All Questions
Tagged with reference-request gr.group-theory
700 questions
13
votes
1
answer
1k
views
Is there a name of semidirect product of a group with its automorphism group?
Consider the construction $G \rtimes \text{Aut}(G)$. Here $
G$ is a group, $\text{Aut}(G)$ is the automorphism group and the semidirect product is over the most obvious action.
1) Is there any name ...
- 885
0
votes
0
answers
181
views
Request for a modern Reference for Frobenius' paper "Über die Charaktere der mehrfach transitiven Gruppen"
I'm interested in the paper of Jan Saxl "The Complex Characters of the Symmetric Groups that Remain Irreducible in Subgroups".
I have only (not yet enough!) standard background on the ...
- 1,013
8
votes
4
answers
1k
views
Degree of commutativity of finite groups and subgroups
Recently I started reading some articles about the
degree of commutativity of finite groups. I have some questions:
In "Subgroup commutativity degrees of finite groups" Tarnauceanu
proposes ...
user11178
3
votes
1
answer
226
views
Can MAGMA compute almost projective $kG$-homomorphisms?
Let $G$ be a finite group and $k$ be a finite field (big enough) whith char$(k)=p$ and $p\mid |G|$.
Let $M$ be a finitely generated $kG$-module.
We denote the first syzygy of $M$ by $\Omega(M)$, i.e....
- 1,755
7
votes
2
answers
415
views
Graph which do not satisfy a pseudo-Poincaré inequality
Say that an infinite (connected) graph (with vertices of bounded degree) satisfies a $\ell_1$-pseudo-Poincaré inequality if there is a constant $C>0$ so that for any $n \in \mathbb{N}$ for any ...
- 4,432
13
votes
1
answer
358
views
Cartography of the duals of GL, PGL, SL, etc
A short version of this question could be
What are the duals of $PGL(2,\mathbf{Q}_p)$, $PGL(2,\mathbf{R})$ and $PGL(2,\mathbf{C})$?
I should obviously add some precisions.
there are different ...
- 5,424
6
votes
2
answers
749
views
Explicit computation of the Burnside ring
I would like to see explicit computations of the Burnside ring $A(G)$ when $G$ is a small Abelian group, such as $G=\mathbb{Z}/2,\mathbb{Z}/2^n,\mathbb{Z}/p^n$ where $p$ is an odd prime and $n\...
- 3,173
11
votes
0
answers
382
views
Ascending chain condition for 1-element normal closures in a free group
Let $F$ be a free group of finite rank. Does $F$ satisfy the ascending chain condition on normal subgroups each of which is a normal closure of one element?
In other words, can there exist elements $...
- 3,181
11
votes
0
answers
486
views
Roadmap to homotopical group theory
I have been lurking here for a long time just enjoying the scenery from my beginner's viewpoint. I have a math.SE account but I think this question is appropriate here based on the nature of the ...
- 211
5
votes
1
answer
429
views
Cohomology of linear algebraic groups
Let $R$ be a commutative ring. Let $G\subset \mathrm{GL}_m$ be a linear algebraic subgroup. Has the group cohomology $H^i(G(R),R^m)$ been studied in the literature?
For example, do we know
(1) $H^...
user39380
2
votes
1
answer
181
views
Generators of sandpile groups of wheel graphs
In the paper "On the Sandpile Group of a Graph" by Cori and Rossin one can find a result related to the structure of the sandpile group of $W_n$. Is there a way to provide a set of ...
- 298
15
votes
1
answer
352
views
$p$-groups with trivial $H^3$
Let $Q_8$ be the group of quaternions of order $8$. It is a non-abelian $2$-group such that $H^3(Q_8,\mathbb{Z})=0$, where $\mathbb{Z}$ has the trivial action. For a proof, see the book "Homological ...
- 153
4
votes
0
answers
103
views
Bound of word width in compact $p$-adic analytic group
A theorem proved by A. Jaikin-Zapirain in On the verbal width of finitely generated pro-$p$ groups says that:
If $G$ is a compact $p$-adic analytic group, then every word $w$ of a free group $F$ has ...
- 329
9
votes
1
answer
485
views
A residually finite modification of the wreath product
I have been looking for ways to construct examples of finitely generated residually finite groups that are poly-(locally virtually abelian) but not virtually solvable.
If $K$ is a finite non-solvable ...
- 740
2
votes
1
answer
231
views
A reference to the fact that a topologically transitive action of a group on a compact metrizable space has a dense orbit
I need a proper reference to the following obvious fact:
An action of a group $G$ on a nonempty compact metrizable space $K$ is topologically transitive (= the orbit $GU$ of any nonempty open set $...
- 41.9k
9
votes
1
answer
460
views
Connections between linear representations and permutation representations
A finite group $\Gamma$ might be represented by a linear transformation
$$\rho : \Gamma\to\mathrm{GL}(\Bbb R^d),$$
or by permutations
$$\phi :\Gamma\to\mathrm{Sym}(n).$$
Of course, latter ones can ...
