All Questions
Tagged with finite-fields finite-groups
57 questions
56
votes
14
answers
21k
views
Fantastic properties of Z/2Z
Recently I gave a lecture to master's students about some nice properties of the group with two elements $\mathbb{Z}/2\mathbb{Z}$. Typically, I wanted to present simple, natural situations where the ...
32
votes
9
answers
5k
views
How many groups of size at most n are there? What is the asymptotic growth rate? And what of rings, fields, graphs, partial orders, etc.?
Question. How many (isomorphism types of) finite groups of size at most n are there? What is the asymptotic growth rate? And the same question for rings,
fields, graphs, partial orders, etc.
...
- 236k
19
votes
4
answers
2k
views
Non-split extensions of $GL_n(F_q)$ by $F_q^n$ ?
A very naive question :
I just learned that there is a non-split extension of $GL_3(F_2)$ by $F_2^3$ (with standard action). It can be realized as the subgroup of the automorphism group $G_2$ of ...
- 9,389
15
votes
1
answer
4k
views
Order of finite unitary group
This may be an easy exercise but I am not getting it. Let $\mathbf F_q$ be a finite field with $q$ elements and $\mathbf F_{q^2}$ be its degree two extension. Define an automorphism $\sigma$ of $\...
- 899
13
votes
2
answers
1k
views
Number of commuting pairs (triples, n-tuples) in GL_n(F_q) (and other groups)?
Question 1 What is the number of pairs of commuting elements in GL_n(F_q) ?
I am aware of many results concerning commuting elements in Mat_n(F_q), but I am interested in GL i.e. non-degenerate ...
- 24.9k
13
votes
4
answers
5k
views
Orthogonal Groups over finite fields
Hello
Let $\mathbb{F}_p$ be the finite field with $p$ elements. One can show that over finite fields, there are just two non-degenerate quadratic forms.
So here I want to pick
any non-degenerate ...
- 2,508
13
votes
1
answer
603
views
Steinberg representation for sporadic simple groups?
The Steinberg representation is a remarkable irreducible representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-...
- 24.9k
11
votes
3
answers
1k
views
Can we always find such an irreducible polynomial of degree n where degree(p(x)-x^n)<= n/2?
Consider unitary polynomials of degree $n$ over $GF(2)$. That is, polynomials of the form $p(x) = \sum_{i=0}^n a_i x^i$ where $a_i\in GF(2)$ and $a_n=1$.
Can we always find such an irreducible ...
- 375
10
votes
8
answers
1k
views
Classifications of finite simple objects
I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "...
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
9
votes
2
answers
731
views
Find the order of a class of finite matrices over finite fields
Consider matrices $M$ of size $L\times L$ over a finite field $\mathbb{Z}_p$, for simplicity focus on $p$ prime. The size $L$ is even. We want to find the order of a specific class of matrices, namely ...
- 293
8
votes
1
answer
1k
views
Cyclic subgroups of GL(n,q)
Let $q$ be a prime power. It is well known that all Singer subgroups (subgroups of order $q^n-1$) in $GL(n,q)$ are conjugate. My question is: If $H$ is a cyclic subgroup of order $m$ in $GL(n,q)$, $m\...
- 123
8
votes
1
answer
534
views
What is the minimum $k$ such that $A^k \equiv I$ mod p for invertible matrices?
Let $F$ be a finite field of order $p$, where $p$ is prime. For any $n\times n$ matrix $A$ that is invertible over $F$, then there would appear to exist integers $k$ such that $A^{k} = I$. My question ...
- 183
8
votes
0
answers
304
views
A strong sum-product "for translates" in finite fields
In the course of some recent research, I've sketched out a proof of the following result. My basis question is: is the result interesting?
Proposition There exists an absolute constant $c$ such ...
- 11.2k
7
votes
2
answers
330
views
Representations of $\operatorname{Sp}(2g,\mathbb{Z}_3)$
Let $V$ be a $2g$-dimensional vector space over $\mathbb{Z}_3 := \mathbb{Z}/3\mathbb{Z}$. First, $\operatorname{Sp}(2g,\mathbb{Z}_3)$ acts on $\Lambda^2(V)$, and this decomposition is reducible, as ...
- 75
6
votes
1
answer
368
views
Number of points on a linear algebraic group over a finite field
Let $G$ be a linear algebraic group defined over a finite field $\mathbb{F}_q$ as a variety of dimension $d$. What would be a good, simple lower bound for $G(F_q)$?
One can get something fairly nice ...
- 20.2k
6
votes
1
answer
1k
views
orthogonal group in characteristic 2
Let $O(2,\mathbb{Z}_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for ...
- 3,779
6
votes
2
answers
113
views
Describing the action of $^2E_6(q)$
One of the constructions of the group $^2E_6(q)$ was presented by Tits in his paper "Les «formes réelles» des groupes de type $E_6$". It is being constructed by looking at the action of $^2E_6(q)$ on ...
- 161
5
votes
1
answer
343
views
Large gaps in Singer planar difference sets?
