Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

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3
votes
0answers
123 views

The radical of $kG$-modules

$\DeclareMathOperator\Rad{Rad}$Let $k$ be a finite field of $p$ elements. Let $G$ be an elementary abelian p-group and $V$ a $kG$-module corresponding to the representation $\alpha:G\rightarrow \...
3
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0answers
69 views

Normalizer of SU(2) x SU(2) in SU(4)

What is the normalizer of SU(2) x SU(2) in SU(4) or how would I find it? Reason for the question: with 2 qubits, if I was interested in conjugation of 2-qubit gates with generic SU(2) elements, ...
1
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0answers
59 views

Limit of radii of convergence of growth series

Consider the Coxeter group $G_n$ generated by a finite set $\{s_1, ..., s_n\}$ with respect to the relations $s_1^2=...=s_n^2=1$ and $s_is_j=s_js_i$ for $|i-j| \geq 2$ and denote the word length with ...
3
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0answers
123 views

Difficulty about Jordan decomposition, (and also an ambiguity about the quadratic forms in indecomposable Jordan components of quadratic modules)

I am trying to understand a concept through solving some exercises, but I can't solve one of them, and I need a hint and guide. I asked my questions in the boxes (See the end of this question). (I ...
20
votes
4answers
2k views

Units in the group ring over fours group after Gardam

Giles Gardam recently found (arXiv link) that Kaplansky's unit conjecture fails on a virtually abelian torsion-free group, over the field $\mathbb{F}_2$. This conjecture asserted that if $\Gamma$ is a ...
12
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5answers
2k views

Group ring and left zero divisor

Let $K$ be a finite field and $G$ be a discrete group. Is it true that for every $a,b\in K[G]$ the condition $ab=0$ implies $ba=0$? It does not seem to be related to zero divisor problem, any ...
0
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0answers
87 views

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, ...
30
votes
6answers
4k views

Is SO(4) a subgroup of SU(3)?

$\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$I want to write a $3 \times 3$ complex-matrix representation of $\SO(4)$, for example, we know that $\SO(5)$ is a subgroup of $\SU(4)$, so we ...
3
votes
1answer
453 views

Is an abelian group of bounded exponent $\aleph_0$-categorical

For an abelian torsion group of finite exponent, i.e. there is an integer $n$ such that $g^n=1$ for all $g\in G$, its theory appears to be $\aleph_0$-categorical by the theorem of Engeler, Ryll-...
3
votes
2answers
203 views

Automorphism groups of simple groups of Lie type

$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PGL{PGL}$In “Automorphisms of finite linear groups”, Steinberg proves that any automorphism of a simple group of Lie type (normal or twisted) is a ...
2
votes
0answers
79 views

Moment of the hitting measure of a subgroup

Given a [finitely generated] group $G$ and a finite generating set $S$, a measure $\mu$ will have finite $\alpha$-moment if $\sum_{g \in G} \mu(g) |g|_S^\alpha$ is finite (where $|g|_S$ is the word ...
6
votes
1answer
95 views

Stabilizers of multilinear forms

Let $\{e_1,\ldots, e_n\}$ be the standard basis of $\mathbb{C}^n$. Consider the $m$-multilinear form $$v=\sum_{i=1}^n e_i^{\otimes m}\in (\mathbb{C}^n)^{\otimes m}$$ and consider the action of $\text{...
3
votes
0answers
103 views

Equivalence of category of internal groups and the category of groups

Is the category of internal groups in set equivalent to the category of groups or isomorphic to it or are they just equal? When we define an internal group in Set since the product is unique only up ...
10
votes
1answer
160 views

Reversals of autonomous subsets in right-angled Coxeter groups

This question has to do with some experimental phenomenon in groups generated by involutions that I cannot explain. Let $G$ be a finite, undirected graph, and let $W$ be the corresponding right-angled ...
2
votes
1answer
117 views

Subgroups of $\mathrm{SO}(A_0, \mathbb{F}_p)$

Let $n \geq 3$. Let $A_0$ denote the $n \times n$ symmetric matrix with $1$'s on the antidiagonal and $0$'s everywhere else. We can define the associated special orthogonal group $$ \mathrm{SO}(A_0, \...
24
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2answers
3k views

In what sense is the classification of all finite groups "impossible"?