- 13.6k
2
votes
0
answers
66
views
Quasi-isometry of solvable minimax groups
[Edits in brackets]
Consider two finitely generated solvable minimax groups $G_i$ ($i = 1,2$) so that $1 \to N_i \to G_i \to Z_i \to 1$ [splits]
with $N_i$ nilpotent, $Z_i$ infinite cyclic and $G_i$ ...
- 4,432
2
votes
0
answers
176
views
Sylow 2-subgroups of finite groups in which every subgroup of order 4 is cyclic
Let $G$ be a finite group. Assume that every subgroup of order 4 in $G$ is cyclic (as happens if $G$ is a cyclic group or a generalized quaternion group). It seems to me that it should follow that a ...
- 71
9
votes
1
answer
311
views
Reference for Schur multiplier identity
Let $G$ be a finite group and $H$ a normal subgroup of $G$. I recently stumbled upon the following identity for the Schur multiplier of $G/H$:
$$\operatorname{H}_2(G/H,\mathbb{Z}) \cong \frac{\...
- 93
2
votes
2
answers
267
views
Irreducible representations of $G_4 = \langle a,b \mid a^{16}, b^{2}, baba^{-7}\rangle$ and other Semidihedral groups
I would like to know the irreducible representations of the group $G_4 = \langle a,b \mid a^{16}, b^2, baba^{-7}\rangle$ and its character table.
More than that, I would like to know the irreducible ...
9
votes
5
answers
2k
views
A catalog of faithful representations of finite groups?
I want a reference that catalogs the smallest-dimensional faithful representation of every noteworthy finite group. Specifically, I want representations on $\mathbb{R}^n$ and $\mathbb{C}^n$.
Where ...
- 7,633
6
votes
1
answer
250
views
Concrete example to illustrate the theory about blocks of groups with cyclic defect groups
I'd like to to have a concrete example to illustrate the theory about blocks of groups with cyclic defect groups.
Thus, I am looking for a finite group $G$ and a prime $p$ dividing $|G|$ satisfying ...
- 1,755
9
votes
1
answer
464
views
Branching Rule for alternating groups
Let $A_n$ be the alternating group of degree $n$. What is the branching rule for the subgroup $A_{n-1}\subset A_n$, i.e., the structure of the restriction of ordinary irreducible representations of $...
- 377
7
votes
1
answer
548
views
The probability that two elements of a finite nonabelian simple group commute
It is mentioned in here (last paragraph of the first page) that Dixon proved the following result: the probability that two elements of a finite nonabelian simple group commute is at most $\frac{1}{12}...
- 71
5
votes
1
answer
390
views
mod p (odd) cohomology of dihedral groups
I've been trying to find the cohomology for the trivial module for $\operatorname{PSL}_2(r^n)$ over $\mathbb{F}_p$ for $2 \neq p \neq r$ and have managed to reduce this to the cohomology of a maximal ...
- 115
7
votes
2
answers
704
views
Reference for nonlinearity of covers of $\operatorname{SL}(2,\mathbb R)$
It is known that no nontrivial connected cover of $\operatorname{SL}(2,\mathbb R)$ admits a faithful finite dimensional linear representation (see, for example, page 143 in Fulton-Harris and Exercise ...
- 1,782
27
votes
1
answer
1k
views
Properties to have matrices that commute in $\mathrm{GL}_n(\mathbb C)$
Let $G$ be a finite subgroup of $\mathrm{GL}_n(\mathbb C)$, $A,B \in G$ whose eigenvalues are thus in the unit circle.
Assume that the eigenvalues of $A$ are included in a circle arc of length $<\...
user49093
11
votes
5
answers
2k
views
Groups as Union of Proper Subgroups: References
There are interesting theorems about groups as union of proper subgroups. The first result in this subject is the theorem of Scorza(1926): "a groups if union of three proper subgroups if and only it ...
- 1,169
5
votes
0
answers
266
views
Character tables of finite groups and isomorphism
I'd like to ask the following question:
Let $G$ and $H$ be finite groups.
Is there a useful criterion involving the ordinary character table which assures that $G$ and $H$ are isomorphic as groups?...
- 1,755
10
votes
1
answer
1k
views
Equivalent descriptions of Coherent Groups
Attending a series of lectures, I have recently been exposed to the notion of Coherent groups, defined as following:
Def: A group $G$ is called Coherent if every finitely generated subgroup $H$ of $G$...
- 493
11
votes
0
answers
345
views
Status of questions in "Group Actions on $\mathbb{R}$-trees"?
Culler and Morgan's "Group Actions on $\mathbb{R}$-trees" lists four questions at the end of the introduction. A few have been famously resolved by work of Rips, Bestvina–Feighn and others.
I'm ...
- 1,996
3
votes
1
answer
95
views
Question concerning Brauer's second main theorem, Brauer correspondent blocks and blocks covered by nilpotent blocks
A version of Brauer's second main theorem is as follows:
Let $G$ be a finite group, $x$ be a $p$-element of $G$, $B\in\mathcal{Bl}(G)$, and $\chi\in$ Irr$(B)$.
If $d_{\chi\mu}^x\neq 0$ and $\mu$ ...