By a classical result of Singer (1938), for a prime number $p$ the cyclic group $C_n$ of order $n=1+p+p^2$ contains a subset $D$ of cardinality $|D|=1+p$ such that $DD^{-1}=C_n$. Such set $D$ is ...
- 41.9k
5
votes
1
answer
285
views
How to make Burnside's formula compatible with point counting for varieties over finite fields?
If $G$ is a finite group acting on a finite set $X$, we have Burnside's formula that counts the number of orbits $|X/G|$ as:
$$ |X/G| = \frac1{|G|} \sum_{g\in G} |X^g|,
$$
with $X^g$ being the set of ...
- 167
5
votes
2
answers
571
views
Exceptional isomorphisms between finite simple Chevalley groups
Steinberg's "Lectures on Chevalley Groups"
https://math.depaul.edu/cdrupies/research/papers/chevalleygroups.pdf
contain ``a complete list of isomorphisms" among the various finite simple Chevalley ...
- 5,050
5
votes
1
answer
603
views
Aschbacher classes and $\mathbb{F}_p$-subspace stabilizers in classical linear groups
I am reading the Kleidman–Liebeck book ("The subgroup structure of the finite classical groups") which is about the Aschbacher classification of maximal subgroups of the classical almost ...
- 731
5
votes
1
answer
764
views
Conjugacy classes in $GL_{n}(Z / pZ)$
Let $p$ be a prime number and $G=GL_n ( \mathbb{Z} / p \mathbb{Z}
)$. Consider the set $U$ of upper-triangular matrices of $G$
having entries of $1$ on the diagonal. The cardinality of $U$ is $p^{\...
- 463
5
votes
0
answers
203
views
Number of elements in $\mathrm{GL}(n,p)$ with maximal order
I learned reading this question that $\mathrm{GL}(n,p)$ elements have at most a multiplicative order of $p^n -1$.
I would like to know how many matrices have an order of exactly $p^n -1$. Do they ...
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
1
answer
287
views
Character values of principal series representations of $GL_n(\mathbb{F}_q)$
Let $P_{\alpha}$ be the principal series representation of $GL_n(\mathbb{F}_q)$, where $\alpha = ( \alpha_1, \alpha_2, \cdots, \alpha_n)$ and $\alpha_i : \mathbb{F}_q^* \rightarrow \mathbb{C}^*$.
...
- 73
4
votes
1
answer
491
views
What is the permutation group generated by those three given morphisms of the affine space $\mathbb{F}_q^3$?
Let $\mathbb{A}^3 = \mathbb{F}_q^3$. Consider the following three functions $\mathbb{A}^3\to\mathbb{A}^3$:
\begin{eqnarray*}
h: (x, y, z) &\mapsto& (x, y, xy - z) \\
u: (x, y, z) &\mapsto&...
user4324
4
votes
0
answers
107
views
Finite transitive linear subgroups
Let $q$ be a prime power and $d$ an integer. I want to understand the classification of the transitive linear subgroups of $GL_d(\mathbb F_q)$. According to the Wikipedia page https://en.wikipedia.org/...
- 509
4
votes
0
answers
1k
views
Representations of general linear groups GL_n(F_q) - decomposition of tensor product?
Let $V$ and $W$ be complex irreducible representations of $GL_n(F_q)$ where $F_q$ is finite field. Is the decomposition of $V \otimes W$ into irreducible representations known?
PS
Same question:
...
- 899
3
votes
2
answers
235
views
Do the irreducible modules of this finite group preserve a tensor product structure?
I am interested in a particular group $G$, where
$$ (A_4\times C_\ell) \lhd G \lhd S_4 \times D_\ell$$
Here, $C_\ell$ is cyclic, $D_\ell$ is dihedral of order $2\ell$, and the two inclusions both have ...
- 11.2k
3
votes
1
answer
298
views
Units in a finite semisimple group algebra
Let $G$ be a finite group and $k$ a finite field, with the characteristic of $k$ not dividing the order of $G$. Then $kG$ is a finite semisimple group algebra with the interesting property that an ...
- 31
3
votes
1
answer
181
views
Centralizers of $\mathbb{F}_q$-rational semisimple elements of a finite group of Lie type
Let $\mathbb{G}$ be a connected reductive $\mathbb{F}_q$ algebraic group over its algebraic closure $\bar{\mathbb{F}_q}$, and $\mathbb{T}$ be an $\mathbb{F}_q$-defined maximal torus. Let $\Phi$ be the ...
- 993
3
votes
1
answer
53
views
describing embedding $U_3(q)<O_6^-(q)$, $q$ even
Let $q=2^k$. I need to explicitly construct $U_3(q)$ as a subgroup of $G=GO_6^-(q)$. It is well-known that
$G\cong U_4(q)$, and as a subgroup of the latter one has $U_3(q)$ fixing a non-isotropic ...
- 14k
3
votes
1
answer
608
views
Representation of GL(n, F_p) over F_p, for n small
The question is related to this post
Representation theory of the general linear group over a finite prime field
However, I am asking for more detailed references for n small, for example, for n=2, ...
