Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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The number of solutions to $ax^2+bxy+cy^2\equiv k\pmod{p^{n}}$, $(x,y)\in\{0,\dotsc,p^{n}-1\}^2$
Let $p^n$ be a prime power and, for integers $a,b,c$, let $Q(x,y)=ax^2+bxy+cy^2$ where $p\nmid b^2-4ac$.
Define
$$N(k,m):=|\{(i,j)\in\{0,\ldots,m-1\}^2: \gcd(i,j,m)=1, Q(i,j)\equiv k\pmod{m}\}|.$$
I ...
- 221
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An infinite profinite group such that any $\overline{\mathbb{F}_{p}((t))}$-adic representation has finite image
This question is a sequel to An infinite profinite group such that any $p$-adic representation has finite image
.
Fix a prime $ p $. We call an infinite profinite group $G$ a Boston group (with ...
- 863
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Fixed points of coset spaces
Let $G$ be a connected algebraic group. Let $\gamma$ be an automorphism of $G$, preserving a connected algebraic subgroup $H \subset G$. We have a map Writing $G^{\gamma} \subset G$ for the fixed ...
- 689
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A system of linear equations with way too many unknowns — constructing a bivariate distribution from marginals and "the diagonal"
Suppose we are given information about distributions of random permutations $\sigma, \tau : \Omega \to S_n$ as follows:
$$p^1_{k,l} = \mathbb P(\sigma(k) = l), p^2_{k',l'} = \mathbb P(\tau(k) = l), p^{...
- 677
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Are generalized symmetric groups maximal finite groups (in a certain sense)? - Part II, Loose Ends
Let $S(m,n)$ be the generalized symmetric group which is a wreath product of the cyclic group of order $m$, denoted here by $\mathbb{Z}_m$, and the symmetric group $S_n$. A standard unitary ...
- 155
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Hamilton decomposition for infinite Cayley graphs
We have a finitely generated group $G$ which is infinite. Does there exist such a finite generating set $S$, $G= \langle S\rangle$, that the corresponding Cayley graph $\mathrm{Cayley}(G,S)$ can be ...
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Are there standard short notations for ascending and descending cyclic permutations?
In a paper I am currently writing I use cyclic permutations of the form
$$
(k,k+1,\dots,\ell)
$$
and
$$
(\ell,\ell-1,\dots,k)
$$
of consecutive elements quite a lot (I added the commas to avoid ...
- 7,127
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New (?) math object. Looking for (if existing) literature [closed]
I am interested in any literature about the following mathematical property.
Let $V$ a vector space and $G$ a group acting on $V$.
What is the name for the property of a set of operators $H=\{h:V\to V\...
- 329
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Characters of upper triangular matrices over finite field - reference request
Let $B_n$ be the group of upper matrices and $U_n$ the subgroup of unipotent upper triangular matrices. I would like some references which discusses complex character theory of $B_n(\mathbb{F}_q)$ for ...
- 2,751
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Is the class of DTI-groups closed under taking quotients?
Recall that a subgroup $H$ of a group $G$ is called a TI-subgroup
if for every $g \in G$ one has $H \cap H^g \in \{1,H\}$.
We say that a group $G$ is a DTI-group if the derived subgroups of all
of its ...
- 347
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The Boolean algebra of all almost invariant subsets of an uncountable locally finite group is contained in every Sub-Boolean that separates points
Let $G$ be a group. A subset $A\subset G$ is said to be almost right invariant if $A\mathbin\Delta A\cdot g$ is finite for all $G$. The family of all almost right invariant subsets $\mathcal{B}_G$ of $...
- 313
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Closed subgroups of totally disconnected Polish amenable groups
Let $G$ be a totally disconnected Polish topological group (e.g., a closed subgroup of the homeomorphism group of the Cantor set). If $G$ is amenable, is every closed subgroup of $G$ also amenable?
...
- 1,322
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Closure of an amenable subgroup
Let $G$ be a topological group, and let $H < G$ be a countable subgroup that is amenable as a discrete group. Is the closure of $H$ an amenable topological group?
- 1,322
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3d slide puzzle group
The 15 puzzle in 2d is associated with the Alternating group A15.
https://en.wikipedia.org/wiki/15_puzzle
Rubik's has produced the Rubik's slide: a 3d slide puzzle. https://www.rubiks.com/en-uk/rubiks-...
- 315
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Number of generators of a group
Given a (finite) group $G$. Is there any bounds on the minimum number of generators $d(G)$?
For example, it is clear that $d(G) \geqslant d(G^{ab})$. Where the right hand side can be easily computed. ...
user131113
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Finite groups of meromorphic functions [closed]
Which finite groups are isomorphic to groups of meromorphic functions on the whole complex plane under composition?
- 2,773
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167
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Minimal degrees of finite simple groups
The minimal projective degrees (minimal degree of an irreducible representation of a central extension) of the finite classical groups are (famously) given by Tiep and Zalesskii [1]. Is there a ...