- 1,755
10
votes
2
answers
1k
views
Kazhdan's property (T) vs. residual finiteness
I have asked this question already on mathstackexchange but got no answer (see https://math.stackexchange.com/questions/1795795/kazhdans-property-t-vs-residual-finiteness) and it was suggested that I ...
- 721
30
votes
1
answer
2k
views
How strong is this conjecture? $(Z/nZ)^*$ is generated by "small" elements
Conjecture: There are constants $c,k$ such that every $(Z/nZ)^*$ is generated by its elements smaller than $k (\log n)^c$.
Where $(Z/nZ)^*$ is the multiplicative group of integers mod $n$. My main ...
- 4,529
3
votes
2
answers
343
views
Good, detailed references for "mod p lower central series"
I am looking for good, detailed references for "mod $p$ lower central series".
So far I only find papers such as (https://core.ac.uk/download/pdf/81193793.pdf, https://www.sciencedirect.com/science/...
- 1,229
10
votes
1
answer
1k
views
Maximal order of elements in SL(n,q)
The maximal order of an element of $\mathrm{GL}(n,\mathbb{F}_q)$ is $q^n-1$, where the characteristic of $\mathbb{F}_q$ is odd $p$. See here for a nice proof that uses the Cayley-Hamilton Theorem.
...
- 8,529
7
votes
3
answers
582
views
Asymptotics for the number of abelian groups of order at most $x.$
The number of abelian groups of order $n$ (call it $a(n)$ is a studied subject (see http://oeis.org/A000688), but I can't seem to find any asymptotic results. Obviously, there is no asymptotic for $a(...
- 96.4k
5
votes
1
answer
184
views
Explicit short presentation of a 2-generated universal group?
A result of Higman states that there exists a finitely-presented group $G$ in which all other finitely-presented groups embed - I'll call such a group universal. Every countable group embeds in a 2-...
- 5,349
4
votes
1
answer
735
views
Shafarevich's theorem about solvable groups as Galois groups
I am seeking references to any proofs of Shafarevich's theorem about solvable groups being Galois groups.
- 101
6
votes
2
answers
457
views
Name of a group-like structure
The late Vladimir Arnold, in
Arnold, V., Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world, Bull. Braz. Math. Soc. (N.S.) 34, No. 1, ...
- 2,901
5
votes
1
answer
316
views
Connected permutation groups and wreath product
Let $G$ and $H$ be subgroups of the symmetric groups $\mathfrak S_m$ and $\mathfrak S_n$. Assume that $n>1$ and that $H$ is a 'connected' permutation group, that is, there is no non-trivial $H$-...
- 5,822
12
votes
1
answer
403
views
Are finite presentations of arithmetic groups computable?
In this famous paper by Borel and Harish-Chandra, Arithmetic Subgroups of Algebraic Groups, it is proved that, in characterisitic zero, arithmetic groups are finitely presented. I have an extremely ...
- 1,033
3
votes
1
answer
224
views
Fixed points of the automorphisms of sporadic groups
Sporadic groups have very few outer automorphisms (in fact, $|\mathrm{Out}(G)|\leqslant2$), so it is very natural to ask what are the fixed points subgroups. For a group of Lie type (and a suitable ...
- 3,186
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 $k$}\},$$
the least common multiple of all ...
- 798
7
votes
1
answer
194
views
Word length zeta function
Let $G$ be a group with a finite symmetric set $S$ of generators.
Let $\ell_S(x)$ denote the word-length of a given $x\in G$.
For $s\in\mathbb C$ set
$$
Z(s)=\sum_{x\in G^*}\ell_S(x)^{-s},
$$
where $G^...
user130903
1
vote
0
answers
107
views
Reference request concerning splitting fields for groups that are related to special symmetric groups
Denote the symmetric group of order $n!$ by $S_n$. Let $H:=S_p$ for an odd prime $p$.
Every finite field $k$ is a splitting field $(^*)$ for $kH$, in particular $k:=\mathbb{F}_p$.
Questions:
Is $k:=\...
- 311
11
votes
0
answers
269
views
Proving a group with two generators is not free that uses the Brahamagupta-Pell equation
Hello I encountered the following while reading a set of notes on free groups. It's not a homework question.
"Does there exist a rational number $\alpha$ with $0 <|\alpha| < 2$ such that the ...
- 111
10
votes
1
answer
534
views
The Tits alternative for $\operatorname{Out}(F_n)$
Not sure if this is the right place to ask this, but the paper I am reading seems to be too specialised for mathstack (if you do not agree, pleas let me know and I will take down this question)
I am ...
- 275
19
votes
0
answers
604
views
How is this group theoretic construct called?
Let $G$ be a finite group, $S\subset G$ a generating set, $|g| = |g|_S = $ word length with respect to $S$. Define the "defect" of $g,h$ to be
$$\psi(g,h) = |g|+|h|-|gh|$$
Then $\psi:G\times G \...
user6671
2
votes
0
answers
99
views
Character degrees in induced blocks
Let $G$ be a finite group and $U\leq H\leq G$ a chain of subgroups.
Presume that $p$ is a prime dividing the order of $U$.
Suppose that $b_1$ is a $p$-block of $U$ and $b_2$ a $p$-block which is ...
- 311