- 31
3
votes
2
answers
1k
views
Cyclic order relation in Zn
The ring Zn:={0,1,..,n-1} under addition and multiplication modulo n.
Suppose a,b,c,x $\in$ Zn are nonzero and the cyclic order R(a,b,c) holds, then under what conditions does R(ax,bx,cx) hold ?
- 1,190
3
votes
1
answer
236
views
Intersections of products of Sylow $p$-subgroups
Motivated by the work of Cohn, Umans and collaborators on bounding the complexity of matrix multiplication, a student and I have been investigating the following problem.
For subsets $X$ and $Y$ of a ...
- 1,300
3
votes
1
answer
169
views
Image of the Lang-Steinberg map on disconnected centralizers of semisimple elements
Let $\newcommand{\dbF}{\mathbb F}\dbF_q$ be a finite field and let $G\subseteq\mathrm{GL}_N(\bar{\dbF}_q)$ be a connected reductive group defined over $\dbF_q$. Let $F$ be the associated Frobenius map,...
- 993
3
votes
1
answer
288
views
Representation of a finite group over a finite field from rational representations
Suppose $G$ is a the cyclic group of $n$ elements and $p$ is a prime not in $n$. $G$ has an action on the cyclotomic field as a $\mathbb{Q}$-vector space $\mathbb{Q}[X] / \langle \Phi_n(X) \rangle$ by ...
- 885
3
votes
1
answer
128
views
embedding of $O_4^-(q)$ in $U_4(q)$
For $q$ an odd prime power, one can construct $H:=O_4^-(q)<U_4(q)$ by treating the $H$-invariant bilinear form as Hermitean. On the other hand $U_4(q)$ is isomorphic to $O_6^-(q)$; I need to ...
- 14k
3
votes
0
answers
147
views
Under what conditions on $A$ and $v$ is the size of the sumset $v \cdot A + A$ over $\mathbb{F}_p$ equal or close to $|A|^2$?
Let $p$ be a prime, let $A$ be a subset of $\mathbb{F}_p$, and let $v \in \mathbb{F}_p \setminus \{0\}$.
Under what conditions is $|v \cdot A + A|$ (that is, $|\{ va + b : a \in A,\ b \in A \}|$) ...
3
votes
0
answers
99
views
Unitary matrices $p$-root of identity such that the Fourier transform matrices are $p$-root of identity
Take a prime number $p$ and $\omega=e^{2i\pi/p}$. Assume we have p complex matrices (in finite dimension $n$) $A_0, \dotsc, A_{p-1}$ such that $\forall i, A_i^p=I$.
Define the $p$ fourrier transform ...
- 583
2
votes
1
answer
380
views
Dihedral subgroups of $\mathrm{PSL}_2(\mathbb{F}_q)$
Let $\mathbb{F}_q$ be a finite field with $q=p^f$ elements. I need to know when $\mathrm{PSL}_2(\mathbb{F}_q)$ contains the group $D_{(q+1)/2}$, where by $D_n$ I mean the dihedral group of order $2n$. ...
- 2,508
2
votes
1
answer
197
views
Invariant subspaces of an $F_2$-representation of the affine linear group of dimension 1
Let $p$ be an odd prime (large if it matters) and let $G= Aff(\mathbb{F}_{p^2}) \cong \mathbb{F}_{p^2} \rtimes \mathbb{F}_{p^2}^*$ be the affine linear group acting on $\mathbb{F}_{p^2}$ by $x\mapsto ...
- 3,362
2
votes
1
answer
166
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, \...
- 1,570
2
votes
2
answers
220
views
How does one calculate/estimate/guarantee the girth of a non-Abelian Cayley graph?
This question is in reference to this other question,
Can someone point out references (or explain!) which give techniques of being able to prove for any Cayley graph this property of having a girth ...
- 617
2
votes
0
answers
168
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
154
views
Reference request - obtaining finite simple groups from algebraic groups
I'm looking for references for the following statements, which I believe are true:
Let $G$ be a simply connected simple linear algebraic group over a finite field $k$ of cardinality $q\ge 4$. Let $Z\...
- 7,527
2
votes
0
answers
203
views
Can the relation between count of commuting pairs and conjugacy classes for finite groups be generalized to semigroups?
It is well-known that number of pairs of commuting elements in finite group G is equal to number of conjugacy classes multiplied by cardinality of G.
More generally here (MO275769) Qiaochu Yuan ...
- 24.9k
2
votes
0
answers
116
views
Loewy structure of $S_4$
How to deduce the Loewy Structure of $kS_4$ where $k$ has characteristic 2. I can compute the Cartan matrix and Decomposition matrix with Brauer Characters without difficulties. But when it comes to ...
- 51
2
votes
0
answers
186
views
Sum of reciprocals in finite fields
Let $p$ be an odd prime number which large enough. I am interested in the study of the sums of reciprocals in the field $\mathbb{F}_p$.
In particular, I have the following question:
which primes $p$ ...
- 347