Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
8,182 questions
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Prime index subgroups of $\langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle$ that is invariant under matrix $Q$
Let $Q $ be a matrix in $ \operatorname{GL}(2, \mathbb{Q}) $ and consider the group
$G = \langle Q^{i}(\mathbb Z^{2}) \mid i \in \mathbb Z \rangle := \langle Q^{i}(v) \mid i \in \mathbb Z, v \in \...
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Hopf algebra of representative k-valued functions of an abstract group
Let $G$ be an abstract group. If we can embed $G$ into a group $H$, in a way that we had $G$ and $H$ of the same Hopf algebra of representative $k$-valued functions ($R(G)\sim R(H)$ as Hopf algebras). ...
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Intersection of identity components
Let $e_{1}$ and $e_{2}$ be involutions in the algebraic group $G=\operatorname{PGL}_{n}(\mathbb{C})$. Do we have $$C_{G}(\langle e_{1},e_{2}\rangle)^{\circ} = C_{G}(e_{1})^{\circ}\cap C_{G}(e_{2})^{\...
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A Newton identity and the primes--the Faber partition polynomials and modular arithmetic
[Edit, July 6, 2022: Removed erroneous characterization of Faber polynomials as an Appell sequence.]
Dress and Siebeneicher in their tale of the Burnside family express an opinion (1.2) that, if I ...
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Computationally intractable orbit of a monoid action on a finite set
Suppose for each integer $n\geq 1$ we have a submonoid $M_n\subset \mathrm{Self}(\{1, \dots, n\})$ of self-maps of $\{1, \dots, n\}$.
A characterization of $M_n$ is an algorithm that takes an integer $...
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Examples of infinitely presented non-LEF groups
A group is LEF (locally embeddable in the class of finite groups) if it embeds into an ultraproduct of finite groups. Residually finite groups are LEF and finitely presented LEF groups are residually ...
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Group algebras and group automorphisms
Say, we have a countable ICC group $G$, a Hilbert space $H$ with a basis indexed by the group elements, the group algebra generated by the left regular representation of $G$ on this Hilbert space, and ...
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A certain class of representations
Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$?
(The word "finite-dimensional" was ...
- 2,118
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Does every character from group factor through largest central subgroup?
Let $G$ be a coonected reductive algebraic group over $\mathbb{Q}$ and $A_G$ its largest $\mathbb{Q}$-split central torus over $\mathbb{Q}$.
Let $X(G)_{\mathbb{Q}}$ be the addtive group of ...
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Complements in $\text{Sub}(\text{Sym}(\omega))$
For any group $G$, we let $\text{Sub}(G)$ be the complete lattice of subgroups of $G$. Let $\text{Sym}(\omega)$ be the group of all bijections $f:\omega\to\omega$.
What is an element of $U\in\text{...
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Harmonicity of the Martin kernels
Let $\Gamma$ be a finitely generated group and let $\mu$ be a probability measure on $\Gamma$. Consider the Green function $G(x,y)=\sum_{n\geq 0}\mu^{*n}(x^{-1}y)$, where $\mu^{*n}$ is the $n$th ...
- 2,090
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Finite group cohomology with roots of unity as coefficients
Let $G$ be a finite group of order $n$, and let $L := (\mathbb{Q}/\mathbb{Z})^d$ be a (not necessarily trivial) $G$-module (we assume that $d$ is finite).
By a direct limit argument, there must be a ...
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locally compact abelian normed group
Why is locally compact abelian normed group always complete? It seems to be something very simple, but I'm unable neither to prove it, nor find any references in the literature.
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about maximal subgroup of p-groups [closed]
Thanks for any help or comments.
Suppose $G$ is a meta cyclic p-group, i.e. $G$ is an extension of cyclic by cyclic group, Is it true that every nonabelian maximal subgroup of $G$ is meta cyclic?
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Question about finite G-sets [closed]
Let G be a finite group with subgroups H and K. Then the set of not necessarily equivariant maps from G/H to G/K is itself a finite G-set under the conjugation action. Is there a good description of ...
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Different computation methods to determine the conjugacy classes of a finite extension group N.G
I am looking for methods to compute the conjugacy classes of any finite extension group N.G from the classes of G.
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About $c(A)$ in $c(A)|A|\leq |A^{-1}A|$
Let $G$ be a finite group, $\emptyset\neq A\subseteq G$, $A^{-1}:=\{ a^{-1}:a\in A\}$, and put $$c(A):=\max\{t\in \mathbb{Z}: t|A|\leq |A^{-1}A|\}$$
It is clear that $1\leq c(A)\leq \frac{|G|}{|A|}$, ...
