Questions tagged [gr.group-theory]
Questions about the branch of algebra that deals with groups.
1,095 questions
3
votes
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answer
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Centralizer of a generator in a braid group
Given a braid group
$$
B_n \simeq
\left\langle
x_1,\ldots,x_{n-1}
\middle|
\begin{array}{l}
x_ix_j = x_jx_i, \;\text{for } |i-j|>1 \\
x_ix_{i+1}x_i = x_{i+1}x_ix_{i+1}
\end{array}
\right\rangle
$$...
- 123
3
votes
2
answers
628
views
Frobenius-Schur indicator and character table of finite groups
Let $G$ be a finite group and $\pi$ an irreducible complex representation. The Frobenius-Schur indicator of $\pi$ is defined as:
$$ \nu_2(\pi):=\frac{1}{|G|} \sum_{g \in G} \chi_{\pi}(g^2) $$
with $\...
3
votes
1
answer
2k
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Automorphisms of quotient groups
I have a general question about automorphism groups. Sorry in advance if I'm talking about well known facts, but I didn't find much in the literature.
Let $G$ be a group and let $N$ be a ...
- 731
3
votes
2
answers
688
views
When is the semidirect product of profinite groups a profinite group?
Following the discussion I have with Yves Cornulier in the following question Finiteness theorems for profinite groups, I would like to ask the following: Suppose $K$ and $N$ are two profinite groups ...
- 5,450
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votes
1
answer
489
views
Finitely generated infinite p-group
I'd like to ask can we characterize the structure of finitely generated infinite p-group which has a unique subgroup of order p?
Can we say that these group are residually nilpotent?
Any comments are ...
- 87
3
votes
1
answer
527
views
How do Dehn functions of special linear and mapping class groups behave?
Hi,
I apologize for the basic questions. I am looking for good references on the following problems:
1) What is known about the Dehn function of $SL_n(\mathbb{Z})$?
2) What is known about the Dehn ...
- 33
3
votes
1
answer
740
views
Bases of free groups
Let $F$ be a free group on a finite set $X$. Let $A \subseteq X$ be a subset of $X$ contained in some $H \leq F$, a subgroup of finite index in $F$. Must there be a basis (free generating set) for $H$ ...
- 11.3k
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votes
0
answers
302
views
What's the ratio of inclusions of finite groups with a distributive lattice?
Definition: Two inclusions of finite groups are equivalent, $(A \subset B) \sim (C \subset D)$, if: $(A/A_B \subset B/A_B) \simeq (C/C_D \subset D/C_D)$ with $A_B$ the normal core of $A$ in $B$.
...
3
votes
1
answer
451
views
Finite group of units in quaternion orders
Let $F$ be a totally real number field, $R$ is a quaternion algebra over $F$ ramified in all infinite places of $F$. Let $\mathcal{O}\subset R$ be an order. By assumption on $R$ its group of units $\...
- 7,377
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votes
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503
views
Is the representation of finite simple groups fully understood?
Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such ...
- 203
3
votes
1
answer
220
views
When does a characteristic subgroup remain characteristic in the profinite completion?
Suppose you have a group $F$, and a characteristic subgroup $K\le F$. Under what conditions on $F$ and $K$ is the closure $\overline{K}$ inside the profinite completion $\widehat{F}$ also ...
- 7,527
3
votes
2
answers
412
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, ...
- 14.2k
3
votes
1
answer
130
views
Enlarging a subdegree-finite "almost transitive" permutation group to a transitive one?
Consider a permutation group $G$ acting on an infinite set $X$. Assume $G$ has finitely many orbits, and every point stabiliser $G_x$ has finite orbits. Now consider a permutation $\tau\in\...
- 327
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votes
2
answers
2k
views
Extension problem
As I understand, if $0\rightarrow A\rightarrow X\rightarrow B\rightarrow 0$ is a short exact sequence of abelian groups, $\mbox{Ext }_{\mathbb{Z}}^{1}(B,A)$ gives all the isomorphism classes of what ...
- 2,857
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votes
3
answers
1k
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A table for irreducible integral representation of finite cyclic groups
Is there such a table where the irreducible integral representations of finite cyclic groups
are listed?
Edited:
Thanks for Todd Leason's comment.Acutally,i want to know all inequivalent ...
- 593
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votes
1
answer
869
views
Regular semisimple elements in $SL(n,q)$
Consider $G=GL(n,q)$. A regular semisimple element of this group, is a matrix, whose characterestic polynomial is square-free and same as minimal polynomial. Results show that the number of such ...
- 117
3
votes
1
answer
298
views
Minimal polynomial of unipotents in orthogonal group
Consider split orthogonal group $O(2l)$ over a field of characteristic zero. We may assume the matrix of the bilinear form to be $\begin{pmatrix} O&I\\ I&O\end{pmatrix}$.
Let $u$ be a ...
- 661
3
votes
3
answers
714
views
Cohomology of elementary abelian $p$-groups, i.e. $H(G,{\mathbb F}_p)$ with $G\cong{\mathbb F}_p^r$
I have two questions.
