All Questions
Tagged with finite-groups reference-request
65 questions with no upvoted or accepted answers
32
votes
0
answers
993
views
Is there a Mathieu groupoid M_31?
I have read something which said that the large amount of common structure between the simple groups $SL(3,3)$ and $M_{11}$ indicated to Conway the possibility that the Mathieu groupoid $M_{13}$ might ...
- 3,645
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
15
votes
0
answers
885
views
How much has been written down about Deligne's geometric approach to the order formula for a finite group of Lie type?
This is a follow-up to a recent mathoverflow question
34387
about computing the orders of finite unitary groups and the comments made there.
Between 1955 (Chevalley's Tohoku paper) and 1968 (...
- 52.9k
9
votes
0
answers
297
views
An abstract zero-sum problem
I would like to know whether the problem described below has appeared in the literature and/or whether similar questions have been studied. I would be very happy to find some references or, if none ...
- 1,169
8
votes
0
answers
274
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Integral representations of finite groups and lattice point geometry
See the edit at the bottom (April 2021)
This contains both a reference request, and a specific problem.
Let $K$ be a finite group, and let $\theta: K \to {\rm GL}(d,{\bf Z})$ be a (faithful) group ...
- 4,679
7
votes
0
answers
235
views
Classification of octonionic reflection groups
I know that there exist classification theorems for real, complex, and quaternionic, reflection groups.
There are presentations for the real reflection groups, as well as further presentations for the ...
- 113
7
votes
0
answers
405
views
How can I get my hands on McKay's "Finite p-groups" lecture notes?
How can we find Susan McKay's "Finite $p$-groups" lecture notes?
The notes I'm talking about are these.
I emailed Peter Cameron, but he has since moved to a different university, and has no ...
- 4,425
7
votes
0
answers
229
views
Computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$
Do you have a nice modern reference where I could find the computation of $H^2(S_n,\mathbb{Z}/2\mathbb{Z})$, where the action is trivial ?
I have looked at the very few books on cohomology of groups ...
- 1,766
7
votes
0
answers
430
views
The maximal order of an element in orthogonal groups over finite fields of characteristic 2
Let $q$ be a power of $2$ and let $(V,Q)$ be a
quadratic space of dimension $2m$ over $\mathbb{F}_q$. Up to isometry, we know that we have exactly two classes of such quadratic spaces: the plus type ...
- 7,609
6
votes
0
answers
121
views
Sylow subgroups of the restricted Burnside group $\mathrm{RB}(d,n)$?
$\DeclareMathOperator\RB{RB}$What is known about the Sylow subgroups of the restricted Burnside groups $\RB(d,n)$ ?
I am looking for a reference.
In fact my question is slightly more general. Recall ...
- 61
6
votes
0
answers
122
views
Schur indices for 2-groups
I am looking for any results on Schur indices over $\mathbb{Q}$ for 2-groups. By a theorem of Roquette (corollary 10.14 in Isaacs) these numbers are at most 2. I am interested in 2-groups for which ...
- 81
6
votes
0
answers
455
views
category theoretic approach to Sylow theorems and finite group theory?
Is there a category theoretic approach to Sylow theorems?
More generally, is there an exposition of finite group theory in terms of category theory which would include Sylow theorems and little facts ...
- 161
5
votes
0
answers
351
views
Adjoint identity on finite nilpotent groups
Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]:
$$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
5
votes
0
answers
179
views
When is a Hermitian matrix of the form $g^*g$ for some matrix $g$
I tried asking this question on Math StackExchange and didn't get any replies. I was hoping that maybe someone here could help, sorry for duplicating.
I'm trying to figure out some properties of ...
- 993
4
votes
0
answers
180
views
Subgroups that conjugate-cover the ambient group
Let $G$ be a finite group, and suppose that a set of proper subgroups $H_1,\dotsc,H_n$ satisfy $G=\bigcup_{g\in G}\bigcup_{i=1}^nH_i^g$, where $H_i^g$ is the conjugate of $H_i$ by $g$. In this case, ...
4
votes
0
answers
107
views
Complex reflection groups: reference request
Suppose that $V$ is a finite-dimensional complex vector space, that $m\ge 2$ is an integer and that $G\subset \operatorname{GL}(V)$ is a finite subgroup such that $V$ is an irreducible ${\mathbb{C}}[G]...
- 3,137
4
votes
0
answers
174
views
Algebraic varieties associated to finite groups
Have the following equations been studied in the literature?
Let $G$ be a finite group.
Then I am looking for functions $f : G \rightarrow \mathbb{C}~ \backslash \left\lbrace 0 \right\rbrace $ such ...
