All Questions
286 questions
12
votes
0
answers
340
views
Does every finite group have a small projective representation (over some ring)?
Question. Let $G$ be a finite group. Can we find some (commutative) ring $R$ and some positive integer $d=O(\log\lvert G\rvert)$ such that $G$ can be found as a subgroup of $\operatorname{PGL}_d(R)$?
...
- 602
2
votes
2
answers
206
views
Software library for complex irreducible representations of $\mathrm{PSL}_n(q)$
I came across an extremely useful Python software library for the Monster group: https://github.com/Martin-Seysen/mmgroup which allows for all sorts of manipulations involving the sporadic finite ...
- 1,150
2
votes
1
answer
161
views
Smallest dimensional faithful complex representation of $\mathrm{PSL}(k,q)$
For given $k>1$ and $q$ a prime power, what is the minimal dimension, as a function of $(k,q)$, for which a faithful complex representation of the projective special linear group over $\mathbb{F}_q$...
- 1,150
37
votes
0
answers
1k
views
Is this generalized character always a character?
Let $G$ be a finite group, and $p$ be a prime. Then there is a generalized character $\Psi$ of $G$ which takes value $0$ on all elements of order divisible by $p$, and has $\Psi(y)$ equal to the ...
- 44.4k
1
vote
0
answers
92
views
Finite groups whose center nontrivially represented in irreps with coprime dimensions
I have been searching for a finite non-abelian group $G$ with the following properties:
Its center $Z(G)$ acts as the identity in all dimension one irreps (i.e., $Z(G)$ is a subgroup of the ...
- 11
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 ...
- 631
2
votes
1
answer
273
views
Equivariant Smith normal form?
Let $F$ be a finitely generated free $\mathbb{Z}$-module on which the group $G$ of two elements acts via group homomorphisms. Let $F'$ be a $G$-invariant submodule. By Smith normal form we know that ...
- 3,031
9
votes
2
answers
438
views
Irreducible tensor product representations in finite simple groups
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\PSp{PSp}\DeclareMathOperator\PSU{PSU}$Background:
A representation $ \rho: G \to \GL(V) $ of a group $ G $ on a (complex) ...
4
votes
0
answers
160
views
Largest primitive subgroup of $\mathrm{GL}_8(\mathbb{C})$ of order $2^a 3^b 5^c$
The paper "Bounds for finite primitive complex linear groups" by M. Collins computes the largest possible value of $[G:Z(G)]$ for $G$ a primitive subgroup of $\mathrm{GL}_N(\mathbb{C})$, for ...
- 123
4
votes
1
answer
249
views
The number of irreducible characters of simple groups of Lie type
Let $S$ be a simple group of Lie type defining over $\mathbb{F}_q$ where $q$ is a power of a prime $p$ and let $\sigma$ a nontrivial field automorphism of $S$.
Set $\mathrm{C}_{S}(\sigma)$ the ...
- 613
4
votes
2
answers
254
views
Order of abelian subgroup of the automorphism group of an abelian group
Suppose $A$ is a finite abelian group and $B\leq\operatorname{Aut}(A)$ is abelian. Is it possible for the order of $B$ to be strictly greater than the order of $A$? What if I additionally impose that ...
- 1,338
1
vote
0
answers
178
views
Applications of Artin's theorem on induced representations
Let $G$ be a finite group and let $R(G)$ be the (complex) representation ring of $G$. As stated in Serre's book on representation theory, Artin's theorem says the following:
Theorem: Let $X$ be a ...
- 11
14
votes
0
answers
527
views
Is the monster group maximal in SO(196883)?
$\DeclareMathOperator\SO{SO}$The smallest degree of a nontrivial complex representation of the monster group $ M $ is $ 196883 $. This irrep has Schur indicator $ 1 $, so the image must lie in the ...
- 4,081
3
votes
1
answer
296
views
Distinct characters with the same character values, outer automorphisms and Galois conjugation
Given an (irreducible complex) character of a finite group the following three construction all yield another irreducible character of the same degree:
multiplying by a degree 1 character
applying an ...