$\bf 1.$ First, a reference request. Let $G\cong{\mathbb F}_p^r$ for some integer $r\geq 0$ and let $V=G^*={\rm Hom}(G,{\mathbb F}_p)$. Then $(H(G,{\mathbb F}_p),+,\cup )$ is a ...
3
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0
answers
335
views
Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions
Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups:
$$
1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1
$$
There exists a ...
- 3,652
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votes
1
answer
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Number of involutions in a finite group [closed]
Let $G$ be a finite group. Does it possible to determine number of involutions in it? If not, is there any bound for it?
- 67
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votes
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answers
2k
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Casimir operator of a given Lie algebra and relation with its matrix representation
I'm following Gilmore's recipe to compute the abstract Casimir operator of a given algebra (in this example, I refer to algebra su(2)). This recipe bring up a matrix representation of the algebra and ...
- 255
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votes
0
answers
282
views
Galois correspondence subgroups/subsystems
In this paper (1998) by M. Izumi, R. Longo, S. Popa, there is the following result (page 49) on compact groups:
Lemma 3.16. Let $G$ be a compact group and $Rep(G)$ the category of finite ...
3
votes
2
answers
949
views
Matrix groups transitive on the sphere
I would like a reference with the classification of the subgroups of SO($d$) which are transitive in the unit sphere of $\mathbb{R}^d$ when acting linearly (i.e. as matrices from SO($d$) act on ...
3
votes
1
answer
204
views
Assigning a finite number, $n(G)$, to a finite group $G$ with this property
I want to assign a finite number, $n(G)$, to a finite group $G$ such that if $H$ is a proper retract of $G$, then $n(H)\lneq n(G)$. By a retract of $G$, I mean a subgroup $H$ of $G$ for which there is ...
- 287
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votes
0
answers
179
views
uniqueness of quotients of principal congruence subgroups
For each $n \geq 2$, is $\Gamma(2^{n})$ the unique normal subgroup of $\Gamma(2)$ with quotient isomorphic to $\Gamma(2) / \Gamma(2^{n})$ (here we are talking about principal congruence subgroups of $\...
- 1,298
3
votes
1
answer
423
views
Conjugation of group extensions
Let $H$ be a finite group. We write ${{\mathbb{C}}}^{*n}$ for the $n$-dimensional complex torus $({{\mathbb{C}}}^*)^n$.
We have a short exact sequence
$$ 0\to {{\mathbb{Z}}}^n\to {{\mathbb{C}}}^n\to{{\...
- 14.2k
3
votes
2
answers
1k
views
Cohomology of SL(2,R) with coefficients given by linear action
Let $SL(2,{\mathbb R})$ act on ${\mathbb R}^2$ by matrix multiplication.
What is known about group cohomology $H^*(SL(2,{\mathbb R}),{\mathbb R}^2)$?
And about $$H^*(\Gamma,{\mathbb R}^2)$$ for a ...
- 10.4k
3
votes
4
answers
654
views
A generalization of Landau's function
For a given $n > 0$ Landau's function is defined as $$g(n) := \max\{ \operatorname{lcm}(n_1, \ldots, n_k) \mid n = n_1 + \ldots + n_k \mbox{ for some $k$}\},$$
the least common multiple of all ...
- 798
3
votes
1
answer
214
views
What is the growth of the rank of a power of a finite simple group?
Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
- 11.3k
3
votes
1
answer
379
views
How often does algebraic-conjugacy imply conjugacy?
Recall that two elements $h_1,h_2$ of a finite group $G$ are called conjugate when $h_2 = gh_1 g^{-1}$ for some $g \in G$, and algebraic-conjugate when $h_2 = gh_1^a g^{-1}$ for some $a \in (\mathbb{Z}...
- 54.6k
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0
answers
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views
A bridge between the algebraic / differential geometry of $\frak{sl}_2(\mathbb{C})$ and the Sheffer-Appell calculus and combinatorics
In "Four examples of Beilinson-Bernstein localization", Anna Romanov introduces the basis
$m_k = \frac{(-1)^k}{k!} \partial^k \delta $
on p. 9, where $\partial$ is a partial derivative and $\...
- 10.5k
3
votes
1
answer
222
views
Real automorphisms of the "quaternionic" real group ${\rm SO}^*(4m)$
Let $m\ge 2$, and let $G={\rm SO}^*(4m)$ denote the "quaternionic" real form of the special orthogonal group ${\rm SO}(4m,\mathbb C)$ of type ${\sf D}_{2m}$.
Let $\tau\in{\rm Aut}_{\Bbb R}(G)$ be a ...
- 14.2k
3
votes
1
answer
374
views
Is a finitely generated residually free group "almost LERF"?
Let $G$ be a finitely generated residually free group.
(i.e. for each $1 \neq g \in G$ there exists a homomorphism $\tau \colon G \to F$ such that $F$ is a free group, and $\tau(g) \neq 1$.)
