# Questions tagged [gr.group-theory]

Questions about the branch of algebra that deals with groups.

6,899
questions

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### 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**

votes

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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**

vote

**0**answers

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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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

**4**answers

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**

votes

**5**answers

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**

votes

**0**answers

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

**6**answers

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

**1**answer

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

**2**answers

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 ...

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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

**1**answer

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{...

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**0**answers

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

**1**answer

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

**1**answer

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**

votes

**2**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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 ...

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votes

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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^...

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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

**2**answers

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 $\...

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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 \...

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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

**1**answer

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?

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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

**1**answer

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 ...

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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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

433 views

### Multiplicative groups of skew fields

Is every group isomorphic to a subgroup of the multiplicative group of some skew field?

**2**

votes

**1**answer

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 ...

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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 $\mathrm{Sp}_{56}(\mathbb R)$ ...

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votes

**5**answers

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

**1**answer

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 $\...

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vote

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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 ...

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vote

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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 ...