- 9,720
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142
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Relation between C-groups and reflection groups
Take a set of reflections $\{r_1,\ldots,r_k\}$ of $\mathbb R^n$. Sometimes, the group presentation will turn out to be a C-group – this is where the regular planar polytopes in Euclidean space, ...
- 447
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Obstruction to lifting homomorphism of groups
Is there a "cohomology" group that encodes obstructions to constructing a lift in a diagram of groups below? If $X\to Y$ is an extension and the bottom row is the identity map it's just $H^1(...
- 1,374
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104
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Quotienting a virtually cyclic group by an element [closed]
Let $K$ be a group, $G \unlhd K$ be a finite normal subgroup of even order, and let $\langle h \rangle<K$ be an infinite cyclic subgroup, so that they fit into a short exact sequence $$0\to G\to K \...
- 1,177
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Variation of the geometry of a Dirichlet region as the defining point varies
Let $\Gamma$ a Fuchsian group acting on the hyperbolic plane $\mathfrak{H}$. For me, I am most interested in the case where $\Gamma$ has a fundamental domain that is a finite-polygon with all ...
- 5,349
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91
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Diophantine equation about the automorphism group of lattice by constraints
Fixed $\sigma_x=\left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right)$ and $K=\left(
\begin{array}{ccc}
3 & 32 & -64 \\
1 & 32 & -32 \\
-2 & -32 & 64 \\
\...
- 149
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Conway's quaternion notation +1/3[C_3×C_3]∙2^(2) represent C_3h of 4d Point group?
In John Conway and Derek Smith's On Quaternions and Octonions: their Geometry, Arithmetic, and Symmetry, they introduce a way to connect quaternions to 4D Point Group.
Suppose: $[l,r]:x\to \bar lxr\;,\...
- 11
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114
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Which group is the standard group of isometries?
For the classical sequence spaces $\ell_p$ ($p\not=2$) and $c_0$ each surjective linear isometry $U$ has the form $U(a_i)=(\varepsilon_i a_{\pi(i)})$ for a permutation $\pi$ of $\mathbb{N}$ and $\...
- 1,073
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Defining the action of a group on a set in GAP [closed]
Let $H$ be a subgroup of the automorphism group of a group $G$ and $S$ is a collection of subsets of the group $G$ with a given size. The group $G$ is defined via free group and relations in GAP. ...
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Sequences generated from commuted quaternions and general commuted linear transformations
Given a pair of non-commuting linear transformations, $A$ and $B$, define the "next
pair" in a sequence as $A*B$ and $B*A$. I am interested in finite cycles (i.e.,
the sequence eventually ...
- 1,547
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Exact sequences and normal series
Trying to understand exact sequences better, I was searching for a simple context in which they naturally arise. (The exact sequences that occur in algebraic topology occur in a very complicated ...
- 887
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175
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Cochains with multilinear differentials
Let $G$ be a group and let $M$ be a $G$-module. We denote by $(C^*(M,G),d)$ the complex of inhomogeneous cochains, i.e. $C^n(G,M)=M^{G^n}$.
We say that a cochain $a\in C^n(G,M)$ is multilinear if it ...
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Group structure extension
Let $G$ be a finite group and $X$ a finite $G$-set. Let $H$ be the set-theoretical cartesian product of $G$ and $X$.
Is there an homological theory controlling all possible group structure on $H$ (...
- 2,384
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Can we generalize the concept of "characters" in group theory via methods from statistics and probability theory?
$\DeclareMathOperator\Cov{Cov}$Motivation: If $G$ is a finite group and $\phi=X+iY: G\to \mathbb{T}$ is a character of $G$, then $\Cov(X,Y)=0$ where $X$, $Y$ are considered as two real random ...
- 356
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Group where Out(G) acts differently on conjugacy classes and irreps? [duplicate]
$\def\Conj{\mathrm{Conj}}\def\Irrep{\mathrm{Irrep}}\def\Out{\mathrm{Out}}$Let $G$ be a finite group, let $\Conj(G)$ be the set of conjugacy classes of $G$, let $\Irrep(G)$ be the set of isomorphism ...
- 156k
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Embedding a family of groups into a certain $2$-generated group (construction by Olshanskii)
While reading "Chain conditions, elementary amenable groups, and descriptive set theory" by Phillip Wesolek and Jay William I stumbled upon the following statement in the proof of ...
- 309
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Is this equivariant function constant?
Let $G$ be a linear algebraic group (think of $SL_n(\mathbb{R})$), $B$ its Borel (standard minimal parabolic) subgroup (think of upper triangular subgroup), and let $\Gamma \leq G$ be a cocompact ...
- 949
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Ideal Ford domain (for finite index subgroup)
Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices
$
g=
\begin{pmatrix}
\alpha & \overline{\beta} \\
\beta & \overline{\alpha}
\end{pmatrix}
$...