I think there is a general belief that the classification of all finite groups is "impossible". I would like to know if this claim can be made more precise in any way. For instance, if there is a ...
5
votes
0answers
102 views

Permutations of a group that are eventually left translations

$\DeclareMathOperator\FSym{FSym}\DeclareMathOperator\Sym{Sym}$Notation: for $X$ a set, $\Sym(X)$ the group of permutations of $X$, and let $\FSym(X)$ be the subgroup of finitely supported permutations ...
14
votes
1answer
310 views

Identity involving zonal polynomials and $\operatorname O(N)$ irrep dimensions

$\DeclareMathOperator\U{U}\DeclareMathOperator\O{O}$Schur functions $s_\lambda(x)$ with $\lambda\vdash n$ are simultaneously the irreducible characters of the unitary group $\U(N)$ and proportional to ...
4
votes
1answer
183 views

Finite maximal closed subgroups of Lie groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\SO{SO}$ Let $G$ be a Lie group. I am interested in maximal closed subgroups $ G $ which happen to be finite. The ...
0
votes
1answer
108 views

Perfect Cayley graphs for abelian groups have $\frac{n}{\omega}$ disjoint maximal cliques

Let $G$ be a perfect/ weakly perfect Cayley graph on an abelian group with respect to a symmetric generating set. In addition let the clique number be $\omega$ which divides the order of graph $n$. ...
5
votes
0answers
182 views

Group cohomology of $\mathbb{Z}$ vs $\mathbb{Z}_p$

Let $M$ be a continuous representation of $\mathbb{Z}_p$ over $\mathbb{F}_p$, likely infinite-dimensional. There is the inflation map of group cohomology $H^*_{\text{cts}}(\mathbb{Z}_p, M) \rightarrow ...
2
votes
1answer
306 views

Computing conjugacy between two elements of $\mathrm{SL}_2(\mathbb{Z})$

The conjugacy classes of $\mathrm{SL}_2(\mathbb{Z})$ are well characterized (see, e.g., this question). Assuming two matrices $A, B \in \mathrm{SL}_2(\mathbb{Z})$ are conjugate, is there a way to ...
7
votes
1answer
252 views

Existence of abelian group extension relative to group homomorphism

Let $f: A \to B\ $ be an abelian group homomorphism. Are there abelian groups $G,\ H,\ K$ such that $K \subseteq H \subseteq G$ and the map $$\pi \circ i: H \to G/K$$ which is the composition of ...
0
votes
0answers
147 views

Subgroups generated by two random elements

Suppose that we have a finite group $G$ and choose elements $a, b \in G$ at random. What can be said about the order of the subgroup generated by $a$ and $b$? Mainly, what is the expected order, $\...
3
votes
1answer
124 views

Injectivity of certain homomorphisms on free groups

Consider free groups $F(A)$ and $F(B)$ on finite generating sets $A,B$. Write $A$ and $B$ as the disjoint unions $A=A_1\sqcup A_2$ and $B=B_1\sqcup B_2$. We consider the free groups $F(A_i)$ and $F(...
8
votes
1answer
199 views

When are biautomatic groups hyperbolic?

This list of open problems from http://grouptheory.info/ includes the question: "Is every biautomatic group which does not contain any $\mathbb{Z} \times \mathbb{Z}$ subgroups, hyperbolic?" ...
11
votes
1answer
384 views

Asking whether there is a compact Lie group containing affine symplectic group

The affine symplectic group is interesting and important in physics. However, the Lie group is noncompact. In order to have some good properties (Basically, we need some good behavior of Haar measure) ...
1
vote
1answer
230 views

Twisted forms of $\mathrm{SL}(2,q)$

$\DeclareMathOperator\SL{SL}$Let $q = p^r$ be a prime power. Let $H$ denote the subgroup of $\SL(2,\overline{\mathbb{F}}_q)$ consisting of matrices of the form $\begin{pmatrix}a & b\\ b^q & a^...
1
vote
0answers
32 views

Congruence closure of principal congruence subgroup of the symplectic group over the integers

This question is a continuation of the question that I asked here: The principal congruence subgroup of the symplectic group over the integers Denote by $\Delta$ the group generated by $T=\{A\in \text{...
3
votes
2answers
225 views

Outer automorphism of a finite simple group which is isomorphic to a subgroup of $S_p$

Here is a statement having a proof that involved the CFSG. Let $p$ be a prime, and $S$ be a nonabelian finite simple group such that $S$ is isomorphic to a subgroup of $S_p$ with $p\mid |S|$. Then $\...
1
vote
0answers
90 views

Quotienting a virtually cyclic group by an element

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 \...
0
votes
0answers
87 views

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
vote
1answer
133 views

Elements of prime power order in finite groups [closed]

Let $G$ be a finite group and $N\triangleleft G$,assume that $xN$ is an element of prime power order in $G/N$. Then in the coset $xN$, does there exist an element of prime power order?
4
votes
0answers
259 views

Finite subgroup of $\mathrm{SO}(4)$ which acts freely on $\mathbb{S}^3$

Let $\Gamma$ be a finite subgroup of $\mathrm{SO}(4)$ acting freely on $\mathbb{S}^3$. It is known that all such $\Gamma$ can be classified. Is there any characterization of $\Gamma$ such that $\Gamma$...
4
votes
1answer
391 views

What is a "cusp" ("кусок") in relation to Guba's embedding theorem?