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All $2$-designs arising from the action of the affine linear group on the field of prime order
Let $p$ be a prime and $\mathbb{Z}_p$ denote as usual the field of order $p$. Let $AL(p)$ be the affine linear group $\{x\mapsto ax+b \;|\; a\in \mathbb{Z}_p\setminus \{0\}, b\in\mathbb{Z}_p\}$. For a ...
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Goldbach-type problem: the valuation of irreducible elements of a subgroup of $\mathbb{Q}_{> 0}$
Let $\phi : \mathbb{Q}_{>0} \to \mathbb{Z}$ be the group morphism defined by $\phi(p) = p$ for $p$ a prime number.
It follows that $\phi(\prod_i p_i^{n_i}) = \sum_i n_i p_i$, with $p_i$ a prime ...
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Semi direct product group
Suppose $G=V \rtimes M$, is a semi product group of an elementary abelian p-group of size $|V|=p^e$ and $M$
is a subgroup of $G$. If $f$ is the natural projection from $G$ onto $M$.
$C_x=\{x^G\}$ is ...
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Direct limit of primitive integral matrices
I'm interested in computing the direct limit of an arbitrary $2\times 2$ primitive matrix over $\mathbf{Z}$. That is for a fixed primitive matrix $M$, the colimit $\displaystyle\lim_{\stackrel{\...
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Why are all involutions conjugate in the special linear group of degree 2?
It appears to be standard that the set of non-identity involutions in $SL(2, 2^n) = PSL(2, 2^n)$ forms a single conjugacy class. What is the best reference for this?
I note that
https://math....
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Isometry group of an integer as of the corresponding $\Omega(n)$-parallelotope
Let $\prod_{i}p_{i}^{a_{i}}$ be the prime factorization of a positive integer $n$ and let's consider the $\Omega(n)$-parallelotope built with, for all $i\in I$, $a_{i}$ pairwise orthogonal vectors of ...
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Centralizer of derived subgroup
In all questions suppose $G$ metabelian p-group such that
G is not regular ( so $cl(G) \geq p$ ), G is not a wreath product;
$Z(G) \leq \phi(G)$.
1) Let $M$ normal abelian subgroup of $G$ such that ...
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Negated varieties and their relatively free algebras
During the past days, I asked some questions in order to gain a clear understanding of the notion of "free algebras". I suppose that the question below is the most clear image of the concept I have ...
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For any n and some prime p there is an elemnet in Zp* of order n [closed]
How can I prove, that for any positive integer $n>0$ there is a prime $p$, such that the multiplicative group of the residue ring $Z_p^*$ contains an element $a$ of order $n$? No ideas at all...
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Embedding a semigroup into a divisible semigroup
The following is motivated by the fact that I'd like to have a way, much better if canonical, to isometrically embed a normed group into a normed divisible group. But semigroups are a much more ...
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Dual concept for the p-primary component
Is there a dual concept for the p-primary component of an abelian group? Please name some books/papers where it is studied.
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subgroups of a $p$-solvable group and complete reducibility
1.
Let $G$ be a $p$-solvable group and $V$ be a finite dimensional
faithful $kG$-module, where the characteristic of $k$ is $p$. But
$V$ is not a semisimple $kG$-module. For every $n\geq 0$, we ...
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Can the order of a rational number in Z/pZ be as large as we want
Suppose $d_1, d_2$ are two fixed coprime integers, $\frac{d_1}{d_2} \neq \pm 1$. Given any $n > 0$, can we find a prime number $p$ such that the order of $d_1d^{-1}_2$ in the multiplicative group ...
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761
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Action of Isometries on a Line in the Plane
I'm trying to determine the stabilizer of a line in a plane when acted upon by the group of isometries of the plane. Please note that I'm using the notation found in the Wikipedia article on Euclidean ...
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Question about the proof of the fact that IR is not quasi-isomtric to IR^2 [closed]
Hello.
Yesterday we proofed, that $\mathbb{R}$ is not quasi-isometric to $\mathbb{R}^2$ (both endowed with the standard Euclidean metric).
Step 1.: $\mathbb{R}$ is q.i. $\mathbb{Z}$ and $\mathbb{R}^...
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Relative invariants of prehomogeneous vector space
Let $(G,\rho,V)$ be a prehomogeneous vector spaces with $f_1,\dots,f_N$ the basic irreducible relative invariants. Suppose that $(G',\rho',V')$ is a second prehomogeneous that is in the same castling ...
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Is the direct product of two primitive unitary groups necessarily a primitive unitary group?