- 4,081
1
vote
0
answers
110
views
Character table of $\mathrm{P\Gamma L}_2(q)$ with $q$ even
Let $q = 2^f$ for some integer $f\geqslant 3$. The character table of $\mathrm{SL}_2(q)\cong\mathrm{PSL}_2(q)$ can be deduced from the character table of $\mathrm{GL}_2(q)$ (see, for example, Exercise ...
- 379
18
votes
2
answers
1k
views
The mysterious significance of local subgroups in finite group theory
EDIT 21/12: Even if there are no conclusive answers to these questions, I would very much like to know if anyone has noted and attempted to explain the mysterious significance of local subgroups: are ...
- 1,433
8
votes
0
answers
383
views
Is this set, defined in terms of an irreducible representation, closed under inverses?
$\DeclareMathOperator\GL{GL}$Let $ H $ be an irreducible finite subgroup of $ \GL(n,\mathbb{C}) $. Define $ N^r(H) $ inductively by
$$
N^{r+1}(H)=\{ g \in \GL(n,\mathbb{C}): g H g^{-1} \subset N^r(H) \...
- 4,081
10
votes
1
answer
237
views
For which finite groups $G$ is $M_n(\mathbb{Q}(\zeta))$ a factor of $\mathbb{Q}[G]$?
I am cross-posting this question from my MSE post here, in case someone here can answer it.
For a finite group $G$, the rational group ring $\mathbb{Q}[G]$ has a Wedderburn decomposition:
$$
\mathbb{Q}...
- 103
9
votes
1
answer
291
views
A question related to Jordan's theorem on subgroups of $\mathrm{GL}_n(\mathbb{C})$
$\newcommand{\C}{\mathbb{C}}$
$\newcommand{\mr}{\mathrm}$
For any positive integer $n$, let $f(n)$ be the minimal integer with the following
property:
For any finite subgroup $G < \mr{GL}_n(\C)$ ...
- 10.5k
5
votes
1
answer
292
views
Extension of base field for modules of groups and cohomology [duplicate]
Let $G$ be a group and let $K/k$ be a field extension. Suppose that $V$ is a $kG$-module, and let $V_K = K \otimes_k V$ be the $KG$-module given by changing the base field.
Is it true that $H^n(G,V_K) ...
- 884
2
votes
0
answers
65
views
Are the integer points of a simple linear algebraic group 2-generated?
Set Up:
Let $ K $ be a totally real number field. Let $ \mathcal{O}_K $ be the ring of integers of $ K $. Let $ G $ be a simple linear algebraic group. Suppose that $ G(\mathbb{R}) $ is a compact Lie ...
- 4,081
0
votes
0
answers
85
views
Is a Lagrangian subgroup of a metric group isomorphic to its quotient?
A metric group is a finite abelian group $G$ with a quadratic function
$$q:G\rightarrow \mathbb R/\mathbb Z\;,$$
that is,
$$M(a,b):= q(a+b)-q(a)-q(b)$$
is bilinear in $a$ and $b$ [edit: and non-...
- 3,001
1
vote
0
answers
179
views
Irreducible module of finite simple groups
Let $G$ be a finite simple group and $p$ be a prime divisor of $|G|$.
Let $V$ be a nontrivial irreducible $\mathbb{F}_p[G]$ module.
I would like to understand the relation of $|V|$ and $|G|_p$ (the $p$...
- 613
2
votes
0
answers
220
views
Characters of alternating groups
I am studying certain properties of characters of alternating groups and I have found precisely three characters not satisfying it up to $A_{15}$. These are:
A character of dimension $3.696$ of $A_{...
- 370
2
votes
0
answers
68
views
Dual representation of a transitive semilinear group
This seems like it should be known, but I couldn't find a reference.
Let $V$ be a finite vector space and let $G$ be a group of semilinear maps on $V$ (i.e. linear composed with a field automorphism). ...