I'm confused by the definition of a "cusp" as found in V.S. Guba, Conditions for the embeddability of semigroups into groups, Math. Notes 56 (1994), Nos. 1-2, 763-769 (link). In the words of Mark ...
0
votes
0answers
97 views

How can Borel-de Siebenthal theory be generalized?

Borel-de Siebenthal theory can be thought of as an algorithm that, given a semisimple compact Lie group $G$, gives all semisimple compact Lie subgroups whose root systems have the same rank as $G$’s. ...
4
votes
1answer
170 views

Adjoint orbits of a finite group of type $G_2$ [reference request]

Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{...
3
votes
1answer
292 views

Normalizers in arithmetic groups

This is a question about the class of arithmetic groups. I am using the definition in Serre's survey: $\Gamma$ is arithmetic if it can be embedded into $G_\mathbb{Q}$ for some algebraic subgroup $G \...
3
votes
2answers
255 views

Indecomposable integral representations of a group of order 2 "by hand"

This question is a duplicate of that 2010 MO question. I am interested in classifying isomorphism classes of $n$-dimensional integral representations of the cyclic group $C_2$ of order $2$. Clearly, ...
5
votes
1answer
433 views

Multiplicative groups of skew fields

Is every group isomorphic to a subgroup of the multiplicative group of some skew field?
2
votes
1answer
187 views

Set of $\mathrm{SU}(6)$ matrices which conjugate $\mathbb{1}_3 \otimes \sigma^3$ subalgebra element into $\mathfrak{su}(2)$

$\DeclareMathOperator\SU{SU}$Consider the Lie group $\SU(6)$, its Lie algebra $\mathfrak{su}(6)$ and the $\mathfrak{su}(2)$ subalgebra spanned by $\mathbb{1}_3 \otimes \sigma^i$, where $\sigma^i$ are ...
4
votes
0answers
146 views

What is known about the cohomology of the U-duality group?

$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$Let $\Es$ denote the split form of $E_7$, which is a real Lie group. It can be characterized as the subgroup of $\mathrm{Sp}_{56}(\mathbb R)$ ...
18
votes
5answers
2k views

How small can a group with an n-dimensional irreducible complex representation be?

More precisely, what is the smallest exponent e such that, for every n, there exists a group of size at most Cn^e for some absolute constant C and with an n-dimensional irreducible complex ...
2
votes
1answer
145 views

Normalizer of SU$(2)$ in SU$(6)$

Consider the $\mathfrak{su}(2)$ subalgebra of $\mathfrak{su}(6)$ embedded as $$\mathfrak{su(2)}=\text{Span}\{\mathbb{1}_3 \times \sigma^i\}, \quad i=1,2,3$$ with $\sigma^i$ the Pauli matrices and $\...
1
vote
1answer
131 views

The principal congruence subgroup of the symplectic group over the integers

Consider the symplectic group $\text{Sp}_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C_g$ and associated root subgroups $U_\varphi$ for $\varphi\in C_g$. These subgroups are ...
7
votes
1answer
333 views

Reference for the Brauer-Nesbitt theorem

In Wikipedia's article on the Brauer-Nesbitt theorem, they state that given a group $G$ and a field $E$, two semisimple representations $\rho_1,\rho_2 : G\longrightarrow \operatorname{GL}_n(E)$ are ...
2
votes
1answer
99 views

Set of $U(6)$ elements which leave a Lie-algebra element invariant under conjugation

Consider the specific element of the corresponding Lie algebra $\mathbb{1}_3 \times \sigma^3$, where $\mathbb{1}_3$ is the unit matrix in 3 dimensions, $\sigma^3$ is the 3rd Pauli matrix and $\times$ ...
3
votes
1answer
192 views

Good algorithmic properties for quotients of braid groups

I'm trying to understand some things about quotients of braid groups, and particularly I'd like to solve the word problem for some elements of these quotients. I'm using MAGMA to try to access this, ...
3
votes
0answers
126 views

Amenability, growth and asymptotic dimension

I recently found this question on MSE, relating growth of groups with whether they are amenable, elementary amenable or not. I would like to know if there is an extra relation to finite or infinite ...
1
vote
0answers
43 views

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 ...

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