- 33
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340
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Random walk on non-abelian free group
Let $F_2$ be the free non-abelian group with generators $a, b\in F_2$.
Has the "random walk" where we start with the identity and then multiply it by $a$ or $b$ or $a^{-1}$ or $b^{-1}$ ...
- 11
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Rack cohomology as derived functor cohomology
Let $X$ be a rack and $A$ be an $X$-module. By this paper, p. 33, we can associate a cochain complex $C^\bullet(X,A)$ to the pair $(X,A)$. This complex is explicitly defined by a differential $d$. I ...
- 1,066
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Does solubility imply nilpotency?
Def. If $X\subseteq G$, we say that $X$ is $k$-large in $G$ if the intersection of any $k$ left translates of $X$ is non-empty. (See the notion of largeness which was introduced in "Largeur et ...
- 2,118
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Non-discrete subgroups acting on Euclidean spaces
I'm curious about finitely generated subgroups in the isometry group of Euclidean spaces. I know the isometry group is the semi-direct product of translation groups with orthogonal groups. Also ...
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A question concerning finite metacyclic groups
Consider a finite metacyclic group with presentation
$$G = \langle x,y \, |\, y^n=1, x^u=y^r, x^{-1}yx = y^k \rangle,$$ where $r \mid n$.
Is it true that if $G$ does not split (i.e. $G$ is a not a ...
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What does square bracket superscript star mean in basic group theory typically?
I'm reading some paper where they haven't really defined their notation very well (or I've missed something). You can see the image below.
What does the square bracket and star mean precisely? The ...
- 322
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73
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Pro-p completion of a quotient of $U/w(U)$ is virtually nilpotent for a finitely generated free group $U$
Let $w$ be a word of a free group. Assume that $H/\overline{w(H)}$ is virtually nilpotent for every finitely generated pro-$p$ group $H$. Let $U$ be a finitely generated free group and $T$ the maximal ...
- 329
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What is the automorphism group of this graph?
Of all the 1-skeleton graphs of finite regular polytopes, exactly one is more symmetric than the polytope or polytopes it is derived from: that of the small stellated 120-cell and great grand 120-cell....
- 2,773
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Irreducible representation of transposed Young diagram of $\mathfrak{S}_n$ [closed]
For a Young diagram $\lambda$, let $V_\lambda$ be an irreducible representation of $\mathfrak{S}_n$ corresponding to $\lambda$ (over $\mathbb{C}$). And denote the transpose of $\lambda$ by $\lambda^T$....
- 177
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Diameter of general linear group wrt monomials and fixed root subgroup
Let $G=GL_n(F)$ where $F$ is a field.
Let $S\subset G$ be the collection of monomial matrices in $G$ union a fixed root subgroup $U_{\alpha}$ of $G$. I.e.
$$
U_{\alpha}:=\{I_n+\lambda E_{i,j} : \...
- 121
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203
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About non-abelian finite simple groups
Let $S$ be a non-abelian finite simple group. Also let $\pi(n)$ be the set of prime divisors of a positive integer $n$. Is it true that $$\big|\pi(|\mathrm{Out}(S)|)-\pi(|S|)\big|<\big|\pi(|S|)\big|...
- 19
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Is there any research on the action of a subgroup on the whole finite group by conjugation?
I want to know whether there are any research on the orbits of the action of a subgroup by conjugation on the whole group, when the group is finite. (Especially whole symmetric group.)
I'm especially ...
- 1,013
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129
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Uniform divergence of geodesics in RAAGs
$\DeclareMathOperator\div{div}$Let $(X,d)$ be a metric space and let $\gamma$ be a geodesic in $X$. Roughly speaking, the divergence of $\gamma$ at a point $x\in \gamma$ is a function $\div:\mathbb{R}...
- 2,090
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If the union of finitely many conjugacy classes hits large enough difference sets, are ther finitely many conjugacy classes?
Not long ago I asked a question about groups where some finite union of conjugacy classes is left-syndetic. Here's a stronger variant.
Suppose $G$ is a finitely-generated group such that there exists ...
- 6,652
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276
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How many elements of each order are there in this $p$-group? [closed]
Let $G$ be a countable Abelian $p$-group which equals a direct sum of at most countably many finite cyclic groups and at most countably many copies of the Prüfer $p$-group, where these finite cyclic ...
- 4,597
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144
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Simultaneous similarity classes of pairs in $\mathrm{GL}_{n}(\Bbb Z / p\Bbb Z)$?
$\DeclareMathOperator{\GL}{\operatorname{GL}}$Let $G$ be an elementary abelian $p$-group of rank $2$. Let $\alpha, \beta :G\rightarrow \GL_{n}(\Bbb Z / p\Bbb Z)$ be two injective homomorphisms. The ...
- 181