Questions tagged [finite-groups]

Questions on group theory which concern finite groups.

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On the finite simple groups with an irreducible complex representation of a given dimension

This answer of Geoff Robinson shows that a finite simple group admits an irreducible complex representation (irrep) of dimension $3$ if and only if it is isomorphic to $A_5$ or $\mathrm{PSL}(2,7)$. ...
Sebastien Palcoux's user avatar
0 votes
1 answer
240 views

Number of commuting pairs in p-group

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 set $U$ is a Sylow $p$-...
Nourr Mga's user avatar
  • 171
11 votes
0 answers
248 views

Which irreducible representations of the symmetric group are eigenspaces of class sums?

In the setting of complex representations of finite groups, a class sum $1_C=\sum_{g\in C} g$ acts on an irreducible representation $V$ as $\lambda(C,V)\operatorname{Id}$, where $\lambda(C,V)=|C|\...
Hjalmar Rosengren's user avatar
1 vote
0 answers
171 views

For a finite subgroup of an infinite group $G$, does its normal closure have infinite index in $G$?

For a finite subgroup of an infinite group $G$, does its normal closure have infinite index in $G$? Or if not, is it true when we replace $G$ by some subgroup? That is: Let $H$ be a finite ...
Lukas Braun's user avatar
1 vote
0 answers
33 views

Do all maximal verbal series have the same length?

Suppose, $G$ is a finite group. Let’s call a series of subgroups of $G$ $\{H_k\}_{k=1}^n$ a verbal series of length $n$, iff $H_1 = E$, $H_n = G$ and $\forall k < n$ $H_k$ is a verbal subgroup of $...
Chain Markov's user avatar
  • 2,618
4 votes
0 answers
77 views

Finite groups of cyclicality index $3$

Suppose $G$ is a group. Let’s define the cyclicality index of $G$ using the following recurrent relation: $$CI(G) = \begin{cases} 1 & \quad G \text{ is cyclic} \\ \max_{H < G} CI(H) + 1 & \...
Chain Markov's user avatar
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12 votes
2 answers
406 views

Does asymmetric fraction of finite groups tend to $0$?

Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
Chain Markov's user avatar
  • 2,618
7 votes
1 answer
521 views

Wreath product $S_k\wr S_n$ inside $S_{kn}$

I want to understand wreath products a little better. Currently, my intuition about them is as follows. Take $nk$ disks, pile them in order forming $n$ piles of $k$ disks (one pile contains disks {1,...
thedude's user avatar
  • 1,417
3 votes
0 answers
236 views

The maximal order of an element in a Coxeter group

Let $W$ be a finite Coxeter group. Let $$ N_W=\operatorname{max}_{g\in W}\operatorname{ord}(g) $$ where $\operatorname{ord}(g)$ denotes the order of an element $g$. By Fermat's little theorem, we ...
Christoph Mark's user avatar
4 votes
0 answers
173 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 ...
jjcale's user avatar
  • 2,768
25 votes
3 answers
3k views

In what sense is SL(2,q) "very far from abelian"?

I am far from an expert in this area. I'd be grateful if someone could explain in what sense $\mathop{SL}(2,q)$ is "very far from abelian," to quote Emanuele Viola? Why does Theorem 1 (below) justify ...
Joseph O'Rourke's user avatar
3 votes
0 answers
93 views

A question to the derived length in modular group algebras

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...
Sven Wirsing's user avatar
1 vote
0 answers
22 views

Exponents in unit groups of modular group algebras

Let $p$ be a prime number, $G=:G_1$ a (non-Abelian) finite p-Group and $K$ a finite field with $\operatorname{char}(K)=p$. It is well-known that the group $G_2:=1+\operatorname{rad}(KG)$ is a p-group ...
Sven Wirsing's user avatar
1 vote
0 answers
255 views

On normalized 2-cocycle

Let $G$ be a group acts trivially on an abelian group $A$. Let $\varepsilon $ be a normalized 2-cocycle in $ Z^{2}(G,A)$. Assume that $G=H_{1} \times H_{2}$ and let $\varepsilon_{1}=res_{H_{1}\times ...
Nourr Mga's user avatar
  • 171
2 votes
0 answers
100 views

Check rapidly if a map of finite groups is a homomorphism

I have two non-abelian finite groups $G$ and $H$ of equal size $n$ and a map $f$ from the underlying set of $G$ to the underlying set of $H$. I store the groups as their multiplication tables (i.e. ...
Zhang's user avatar
  • 21
6 votes
0 answers
254 views

Diameter of finite rational matrix groups

Suppose $G$ is a finite subgroup of $\mathrm{GL}(n,\mathbb{Q})$. For a set $\mathcal{M} \subseteq G$ that generates $G$, define the $\mathcal{M}$-diameter $\mathit{diam}(G, \mathcal{M})$ of $G$ to be ...
Stefan Kiefer's user avatar
1 vote
0 answers
65 views

Number of conjugacy classes of unit groups of modular group algebras

Let $n$ be a natural number, $p$ a prime number, $G$ a finite $p$-group and $K$ a finite field with $p^n$ elements. We focus on the group $1+J(KG)$, where $J(KG)$ is the Jacobson radical of $KG$, ...
Sven Wirsing's user avatar
0 votes
0 answers
101 views

In a Group as a category C with one object, How is the bifunctor ⊗ : C × C → C defined on morphisms?