- 4,728
1
vote
0
answers
179
views
Character extension about $Q_8$
Recently, I am studying the book Navarro - Character Theory and the McKay Conjecture. I am trying to solve the following exercise:
(Exercise 5.9)
Let $G$ be a finite group and $N\unlhd G$, suppose ...
- 195
6
votes
0
answers
320
views
(CFSG-free) Finite simple groups whose character degrees square divide its order
Let $G$ be a finite group. It is well-known that for all irreducible complex character $\chi$ then $\deg(\chi)$ divides $\lvert G\rvert$. Motivated by some problems with modular tensor categories, we ...
4
votes
1
answer
274
views
Wedderburn decomposition of special linear groups
$\DeclareMathOperator\SL{SL}\newcommand\card[1]{\lvert#1\rvert}$I want to study about Wedderburn decomposition of group algebra $k\SL(n,\mathbb{F}_p)$ where $k$ is either an algebraically closed field ...
- 249
4
votes
1
answer
343
views
Converse of Clifford's theorem for a semidirect product
Suppose that a group $G$ is a semidirect product $G = N \rtimes H$ with $N \trianglelefteq G$.
Let $\mathbb{F}$ be a field.
Say $V$ is a finite-dimensional $\mathbb{F}[G]$-module such that $V \...
- 2,821
12
votes
2
answers
926
views
Finite groups with integral character table
The character table of a finite group will be called integral if all its entries are integers. There are $11$ such groups up to order $16$, namely $C_1$, $C_2$, $C_2^2$, $S_3$, $D_8$, $Q_8$, $C_2^3$, $...
2
votes
0
answers
244
views
Existence of $\sqrt{2}$ in a finite group algebra over $\mathbb{Q}$
I cannot find a finite group $G$ such that $\exists x\in \mathbb{Q}[G]$ with $x^2=2e$, where $\mathbb{Q}[G]$ is the group algebra of $G$ over $\mathbb{Q}$.
I also could not prove it does not exist. ...
- 188
3
votes
1
answer
301
views
The torsion subgroup of the coinvariants for a $G$-module
Let $G$ be a finite group and $M$ be a finitely generated $G$-module,
that is, a finitely generated abelian group on which $G$ acts.
Consider the functor
$$ (G,M)\rightsquigarrow F(G,M):= (M_G)_{\rm ...
- 14.2k
5
votes
1
answer
253
views
Irreducible deleted permutation module for a finite group
Let $G$ be a subgroup of the symmetric group $S_n = \operatorname{Sym}(X)$.
Let $k$ be a field, and let $V$ be the permutation module corresponding to $X$.
Then $V$ is not irreducible, it has a $1$-...
- 2,821
5
votes
1
answer
244
views
What are the indecomposable modules over $\mathbb{F}_2(C_2\times C_2)$?
Let $C_2$ be the cyclic group of order $2$ and $\mathbb{F}_2$ the field with $2$ elements. Consider the group algebra $A:= \mathbb{F}_2 (C_2\times C_2)$. It is well-known that $A$ has infinite ...
- 696
4
votes
1
answer
237
views
Aschbacher classes for compact simple group
Posted this to MSE several weeks ago and it got 3 upvotes but no answers or even comments so I'm cross-posting to MO
Aschbacher's theorem says that every maximal subgroup of a finite simple classical ...
- 4,081
21
votes
0
answers
473
views
Is there a "direct" proof of the Galois symmetry on centre of group algebra?
Let $G$ be a finite group, and $n$ an integer coprime to $|G|$. Then we have the following map, which is clearly not a morphism of groups in general: $$g\mapsto g^n.$$
This induces a linear ...
- 1,949
6
votes
1
answer
567
views
Finite simple groups and $ \operatorname{SU}_n $
A follow-up question to Alternating subgroups of $\mathrm{SU}_n $.
$\DeclareMathOperator\PU{PU}\DeclareMathOperator\PSU{PSU}\DeclareMathOperator\PSL{PSL}$Let $ \PU_m $ be the projective unitary group, ...