Let $...
- 11.3k
3
votes
1
answer
396
views
Are there quasiconvex normal subgroups?
Let $G$ by a hyperbolic group, and let $H \lhd G$ be a normal quasiconvex subgroup. Is it possible that $|H| = [G : H] = \infty$ ?
- 11.3k
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votes
0
answers
210
views
How many conjugacy classes of elementary abelian subgroups of order $p^2$ does $\operatorname{GL}_{4}(\Bbb Z / p\Bbb Z)$ have?
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Aut{Aut}$Let $f\in \Hom((\Bbb Z/p\Bbb Z)^2,\GL_{4}(\Bbb Z / p\Bbb Z))$ be an injective homomorphism. What is the number of ...
- 463
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votes
1
answer
384
views
Using the Lehmer quintic to solve $11$-degree equations and higher?
(This is a natural continuation of a previous post.)
I. Quintic method
Given the Lehmer quintic,
$$x^5 + n^2x^4 - (2n^3 + 6n^2 + 10n + 10)x^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)x^2 + (n^3 + 4n^2 + 10n + ...
- 12.6k
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0
answers
217
views
References on a certain generalization of Dedekind groups
Recall that a group is called a Dedekind group if all of its subgroups
are normal. Also recall that a weaker property of a subgroup than normality
is that of being a TI-subgroup: a subgroup $H$ of a ...
- 19.6k
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votes
4
answers
1k
views
Automorphisms of the Selberg class
Hello,
assuming Selberg's orthonormality conjecture, let's consider bijective maps $f$ from Selberg's class $\mathcal{S}$ to itself such as:
1) $f$ maps a primitive function of $\mathcal{S}$ to a ...
- 195
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votes
1
answer
588
views
Infinite groups of finite exponent inside of SL(2,C)
Fix an integer $n>0$. Are there infinite subgroups of $SL_2(\mathbb{C})$ such that every element is $n$-torsion?
- 790
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votes
0
answers
421
views
Marshall Hall's theorem for surface groups [closed]
Let $\Gamma_g$ be a surface group of genus $g \geq 2$, that is we have a presentation: $$\Gamma_g = \langle x_1,y_1 \dots, x_g,y_g \vert \prod_{i = 1}^g [x_i,y_i] = 1\rangle$$
Let $H \leq \Gamma_g$ ...
- 11.3k
2
votes
1
answer
307
views
Automorphisms of locally finite countable posets
Hi,
Is the automorphism group of a countable locally finite connected poset finite or countable?
If not, is there a way to equipp it (the uncountable group) with a topology and a measure?
Need this ...
user16974
2
votes
1
answer
219
views
Existence of a special ordering of the elements of a finite group (II)
Let $G$ be a finite non-abelian group of order $n$. Given $g\in G$ we denote its order by $\mathrm{ord}(g)$.
Consider the group algebra $\mathbb{F}[G]$ for some field $\mathbb{F}$.
Given an ordering $...
- 14.6k
2
votes
2
answers
528
views
Characterizations of amenable groups which use the space $\ell_1(G)$ and convolution
Let $G$ be a discrete group.
Do you know characterizations of amenable groups which use the space $\ell_1(G)$ and convolution?
I only know Johnson's theorem:
A group is amenable if and only if the ...
2
votes
1
answer
220
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 ...
- 155
2
votes
1
answer
191
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 $\...
- 155
2
votes
1
answer
551
views
Product of an elementary abelian group and a cyclic group of coprime order
I asked the same question on stackexchange, but I didn't get an answer, so I am reposting it here in hope of one (or an appropriate reference to a textbook or otherwise). I am assuming all groups ...
user13040
2
votes
1
answer
223
views
Possible symmetry groups of power terms
Previously asked and bountied at MSE:
Let $\mathfrak{E}=(\mathbb{N};\mathit{exp})$ be the algebra in the sense of universal algebra consisting of the natural numbers with just exponentiation. To each ...
- 20.5k
2
votes
2
answers
474
views
Non-archimedean group over the reals
I have a totally ordered group $(\mathbb{R};\leq,\oplus,0,-)$ with the reals as base set satisfying monotonicity, i.e.
for all $x,y,z$ we have that if $y\leq z$ then $x\oplus y \leq x\oplus z$, and I ...
- 43
2
votes
1
answer
327
views
Groups (not necessarily finite) with a given number of maximal subgroups
It is somewhat easy to see that a group $G$ with exactly one maximal subgroup $M$ must be cyclic: any element in $G\setminus M$ generates $G$.
EDIT: @YCor pointed out in the comments that this ...
- 1,433
2
votes
1
answer
1k
views
Cancellation theorem for direct and other kinds of products between groups
Cancellation theorem in group theory (for direct product) says that if $B$ is a finite group and $A \times B \simeq A_1 \times B_1$ and $B \simeq B_1$ then $A \simeq A_1.$
Of course, if $B$ is not ...
- 1,661