- 2,753
4
votes
0
answers
100
views
$\mathrm{Sp}_n(q)$-conjugacy classes in $\mathrm{GL}_{2n}(q)$
The symplectic group $\mathrm{Sp}_n(q)$ acts on $\mathrm{GL}_{2n}(q)$ by conjugation. All the literature I have found concerning the orbits of action of this kind is "Unipotent conjugacy classes in ...
- 295
4
votes
0
answers
218
views
Conjugacy class representatives for the automorphism group of a finite abelian group
Given a finite abelian group $A$, I'd like a list of conjugacy class representatives for its automorphism group ${\rm Aut}(A)$.
In fact, it's not important that I have exactly one representative from ...
- 286
4
votes
0
answers
135
views
Improvements of the Reidemeister-Schreier index formula for particular classes of groups
I have a couple of questions regarding possible improvements of the Reidemeister-Schreier index formula: let $G$ be a $d$-generated group and let $H$ be a subgroup of $G$, then
$$d(H) \le (d-1) \...
- 116
3
votes
0
answers
125
views
Short path problem on Cayley graphs as language translation task (from "Permutlandski" to "Cayleylandski"(s) :). Reference/suggestion request
Context: Algorithms to find short paths on Cayley graphs of (finite) groups are of some interest - see below.
There can be several approaches to that task. One of ideas coming to my mind - in some ...
- 24.9k
3
votes
0
answers
190
views
Efficient implementation of the Clifford group for $n$ qubits
I'm looking for an efficient implementation of the Clifford group $\mathcal{C}_n$
of $n$ qubits.
The Clifford group $\mathcal{C}_n$ has stucture $(2_+^{1+2n} \circ C_8).Sp(2,n)$,
where $2_+^{1+2n}$ ...
- 558
3
votes
0
answers
170
views
What is $G_2(2^m)$, and how is it embedded in $\Gamma L_6(2^m)$?
I am trying to understand the classification of doubly transitive groups, specifically the nonsolvable affine case. Dixon and Mortimer (p.244) says there are three infinite families, one of which is $\...
- 7,633
3
votes
0
answers
62
views
Torus in the small Ree group ${}^2G_2$ over an infinite field
In “Simple group of Lie type” by R. W. Carter there is a remark (after Theorem 13.7.4):
It is not known whether $H^1$ coincides with the set of $\sigma$-invariant elements of $H$ if $\mathfrak{L}$ ...
- 3,186
3
votes
0
answers
101
views
Symmetries of irregular simplices
On the wikipedia page of tetrahedron, there is a list of eight symmetry groups for a (possibly irregular) $3$-simplex (with unmarked faces). There is also a list on the page of 5-cell but doesn't ...
- 2,581
3
votes
0
answers
269
views
Reference for the rank of the BN-pair of the finite simple groups of Lie type and not Chevalley
The rank of the BN-pair of a Chevalley group is the number of simple roots of its Lie algebra, which is the index of the name of its Dynkin diagram ($A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4,G_2 $).
...
3
votes
0
answers
186
views
Which Dihedral Groups are $\text{CI}$-Groups?
Let $D_{n}$ denotes the dihedral group of order $2n$. Firstly, for self-referencing of the question, I give some definitions which are standard.
Let $G$ be a finite group. A subset $S$ of group $G$ ...
- 4,784
3
votes
0
answers
102
views
Localized at $p$ integral representations of finite elementary $p$-groups
Let $C_p$ be a cyclic group of prime order $p$.
Let $F=C_p^n=C_p\times\dots\times C_p$ ($n$ times).
I would like to to classify finite dimensional representations of $F$ over ${\mathbb{Z}}$.
However, ...
- 14.1k
3
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
0
answers
127
views
"A locally dual polar space for the Monster"
I am currently looking at Ronan and Stroth's 1984 paper Minimal Parabolic Geometries for the Sporadic Groups. When considering the $3$-minimal parabolic system of $F_{1}$, they cite a preprint by ...
- 295
3
votes
0
answers
209
views
What is known about 2-modular representations of Ree groups of type $F_4$?
A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite ...
- 52.9k
3
votes
0
answers
153
views
On divisors occurring as subgroup sizes
Given a finite group $G$ define $D(G)$ to be the number of divisors $r$ of $|G|$ for which there exists a subgroup of $G$ of order $r$.
Clearly $D(G) \leq d(|G|)$, where $d(n)$ denotes the number of ...
- 731
3
votes
0
answers
264
views
How do I determine the smallest dimension of an irreducible $\mathbb{F}_p[G]$-module with a prescribed trivial fixed point space?
This is a crosspost from MSE since I haven't found an answer there yet.
I am not very familiar with modular representation theory or Brauer theory yet, however lately I have needed to use $\mathbb{F}...
