All Questions
Tagged with finite-groups nt.number-theory
88 questions
3
votes
0
answers
89
views
Which elements in $\mathrm{Aut}(\widehat{F_2})$ preserve the procyclic subgroup generated by the commutator $c=[a,b]$?
Let $F_2$ denote the free group over two generators $a,b$, and we denote $c=[a,b]$ as the commutator. It is well-known that any automorphism $\psi$ of $F_2$ preserves the conjugacy class of the ...
- 605
9
votes
2
answers
794
views
Why do these finite group Dedekind matrices seem to have integer spectrum when specialized to the order of group elements?
Let $p$ be a prime and let $f_p$ be the permutation on the set $\{1,2,\cdots,p-1\}$ which is given by taking inverses in $\mathbb{Z}/(p)$:
$$x \bmod(p) \mapsto \frac{1}{x} \bmod (p)$$
So for instance, ...
- 3,064
2
votes
0
answers
98
views
Estimating the cardinality of the set of conjugacy classes of subgroups in a finite group of given order
1. Let $G$ be a finite group of order $n$. I need an estimate for the number $c$ of conjugacy classes of subgroups $D\subseteq G$.
Note that any subgroup of $G$ contains $1_G$, and so the set of all ...
- 14.1k
3
votes
0
answers
172
views
Abelian characters and odd perfect numbers?
This question is about applications of abelian characters to odd perfect numbers:
Context and Definitions:
Let $n$ be a natural number and $D_n$ be the set of divisors.
We can make this set to a ring ...
- 3,064
2
votes
3
answers
345
views
A subgroup of index 2 for a solvable finite group transitively acting on a finite set of cardinality $2m$ where $m$ is odd
Let $G$ be a finite group acting transitively on the finite set $X=\{1,2,\dots, 2m\}$
of cardinality $2m\ge 6$ where $m$ is odd.
Question 1. Is it true that $G$ always has a subgroup $H$ of index 2
...
- 14.1k
1
vote
0
answers
117
views
Reduction mod 2 for orthogonal groups
Setting Let $k$ be a real quadratic field, $\mathbb Z_k$ its ring of integers. Let $n$ be an even integer $A$ a symmetric $n$-by-$n$ matrix with coefficients in $\mathbb Z_k$. Let $L$ be the lattice $\...
- 3,399
9
votes
0
answers
189
views
Cyclic numbers of the form $2^n + 1$
A cyclic number (or cyclic order) is a number $m$ such that the only group of order $m$ is the cyclic group $\mathbb{Z}/m\mathbb{Z}$. The set of cyclic numbers admits a couple of cute number-theoretic ...
- 756
2
votes
1
answer
193
views
Sparsity of q in groups PSL(2,q) that are K_4-simple
One of the problems that has come up during my research concerns $K_4$-simple groups (simple groups with $4$ prime divisors). The only (potentially) infinite family of groups satisfying this condition ...
- 23
12
votes
1
answer
450
views
abelian quotients of permutation groups
Let $G$ be a subgroup of the permutation group $S_n$, and let $H$ be a normal subgroup of $G$ such that the quotient group $G/H$ is abelian. What is the best known upper estimate for the cardinality $...
- 1,294
37
votes
1
answer
1k
views
What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$What is the smallest group not known to be a Galois group over $\mathbb{Q}$?
Variants have been asked here before (e.g. Which small finite ...
- 2,959
3
votes
0
answers
116
views
Ways to tell from residues modulo prime factors if $z$ is below half point
Let $N=\prod_{k=0}^{k=m}{ p_k }$ be a square-free odd integer where $p_k$ is a prime. If we are given any integer $g$ such that $0<g<N$, it is very easy to tell if $g < \frac{N}{2}$ or not. ...
5
votes
1
answer
358
views
The number of polynomials on a finite group, II
This question is follow up of this MO-post.
First let us recall the necessary definitions.
A function $f:X\to X$ on a group $X$ is called a polynomial if there exists $n\in\mathbb N$ and elements $a_0,...
- 41.8k
0
votes
1
answer
213
views
A particular permutation on $\mathbb{Z}_n$
Let us consider $\mathbb{Z}_N=\{0,1,\cdots N-1\}$.
Does there exist any permutation $\sigma$ on $\mathbb{Z}_N$ satisfying :
$$\exists p,q\in\mathbb{N}, ~~p,q\neq0 (mod~N) ~~\forall i,j\in \mathbb{Z}...
- 4,058
1
vote
1
answer
266
views
Strange result of divisibility
I have noticed experimentally that the following question has a positive answer.
Let $p>5$ and $H$ be a subgroup of $(\mathbb Z/p\mathbb Z) ^*$, with $a\in H$ and $a>2$.
Is it true that $$(a-1)\...
- 4,074
8
votes
2
answers
880
views
Moebius function of finite abelian groups
I am wondering if there is any literature on general formula of the Moebius function of subgroup lattices of any finite abelian group $G$? What I know is
When $G$ is cyclic, the Moebius function is ...
