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$m$-fold character product
Here is an interesting and long exercise from the book "Character Theory of finite groups : I.M. Isaacs" (usually handled by wedge products)
Let $V$ be a ${\mathbb{C}}[G]$-module with basis $\{ v_1, \...
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For a finite subgroup of an infinite group $G$, does its normal closure have infinite index in $G$?
For a finite subgroup of an infinite group $G$, does its normal closure have infinite index in $G$?
Or if not, is it true when we replace $G$ by some subgroup? That is:
Let $H$ be a finite ...
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Do all maximal verbal series have the same length?
Suppose, $G$ is a finite group. Let’s call a series of subgroups of $G$ $\{H_k\}_{k=1}^n$ a verbal series of length $n$, iff $H_1 = E$, $H_n = G$ and $\forall k < n$ $H_k$ is a verbal subgroup of $...
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On normalized 2-cocycle
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
...
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Change variable in integration with symmetry
Not sure if I can ask such fundamental problem here.
Let
$G$ be a finite group, $\sigma \in G$. Consider linear group actions fo $G$ on $\mathbb{R}^n$.
$\sigma(K)=\{x\in \mathbb{R}^n: \sigma^{-1}(...
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Number of conjugacy classes of unit groups of modular group algebras
Let $n$ be a natural number, $p$ a prime number, $G$ a finite $p$-group and $K$ a finite field with $p^n$ elements. We focus on the group $1+J(KG)$, where $J(KG)$ is the Jacobson radical of $KG$, ...
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Partitions corresponding to unipotent elements in simple classical algebraic groups
Let $G$ be a simple classical algebraic group with corresponding root system $\Phi$ and natural module $W$ of dimension $n$.
For a (closed) subsystem $\Psi$ of $\Phi$, let $G_\Psi = \{ U_\alpha \mid ...
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Composition factors in subgroups of R. Thomspon's group $F$
Recall Thompson's group $F$, acting by piecewise-affine homeomorphisms on the interval. I wish to make the following conjectures on all subgroups of $F$, which originate from this question on MO.
...
- 2,519
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Perfect groups of infinite order
Let $M$ be a closed, minimal, hypersurface in the sphere $S^{n-1}$, $n\geq 4$. Suppose $M$ has $H^1(M,\mathbb{Z})=0$. What can we say about the cardinality of the first fundamental group of $M$, $\...
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Characterizing the big Bruhat cell of the universal Chevalley groups over $\mathbb C$
Is there a simple characterization of the big Bruhat cell of the universal (simply-connected) Chevalley groups over $\mathbb C$?
For example, it is known that the Borel subgroup of $\mathrm{SL}_n(\...
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A generalization of $p$-groups [closed]
I was wondering if there is a reference studying groups with order $m^k$ where $m,k$ are integers and $m$ is not supposed to be a prime, as a generalization of $p$-groups?
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Does $\Sigma$ generate the variety of all groups?
Suppose $FT$ is the class of all isomorphism classes of finite T-groups (a T-group is a group, where all subnormal subgroups are normal). Define, $\Sigma$ as the set of all functions $f$ from $FT$ to $...
- 2,618
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Lyndon words and free groups [closed]
It is well known that Lyndon words form a basis for free Lie algebras. Is there any analog result for free groups? What is the connection between Lyndon words and free groups? Since groups and Lie ...
- 1,269
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Does $AA^{-1}$ have the unique product property?
Let $A$ be a finite subset of a torsion free group $G$ with $|A|\geq3$. Does the set $AA^{-1}$ have the "unique product" property (i.e. there exist an element $c\in AA^{-1}$ that is uniquely written ...
- 2,118
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Lower and upper bounds of the distance between two Frobenius numbers
I consider two sequences of numbers: $A=\{a_1,...,a_{m-1},n\}$ and $B=\{n-a_{m-1},...,n-a_1,n\}$, where $a_1 < a_2 < ... < a_{m-1} < n$ and $\gcd(A) = \gcd(B) = 1$.
I investigate the ...
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Do small subsets of $S_n$ subgroups cover almost all permutation configurations 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 $G_1,\dots,G_m$.
For ...
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