Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
2,259
questions
14
votes
2
answers
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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 $\mathrm{PSL}(2,7)$.
...
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 $p$-...
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|\...
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 ...
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 $...
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:
$$CI(G) = \begin{cases} 1 & \quad G \text{ is cyclic} \\ \max_{H < G} CI(H) + 1 & \...
12
votes
2
answers
406
views
Does asymmetric fraction of finite groups tend to $0$?
Let’s define asymmetric fraction of a finite group $G$ as the number $$\mathrm{af}(G) = \frac{|\{(g, a) \in G \times \mathrm{Aut}(G)\mid a(g) = g\}|}{|G|\cdot|\mathrm{Aut}(G)|}.$$ Equivalently it can ...
7
votes
1
answer
521
views
Wreath product $S_k\wr S_n$ inside $S_{kn}$
I want to understand wreath products a little better.
Currently, my intuition about them is as follows. Take $nk$ disks, pile them in order forming $n$ piles of $k$ disks (one pile contains disks {1,...
3
votes
0
answers
236
views
The maximal order of an element in a Coxeter group
Let $W$ be a finite Coxeter group. Let
$$
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 ...
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 ...
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 ...
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 ...
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 ...
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
...
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. ...
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 ...
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$, ...
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), ...
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{-...
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}...
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 $S$ ...
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, $|\{ va + b : a \in A,\ b \in A \}|$) ...
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 $\...
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 ...
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 $...
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 ...
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 ...
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 ...
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 ...
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 \...
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 ...
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 ...
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 ...
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 ...
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\...
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 ...
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 ...
-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 ...
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 (...
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|...
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 \...
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 $486$ ...
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 ...
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 ...
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,...
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 ...
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,...
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)...
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 $...
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.)