- 231
5
votes
0
answers
78
views
Conjugacy classes in normalized unit group of a group ring
Let $V(FA_4)$ be the normalized unit group of the group ring $FA_4$, where $F$ is the field containing 4 elements and $A_4$ is the alternating group on 4 symbols. How can I find conjugacy classes of ...
- 381
11
votes
1
answer
416
views
Number of 1's in binary expansion of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$
My question is about the Hamming Weight (or number of 1's in binary expansion) of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$ A152007
For example, $a_3 = 9709 = (10110111101001)_2 $ has nine 1's in binary ...
- 113
1
vote
0
answers
94
views
A question concerning finite metacyclic groups
Consider a finite metacyclic group with presentation
$$G = \langle x,y \, |\, y^n=1, x^u=y^r, x^{-1}yx = y^k \rangle,$$ where $r \mid n$.
Is it true that if $G$ does not split (i.e. $G$ is a not a ...
4
votes
1
answer
196
views
Generalization of $\lim_{n \rightarrow \infty} \prod_{i=1}^{n}\frac{2i-1}{2i}$ for a character $\chi:\mathbb{Z}/s \mathbb{Z} \rightarrow \mathbb{C}^*$
Playing with some infinite products I came up with this problem, that I'm not able to figure it out by myself. Moreover in the internet it doesn't seem to appear anywhere.
Maybe it is just an easy ...
- 1,343
5
votes
1
answer
368
views
Boolean ring of unitary divisors / Structure of unitary divisors?
I hope this question is appropriate for MO:
Let $n$ be a natural number, $U_n := \{ d | d \text{ divides } n, \gcd(d,n/d)=1\}$ be the set of unitary divisors.
We can make $U_n$ to a boolean ring:
$$a \...
user6671
8
votes
2
answers
479
views
Abundancy index and non-solvable finite groups
Let $\sigma$ be the sum-of-divisors function. A number $n$ is called abundant if $\sigma(n)>2n$. Note that the natural density of the abundant numbers is about $25 \%$. The abundancy index of $n$ ...
13
votes
2
answers
668
views
On the sum of the subgroup orders of a finite group
Let $G$ be a finite group. Consider the function providing the sum of the subgroups orders
$$\sigma(G) = \sum_{H \le G} |H|.$$
Note that if $C_n$ is cyclic of order $n$ then $\sigma(C_n) = \sigma(n)$, ...
1
vote
1
answer
218
views
Average size of iterated sumset modulo $p-1$,
Given a prime $p$, what is the average size of the iterated sumset, $|kA|$, modulo $p-1$, with $p$ a prime, and $k$ given, with $A$ chosen at random?
You can pick any type of prime you like for $p$, ...
- 221
7
votes
1
answer
145
views
How to prove the relationship between Stern's diatomic series and Lucas sequence $U_n(x,1)$ over the field GF(2)?
I found the bit count of Lucas sequence $U_n(x,1)$ over the field GF(2) is Stern's diatomic series, I want to know the reason?
https://oeis.org/A002487 : Stern's diatomic series
https://oeis.org/...
- 317
1
vote
1
answer
267
views
Adding $n$-tuples over groups
Consider a finite abelian group $\mathcal{G}$. Let $S_0$ be a $n$-tuple of elements of $\mathcal{G}$, and let $S_i$ be the cyclically shifted version of $S_0$ by $i$ indices to the right. So for ...
- 431
0
votes
0
answers
112
views
Definition of "tame $p$-part of $\chi$"
At the beginning of Haruzo Hida's article "Big Galois representations and $p$-adic L functions", he has defined $$\chi_1= \textrm{the } N_0 \textrm{-part of} \chi \times \textrm{the tame } p\textrm{-...
- 337
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
115
views
On group varieties and numbers
Suppose $\mathfrak{U}$ is a group variety. Let’s define $N_{\mathfrak{U}} \subset \mathbb{N}$ as a such set of numbers, that for any finite group $G$, $|G| \in N_{\mathfrak{U}}$ implies $G \in \...
- 2,618
6
votes
1
answer
2k
views
Are there infinitely many insipid numbers?
A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ ...
12
votes
1
answer
455
views
Is there a classification of finite simple groups of perfect power order?
The finite simple group $\operatorname{PSp}(4,7)$ has order $138297600 = 11760^2$.
There also seems to be a description of the $q$ such that $\operatorname{PSp}(4,q)$ has square order, see for example ...
- 2,821
1
vote
0
answers
127
views
Lower and upper bounds of the distance between two Frobenius numbers
I consider two sequences of numbers: $A=\{a_1,...,a_{m-1},n\}$ and $B=\{n-a_{m-1},...,n-a_1,n\}$, where $a_1 < a_2 < ... < a_{m-1} < n$ and $\gcd(A) = \gcd(B) = 1$.
I investigate the ...
- 13
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
0
votes
1
answer
223
views
Greatest common divisor of two specified sequences of numbers (search for equality)
I consider two sequences of numbers $A=\{a_1,...,a_n\}$ and $B=\{k-a_1,...,k-a_n\}$, where $a_1 \le a_2 \le ... \le a_n \le k$.
I am looking for such conditions under which: $\gcd(a_1,...,a_n) = \gcd(...