Let $G$ and $H$ be two primitive unitary groups, is $G\times H$ necessarily a primitive unitary group? If not, is there any counterexample?
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Find elements $\rho_i$ such that $H < B : [\langle H, \rho_i \rangle : H ] = 2$
Let $\Gamma (G;(G_i)_{i \in I})$ be a coset geometry (in the sense of Buekenhout) firm, residually connected and flag transitive with Borel subgroup $B$. Consider $H$ any subgroup of $B$. I want to ...
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Sequences, semigroups, addition formulae.
I am interested in the efficient computability of sequences.
Is it possible some ``interesting sequences'' be computed via addition formulae/semigroup operation?
Here is an artificial example.
...
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Calculating norms over a finite field (orthogonal groups).
I'm trying to work through calculating the order of orthogonal groups in characteristic $\neq 2$. However there is one proof by induction used that i can't quite follow. Could someone help me ...
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107
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Subgroup Groups and Coordinate Algebra Subalgebras
Let $G$ be a (complex algebraic) group, $H$ a subgroup, and ${\cal O}(G)$ and ${\cal O}(H)$ the coordinate algebras of complex regular functions of $G$ and $H$ respectively. Can ${\cal O}(H)$ ever ...
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Intersection/Union of Rectangles as a Group (or Monoid or...?)
In a computer graphing library, a rectangular region of the Cartesian plane may be defined by {x, y, w, h} (where w,h are width and height).
Intersection (lets say '^') is defined as the overlapping ...
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Co-boundary crossed homomorphism & "sign" preserving. Why 2-valued components is special?
Suppose $h_{g}: \mathbb{R}^n \to \mathbb{R}^{n-1}$ be a coboundary crossed homomorphism with action $g$ as a cyclic permutation of coordinates on $\mathbb{R}^n$ vectors. So, the acting group is a ...
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What are the automorphisms of finite commutative groups？ [migrated]
What are the automorphisms of finite commutative groups？Is there a relatively complete conclusion? Although it can be decomposed into the direct product of cyclic groups, this question still seems ...
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Class multiplication coefficients of symmetric groups
My question is that I was working with some counting problems, and finally the answer should be
$$
\nu_{\mu_1,\mu_2,\mu_3}=\#\{(\sigma_1,\sigma_2,\sigma_3): \sigma_1\sigma_2\sigma_3=1, \sigma_1\in C_{\...
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Defining rank of an abelian subgroup using the second centralizer
I recently posted this on MSE, but didn't receive any feedback; so I'm posting it on MO.
I recently came across this article which explored the maximal abelian subgroups of the symmetric group $S_n$. ...
- 195
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Higher-order obstructions in thin group orbits
Let $G$ be a finitely generated group acting on the integers $\mathbb{Z}$. Let $O_a = \{g \cdot a : g \in G\}$ be the orbit of an integer $a$ under this action. Assume that $O_a$ is a thin orbit, ...
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Question on definition of closed embedding of affine group schemes
$\DeclareMathOperator\Hom{Hom}$I am reading Introduction to affine group schemes by Waterhouse. In the second chapter he defines a closed embedding in the following way. Let $G = \Hom_k(A,-)$ and $H'= ...
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Which finite group schemes over positive characteristic have projectivity detected by a set of (generalized) cyclic subgroups?
A finite algebra $B$ over a field $K$ has projectivity detected by a set $S = \{f_i : A \to B \mid i \in I\}$ of maps into $B$ from a fixed algebra $A$ if for any finite module $M$, $M$ is projective ...
- 542
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A reference for this statement (representations of universal central extensions)
Let $G$ be a connected Lie group with universal cover $\tilde{G}$. I would really appreciate if someone could give me a reference for the proof of the following fact:
"Every projective unitary ...
- 287
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138
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Model-theoretic construction of Gromov boundaries on groups
For context, I'm only a second year undergraduate mathematician, so I won't know much.
For third year, I'm hoping to do a research project. I met up with a professor who might be my supervisor today, ...
- 101
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58
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A stronger(?) notion than uniform contractibility
Let's call a metric space $ X $ strongly contractible if there exists a
function $ \rho : \mathbb{R}_+ \to \mathbb{R}_+ $ such that for every ball $
B(x;r) $ around a point $ x \in X $ we have:
$ B(x;...
- 169
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127
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When $G$ is amenable is true that for a set $A \in p$ which is left piecewise syndetic then $\{ x: Ax^{-1} \in p \}$ is both sided syndetic?
Let be $G$ be a discrete group. I recall the definition of syndetic sets, thick sets and piecewise syndetic sets.
Definitions:
A set $A$ is left syndetic if there exist a finite $H \subset G$ such ...
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