- 4,081
5
votes
1
answer
211
views
The rank of indecomposable finite abelian 2-group
$\DeclareMathOperator\rank{rank}$Let $P$ be a finite $p$-group. The rank of $P$ is $\log_{p}|P/\Phi(P)|$ where $\Phi(P)$ is the Frattini subgroup of $P$, we write $\rank(P)=\log_{p}|P/\Phi(P)|$.
Let a ...
- 613
3
votes
2
answers
222
views
Equality of subsets of abelian groups
Let $G$ be a finite abelian group, $X$ and $Y$ be two non-empty subsets of $G$ of equal size. Suppose that for each irreducible character $\chi$ of $G$ we have $\sum_{x\in X}\chi(x)=\sum_{y\in Y}\chi(...
- 105
4
votes
0
answers
227
views
Orbits of group representation over $\mathbb{F}_2$
Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ...
- 161
1
vote
1
answer
362
views
Is $\sum_{\rho \text{ irred. }} \deg(\rho) \chi_{\rho}(g)=0$ for every group element $1 \neq g \in G$ of the finite group $G$?
Is $\sum_{\rho \text{ irred. }} \deg(\rho) \chi_{\rho}(g)=0$ for every froup element $1\neq g \in G$ of the finite group $G$?
I have searched for but not found a proof to this. Probably it is not so ...
- 3,064
54
votes
4
answers
5k
views
How many square roots can a non-identity element in a group have?
Let $G$ be a finite group. Let $r_2\colon G \to \mathbb{N}$ be the square-root counting function, assigning to each $g\in G$ the number of $x\in G$ with $x^2=g$. Perhaps surprisingly, $r_2$ does not ...
- 805
5
votes
1
answer
189
views
Question about Suzuki's theory of exceptional characters
$\DeclareMathOperator\Irr{Irr}$As elegant as Suzuki's theory is, the set up requires that the number of conjugacy classes of $p$-elements in a cyclic T.I. (as an example) Sylow $p$-subgroup $P$ of $G$,...
- 191
1
vote
0
answers
167
views
Minimal degrees of finite simple groups
The minimal projective degrees (minimal degree of an irreducible representation of a central extension) of the finite classical groups are (famously) given by Tiep and Zalesskii [1]. Is there a ...
- 9,720
10
votes
1
answer
496
views
How quasirandom are the nonabelian finite simple groups?
A group is $d$-quasirandom if every nontrivial complex representation has dimension at least $d$. Gowers introduced quasirandomness in this paper and proved that every nonabelian finite simple group ...
- 7,633
8
votes
2
answers
448
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 \...
- 393
5
votes
0
answers
115
views
Rigid points of $\mathrm{Co}_0$
Let the rigid points of a matrix group refer to subgroups of it that stabilize a nonzero vector and are maximal among such subgroups.
How many conjugacy classes of rigid points are there under the ...
- 2,773
5
votes
1
answer
508
views
Finite maximal closed subgroups of Lie groups
Cross-posted from MSE
https://math.stackexchange.com/questions/4272017/finite-maximal-closed-subgroups-of-lie-groups
$\newcommand{\G}{\mathcal{G}} \newcommand{\K}{\mathcal{K}} \DeclareMathOperator\SU{...
- 4,081
2
votes
2
answers
123
views
Smallest $\mathbb R$-algebra which contains a subgroup isomorphic to $A_4$
$A_4$ (the alternating group on $4$ elements) can be thought of as the group of direct Euclidean isometries of a regular tetrahedron. This shows that there is a subgroup of the algebra of $3\times3$ ...
- 4,943
4
votes
0
answers
186
views
Subgroups of $\operatorname{GL}(n,q)$ transitive on non-zero vectors
Is there a classification of subgroups $G$ of $\operatorname{GL}(n,q)$ which act transitively on $\mathbb{F}_q^n \setminus \{0\}$, the set of non-zero vectors?
Any $G$ with $\operatorname{GL}(n/m,q^m) ...
- 2,821