- 13
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
1
vote
0
answers
69
views
On equality of two quotients of a congruence subgroup
Related question: Non-torsion part of the abelianisation of congruence subgroups
Let $A = \mathbb{F}_q[T]$ be the ring of polynomials with coefficients in a finite field, with $N$ a nonconstant ...
- 345
1
vote
0
answers
140
views
Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$?
Does there exist an unramified $PSL_2(\mathbb{F}_p)$ extension of a quadratic field $K$ for $p\geq 5$?
user130124
5
votes
1
answer
360
views
Is the following variant of Shafarevich's theorem known?
Let $Q$ be a finite simple group which may be realized as the Galois group of some extension of $\mathbb{Q}$ (like for instance $PSL_2(\mathbb{F}_p)$ for $p\geq 5$, or the monster group) and let $G$ ...
user130124
11
votes
2
answers
577
views
About normal minimal subgroups not in the Frattini
In Neukirch--Schmidt--Wingberg, "Cohomology of Number Fields", Second edition, page 624, Exercise 2, it is stated the following fact.
$\textbf{Claim}$: If $N$ is a normal subgroup, minimal among ...
- 113
7
votes
1
answer
332
views
Conjectured combinatorial non-equality
Let $n,k,\ell$ be integers for which $0\leq k<\ell \leq n-6$. For a fixed $n$, think of $k,\ell$ as being allowed to vary. I believe the values
$$(n-k-5)(k+1)(k+2)\binom n{k+3}~~~\text{and}~~~(n-...
- 1,068
0
votes
1
answer
283
views
Is any abelian subgroup of a semidirect product isomorphic to a direct product of abelian subgroups? [closed]
Let $H$ and $K$ be groups and $V$ an abelian subgroup of the semidirect
product $\ H\rtimes K$. Do there exist abelian subgroups $H^{\prime }\leq H$
\ and $K^{\prime }\leq K$ \ such that $V\cong H^{\...
- 463
5
votes
0
answers
117
views
$m$-thick sets with small $n$-fold sumsets in finite cyclic groups
Problem. Is it true that for every positive integers $n,m$ there exists a subset $A_{n,m}$ of a finite cyclic group $G$ having the following two properties:
$(\Sigma_n)$ the $n$-fold sum $A_{n,m}^{+...
- 41.8k
4
votes
0
answers
159
views
does infinite leinster groups indicates infinite perfect numbers?
The simplest definition i managed to find is: Leinster group is a finite group which her order equal the sum of its normal subgroups order, and as a perfect number is a positive integer that is equal ...
- 143
11
votes
1
answer
364
views
Can we glue characteristic 0 and characteristic p representations of a finite group given equality of (Brauer) characters?
Suppose I have a prime $p$ and a finite group $G$ together with representations $\sigma: G \to GL_n(\mathbb{Q}_p)$ and $\pi: G \to GL_n(\mathbb{F}_p)$. My question is:
When does there exist a ...
- 45.7k
3
votes
1
answer
157
views
On the Upper Density of $C_2$ in finite groups
We define the upper density $\rho (G)$ of a finite group $G$ as the ratio of the number of finite groups of order $<n$ which contain a subgroup isomorphic to $G$ to the number of groups of order $&...
- 205
18
votes
0
answers
2k
views
$G$ a group, with $p$ a prime number, and $|G|=2^p-1$, is it abelian?
During my research I came across this question, I proposed it in the chat, but nobody could find a counterexample, so I allow myself to ask you :
$G$ a group, with $p$ a prime number, and $|G|=2^p-1$, ...
- 4,074
10
votes
1
answer
302
views
Finite subgroups of Lie group over algebraic ring of integers
I have frequently seen results like:
There are 4 isomorphism types of finite subgroups of $SL_2(\mathbb{Z})$, namely $\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_4,\mathbb{Z}_6$.
I wonder what is known of ...
- 1,846
3
votes
1
answer
314
views
Probability in $GL_2(\mathbb{Z}/p^{r}\mathbb{Z})$
My question may be not interesting or easy to answer ! but I am really not familiar with proba.
Let $p$ be an odd prime number. and let $r\geq1$ an integer. choose an element $A\in\mathrm{GL}_2(\...
- 347
2
votes
1
answer
186
views
Number of homomorphism, or number of solution to equations, in finite groups
Let $G$ be a finite group, and let $P$ be a finitely generated group.
Consider the number $$n=\#Hom_{Grp}(P,G).$$
It is known (see Number of solutions to equations in finite groups) that under ...
- 5,039
36
votes
1
answer
2k
views
Tell me an algebraic integer that isn't an eigenvalue of the sum of two permutations
Can you tell me an algebraic integer, with all archimedean absolute values less than 2, which is not an eigenvalue of $\pi_1 + \pi_2$ for any two permutation matrices $\pi_1,\pi_2$?
Is it ...
- 19.2k
5
votes
1
answer
555
views
Which cyclic groups admit a difference set?
Problem 1. For which $n$ does the cyclic group $C_n$ admit a difference set $D\subset C_n$, i.e., a set such that each non-unit element $x\in C_n$ can be uniquely written as the difference $x=ab^{-1}$...
- 41.8k