I know that for the one object {1,2,3}⊗{1,2,3} = {1,2,3}. But what is (1,2) ⊗ (2,3)? hom(A⊗B,C) ≅ hom(A,hom(B,C)) so for A = B = C hom(C⊗C,C) ≅ hom(C,hom(C,C)), so if the group is S3 = hom(C,C), ...
user145873's user avatar
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{-...
tanjia's user avatar
  • 337
2 votes
0 answers
102 views

Lattices with trivial coinvariants for finite groups

Let $G$ be a finite group. A $\mathbb{Z}G$-lattice is a $\mathbb{Z}G$-module that is (as abelian group) a free abelian group of finite rank. Question: Is there a finite group $G$ and a $\mathbb{Z}...
tj_'s user avatar
  • 2,160
5 votes
1 answer
262 views

Diameter of Cayley graphs of finite simple groups

Babai, Kantor and Lubotzky proved in 1989 the following theorem (Sciencedirect link to article). THEOREM 1.1. There is a constant $C$ such that every nonabelian finite simple group $G$ has a set $S$ ...
khers's user avatar
  • 235
3 votes
0 answers
140 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, $|\{ va + b : a \in A,\ b \in A \}|$) ...
Daira-Emma Hopwood's user avatar
6 votes
1 answer
211 views

Number of irreducible representations of $SO_3(\mathfrak{o}/\mathfrak{p}^l)$

$\DeclareMathOperator\SO{SO}$Let $F$ be a finite extension of $\mathbb{Q}_p$, and let $\mathfrak{o}$ denote the ring of integers, with maximal ideal $\mathfrak{p}$. Let $G_l$ denote the finite group $\...
nikola karabatic's user avatar
4 votes
0 answers
268 views

What are the normal subgroups of the finite Coxeter Groups of type Bn?

Let $B_n = \langle \rho_0,\rho_1,\ldots,\rho_{n-1} \rangle$ subject to the relations that $(\rho_i\rho_j)^{m_{i,j}} = id$ with $m_{i,i} = 1$, $m_{i,j} = 2$ for $|i-j|\ge 2$, $m_{i,i+1} =3$ for $0 \le ...
Rob Nicolaides's user avatar
8 votes
2 answers
554 views

How many minimum generating sets are there in a finite group?

Let $G$ be a finite group of order $n$. A generating set in $G$ is said to be minimum if it has minimal size. Is there a known lower bound on number of minimum generating sets in a group of order $...
user avatar
3 votes
1 answer
269 views

Conjugacy classes of rational tori in Symplectic group

Rational conjugacy classes of Frobenius stable tori (in a finite group of Lie type) are in bijection with Frobenius-conjugacy classes of the corresponding Weyl group.When the group is the Symplectic ...
Hans Pohl's user avatar
1 vote
1 answer
107 views

Some quantities associated to finite dimensional Hopf algebras

let $(H,\Delta,m,s)$ be a Hopf algebra. To this Hopf algebra one can associate two obvious linear maps $T_H, S_H: H \to H $ with $T_H=m\circ \Delta,\quad S_H=s$. Are there two finite dimensional ...
Ali Taghavi's user avatar
6 votes
3 answers
405 views

Subgroup generated by a subgroup and a conjugate of it [closed]

Let $H\leq G$ be groups, and $a\in G$ so that $\langle H,a\rangle=G$. Does it follows that $\langle H\cup aHa^{-1}\rangle$ is a normal subgroup of $G$? My hope is that this is true, and my guess is ...
Iras's user avatar
  • 61
5 votes
0 answers
144 views

Explicit description of the smallest class of groups, that contains all finite simple groups and is closed under semidirect products

Suppose $\Pi$ is the smallest class of groups satisfying the following conditions: All finite simple groups lie in $\Pi$ If $G \cong H \rtimes K$ and both $H$ and $K$ are in $\Pi$, then $G$ is also ...
Chain Markov's user avatar
  • 2,618
3 votes
0 answers
114 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 \...
Chain Markov's user avatar
  • 2,618
5 votes
0 answers
196 views

Are finite groups of exponent $d$ rare for $d \neq 4$?

Is there a way to prove, that $\lim_{n \to \infty} \frac{\text{the number of all groups of exponent }d \text{ and order less than }n}{\text{the number of all groups of order less than } n} = 0$ for $d ...
Chain Markov's user avatar
  • 2,618
3 votes
0 answers
135 views

A question on a result of Imre Ruzsa concerning sum-sets

Th main result of this preprint of Imre Ruzsa implies the following Corollary (Ruzsa): For every $r\in\mathbb N$ there exists a real number $\alpha<1$ and a positive integer $m$ such that for ...
Taras Banakh's user avatar
  • 40.7k
5 votes
1 answer
487 views

Is there some sort of classification of finite groups that force solvability?