- 1,173
2
votes
0
answers
167
views
Centralizer of PSL in PGL and of SL in GL: reference request
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\PSL{PSL}$Consider the general linear group $\GL(n,q)$ over a finite field with $q$ elements and ...
- 361
2
votes
0
answers
97
views
Centralizer bound for irreducible representations of $\operatorname{SU}_n(\mathbb{C})$
Let $G$ be a finite group, $χ$ be the character of an irreducible representation $V$ of $G$, and $g ∈ G$. Then a classical bound on the trace of $g$ is given by: $|χ(g)|² ≤ |C_G(g)|$, where $C_G(g)$ ...
- 2,907
2
votes
0
answers
91
views
Reference request: structure of group of units of finite group ring
Let $G$ be a finite group, let $F$ be a finite field and let $F[G]$ be the group algebra of $G$ over $F$.
What is known about the structure of the group of units $F[G]^\times$? Of course, it must ...
- 1,433
2
votes
0
answers
408
views
Conceptual proof of fundamental theorem of finite abelian groups
I'm looking for a conceptual proof of the following statement:
Lemma: Let $G$ be a finite abelian $p$-group. Let $a$ be an element of maximal order. Then $G=\langle a \rangle \times H$ for some ...
- 2,751
2
votes
0
answers
127
views
Positive values of Schur polynomials
Recall that for a given partition $\lambda=(\lambda_1,\ldots,\lambda_r)$, its Schur polynomial in $n$-variables is the sum of monomials
$$s_\lambda(x_1,\ldots,x_n)=\sum_{T\in\operatorname{SSYT}(\...
- 868
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
2
votes
0
answers
194
views
Abstract of talk by Wielandt required
I am searching for Abstracts of short communications of the International Congress of Mathematicians, 1962. In particular, the abstract of Wielandt's talk "Bedingungen für die Konjugiertheit von ...
2
votes
0
answers
87
views
A theory of (or reference for) symmetric point arrangements
I wonder where I can find something written on symmetric point arrangements (see definition below). I am interested in general references, preferably books that introduce (or papers that use) some ...
- 13.6k
2
votes
0
answers
72
views
"Spectral gaps" of commutativity measure
There's a notion of commutativity measure $P(G)$ of a finite group $G$ which is probably folklore: count commuting pairs in $G \times G$ and divide by $|G \times G|$. There are some results:
$P(...
- 4,600
2
votes
0
answers
187
views
Classification of Automorphism set of a Regular graph
Let $A$ be the adjacency matrix of an $r$-regular graph $G$ with $n$ vertices (Not complete or cycle graph) . Also, let $Aut(G)$ be the set of all its automorphisms (i.e. set of permutation matrices)....
- 267
2
votes
0
answers
89
views
explicit matrices for Weil ($p^2$ dimensional) representation of $Sp(4,\mathbb{F}_p)$, $p>3$
I am looking for more-or-less explicit matrices for the $p^2$ dimensional Weil representation of $Sp(4,\mathbb{F}_p)$, suitable for computer implementation. Ideally, I would like the images of the ...
- 1,639
2
votes
0
answers
298
views
Reference request
I am looking for a reference or proof for the following problem:
Problem: Let $r$ be prime, then $2r$ is a Sylow $p$-number if and only if $2r=1+p^{2^n}$.
Thanks in advance.
- 102
2
votes
0
answers
318
views
Pierpont primes
A Pierpont prime is a prime $p$ that can be written as $$p=2^u 3^v + 1.$$
What is known about Pierpont primes? I'm not a number theorist, and the best I can find is
http://en.wikipedia.org/wiki/...
- 21
1
vote
0
answers
172
views
Isomorphism classes of finite $\mathbb{N}$-groups
Where can I find resources on isomorphism classes of finite $\mathbb{N}$-groups, i.e. groups acted on by the monoid $(\mathbb{N}, +)$?
I edited this question to be more focused on what I'm interested ...
- 591
1
vote
0
answers
61
views
Cardinal of finite orthogonal groups
Let $p \neq 2$ and let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$ and maximal ideal $\mathfrak{p}$.
By a quadratic space $V_{\mathcal{O}}$ of dimension $d$ over $\mathcal{O}$, I mean ...
- 81
1
vote
0
answers
127
views
Irreducible projective representations of finite abelian groups
I want to know if there is a description of all irreducible complex projective representations of an arbitrary finite abelian group. I have seen this for particular cases such as those given here and ...
- 249
1
vote
0
answers
64
views
Sylow subgroups of the free product of profinite groups
I am interested in the Sylow subgroups of the profinite completion of a free product of finite groups.
Is the following naive expectation true ? I assume things like this should be well-known, and am ...
- 11