Suppose $G$ is a finite group. We will say, that it force solvability if any finite group $H$, such that $G$ is isomorphic to its maximal proper subgroup, is solvable. Does there exist some sort of ...
Chain Markov's user avatar
  • 2,618
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 ...
Buturlakin Alexander's user avatar
11 votes
2 answers
699 views

A criterion for finite abelian group to embed into a symmetric group

Let $G$ be a finite abelian group. Write $G\approx \mathbb{Z}/p_1^{i_1}\mathbb{Z}\times\dots \mathbb{Z}/p_m^{i_m}\mathbb{Z}$, with $m\ge 0$, $p_1,\dots,p_m$ primes (not necessarily distinct) and $i_k\...
user avatar
9 votes
1 answer
347 views

Is there a way to prove, that $2$-generated groups are rare among finite groups?

Is there a way to prove, that $\lim_{n \to \infty} \frac{\text{the number of all } 2 \text{-generated groups of order less than }n}{\text{the number of all groups of order less than } n} = 0$? This ...
Chain Markov's user avatar
  • 2,618
2 votes
1 answer
275 views

What finite simple groups appear as factors of surface fundamental groups?

Let $\Sigma_g$ be the a closed orientable surface of genus $g$. My somewhat naive question: what is known about simple finite factors of $\pi_1(\Sigma_g)?$ In particular, I know that the composition ...
Dmitry Vaintrob's user avatar
-2 votes
1 answer
181 views

What do you call continous transformations that preserve the finite group structure?

A number of years ago I studied a preon model (Journal of Mathematical Physics 38:3414-3426, 1997) in which the preons interacted like group elements. I thought it curious that you could sometimes ...
James Bellinger's user avatar
9 votes
2 answers
793 views

Groups without factorization

A group G is said to have a factorization if there exist proper subgroups $A$ and $B$ such that $G = AB = \{ ab \ | \ a \in A, b \in B \}$. The paper Factorisations of sporadic simple groups (...
Sebastien Palcoux's user avatar
0 votes
1 answer
92 views

Confusion on translating k-fold transitivity of groups from Endliche Gruppen by Huppert

The definition 1.7 from Endliche Gruppen, B.Huppert, vol-I, Chap.II, Pg.148 is as follows: Die Permutationsgruppe $\mathfrak G$ auf der Ziffernmenge $\Omega$ heißt $k$-fach transitiv $(k \leq |\Omega|...
Siddhartha's user avatar
4 votes
0 answers
190 views

A group-theoretical analogous of Temperley-Lieb-Jones subfactor planar algebras

The Temperley-Lieb-Jones subfactor planar algebra $\mathcal{TLJ}_{\delta}$ admits the following properties: maximal, it exists for every possible index, i.e. $\delta^2 \in \{4cos^2(\pi/n) \ | \ n \...
Sebastien Palcoux's user avatar
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 $486$ ...
Sebastien Palcoux's user avatar
1 vote
1 answer
416 views

The sporadic numbers

Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups. By GAP, the set of all the ...
Sebastien Palcoux's user avatar
3 votes
0 answers
125 views

Are all exceptional Schur covers sub-sporadic?

Famously, all but finitely many finite simple groups are (cyclic or alternating or) of Lie type. The groups of Lie type have central extensions coming from the simply connected covers of the ...
Theo Johnson-Freyd's user avatar
3 votes
1 answer
181 views

Maximal factorization of finite simple groups and no extra intermediate

The book The maximal factorizations of the finite simple groups and their automorphism groups (by Martin W. Liebeck, Cheryl E. Praeger and Jan Saxl) provides a classification of all the triples $(G,A,...
Sebastien Palcoux's user avatar
12 votes
1 answer
432 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 ...
spin's user avatar
  • 2,781
3 votes
1 answer
153 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,...
kneidell's user avatar
  • 993
3 votes
0 answers
133 views

Is there some sort of formula for $t(S_n)$?

Let $G$ be a finite group. Define $t(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > t(G)$ and $\langle X \rangle = G$, then $XXX = G$. Is there some sort of formula for $t(S_n)...
Chain Markov's user avatar
  • 2,618
1 vote
0 answers
107 views

Does $\Sigma$ generate the variety of all groups?

Suppose $FT$ is the class of all isomorphism classes of finite T-groups (a T-group is a group, where all subnormal subgroups are normal). Define, $\Sigma$ as the set of all functions $f$ from $FT$ to $...
Chain Markov's user avatar
  • 2,618
1 vote
1 answer
175 views

On the number of involutions in some groups

How many involutions are there in $O_7(11)$ and $PSp_6(11)$ respectively? (Note that the sizes of the two groups mentioned here are the same.)
user319994's user avatar

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