Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
2,259
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In how many ways does a Lie algebra decompose as an orthogonal direct sum of Cartans?
For a prime $p$, the Lie algebra $\mathfrak{su}(p)$ can be decomposed into an orthogonal direct sum of $p+1$ Cartan subalgebras as follows. Consider the clock and shift matrices — these are a pair of ...
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Classification of complex irreducible representations of $\mathrm{GL}_n(\mathbb{F}_q)$ [duplicate]
Is there a classification of complex irreducible representations of the group $\operatorname{GL}_n(\mathbb{F}_q)$, where $\mathbb{F}_q$ is a finite field with $q$ elements?
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Irreducible subspaces in the space of functions on Grassmannian acted by $\mathrm{GL}_n(\mathbb{F}_q)$
$\DeclareMathOperator\GL{GL}\DeclareMathOperator\Gr{Gr}$Let $\mathbb{F}_q$ be a finite field with $q$ elements. Let $\Gr_{i,n}(\mathbb{F}_q)$ denote the Grassmannian of linear $i$-dimensional ...
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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 ...
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Which groups can be generated by a single conjugacy class?
How can we characterize the finite groups generated by a subset of a single conjugacy class?
This post asks for well-known families of finitely generated groups generated by a single conjugacy class. ...
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Bounds for the orders of second largest subgroups of $\mathrm{SL}_n(\mathbb F_q)$
$\DeclareMathOperator\SL{SL}$By Patton's thesis, except for a finite number of possibilities, the $(n-1, 1)$ parabolic subgroup, $P$ say, has the largest number of elements among all non-trivial ...
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Finite-maximal subgroups of orthogonal groups
I define a finite subgroup $H$ of a group $G$, finite-maximal if for any $g\in G\setminus H$, $\langle H,g\rangle$ is infinite.
My question is now to find the finite-maximal subgroups of $\mathrm{SO}...
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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 ...
3
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270
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Number of conjugacy classes of pairs of commuting elements
Let $G$ be a finite group and denote by $r_G$ the number of conjugacy classes of pairs of commuting elements, i.e. the cardinality of the following set $$ A_G = \{ c(a_1,a_2) \ | \ a_1,a_2 \in G \text{...
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How small can maximal subgroups be?
Given a finite group $G$, let $p(G)$ denote the number of prime factors
of the order of $G$ (counting multiplicities).
Does there exist a function $f: \mathbb{N} \rightarrow \mathbb{N}$
which grows ...
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The direct product of two proper commuting subgroups of a non-abelian indecomposable finite group G can be equal to G?
Let G be a finite non-abelian indecomposable group. Let $G_1$ and $G_2$ be two proper subgroups of $G$. Assume that $G_1$ and $G_2$ commute, i.e., for every $g_1$ in $G_1$ and $g_2$ in $G_2$ we have $...
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Finite groups and noncommutative algebraic geometry
DISCLAIMER: My relationship with noncommutative algebraic geometry is that of a curious, ignorant bystander. I confess that I know very little about noncommutative algebraic geometry, but I am ...
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Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?
Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
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Are the alternating pentultimate and the $2$-alternating pentultimate isomorphic?
Consider a physical puzzle which is in the shape of a dodecahedron where the pieces that move are the face centers and you can slice it through a plane parallel to two opposite faces going through the ...
4
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Polynomials of growth for finite Heisenberg groups
Take a standard finite Heisenberg group with two standard generators and let's consider its growth polynomial - the polynomial which coefficients are equal to the sphere sizes.
For example for $H_3(Z/...
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Embedding $\mathrm{SL}_n(3)$ into $\mathrm{SL}_n(p)$
$\DeclareMathOperator\SL{SL}$Let $p$ be an odd prime. It is easy to show that $\SL_2(3)$ can be embedded into $\SL_2(p)$.
Now, let $n$ be an integer larger than $2$.
Question: In which circumstances, $...
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Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?
Let $G$ be a finite group generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation.
The corresponding Hurwitz representation $V_{\text{Hur}}$ has character $$\...
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56
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Involutions in $\operatorname {PSO}(4,K)$
In the algebraic group $G=\operatorname {PSO}(4,K)$ where $K$ is an algebraically closed field of an odd characteristic, how many different classes of involutions are there and what are the ...
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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 ...
4
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3
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547
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Regular orbits for automorphisms of finite simple groups
Let $G$ be a finite group and $f$ be an automorphism of $G$. We say that $f$ has a regular orbit if there exists $x\in G$ such that $|x^f|=|f|$. If $G$ is abelian it is known that every automorphism ...
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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 ...
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What are double groups mathematically?
In physics and chemistry, there is the concept of double groups. These are double covers of the usual point groups, obtained by "adding an element $R$, which represents rotation by $2\pi$" ...
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Exact structures on representations of a finite group
For simplicity assume $G$ is a (finite) $p$-group, and $k$ is field of characteristic $p$, so that there exists a unique simple $kG$-module the trivial module $k$. I am looking for a class of short ...
3
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Nonisomorphic central products on the same pair of groups?
A central product of two groups $G$ and $H$ is determined as follows. The groups $G$ and $H$ have respective central subgroups $A$ and $B$ which are isomorphic, let $\delta:A\rightarrow B$ be such ...
3
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259
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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 ...
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94
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An explicit matrix form in the symplectic group
In the algebraic group $G=\operatorname {PCSp}(2^{r},K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[...
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An explicit matrix form
In the algebraic group $G=\operatorname {PCGO}(2m,K)$ where $K$ is an algebraically closed field of an odd characteristic, there is a conjugacy class of involutions with representative:
$$
e=\left[
\...
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Distribution of 2-groups
In the family of finite groups of order less than $2000$, there are about 99% of order $1024$, so I have a question about $2$-groups:
Let $f(n)$ be the number of non-isomorphic finite groups of order $...
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Sylow subgroups of the restricted Burnside group $\mathrm{RB}(d,n)$?
$\DeclareMathOperator\RB{RB}$What is known about the Sylow subgroups of the restricted Burnside groups $\RB(d,n)$ ?
I am looking for a reference.
In fact my question is slightly more general. Recall ...
6
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Classification of non-abelian simple groups with cyclic T.I. Sylow p -subgroup
Let $G$ be a finite group and $H$ a subgroup of $G$. We say that $H$ is a trivial intersection (for short T.I.) subgroup of $G$
if $H\cap H^x=1$ for each $x\in G-N_G(H)$.
I read the next result in the ...
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Irreducible projective representations of finite abelian groups
I want to know if there is a description of all irreducible complex projective representations of an arbitrary finite abelian group. I have seen this for particular cases such as those given here and ...
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Are there infinitely many simple integral fusion rings of rank $4$?
$\DeclareMathOperator\ch{ch}$$\DeclareMathOperator\FPdim{FPdim}$We refer to [EGNO15, Chapter 3] for the notion of fusion ring and basic results. The type of a fusion ring $R$ is the list $(\FPdim(b_i)...
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Finite groups with bounded centralizers
Let $G$ be a finite group. For each $x\in G$, the centralizer $\mathbf{C}_G(x)$ must contain $\langle x\rangle$.
QUESTION: What are some interesting results of the following form:
Given some bound on $...
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Automorphisms of (nilpotent) groups : torsion cokernel on the abelianisation implies torsion cokernel on the center?
$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\End{End}\DeclareMathOperator\GL{GL}\DeclareMathOperator\Z{Z}\DeclareMathOperator\Sp{Sp}\DeclareMathOperator\Span{Span}\DeclareMathOperator\ker{ker}\...
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Splitting of $\mathbb{Z}/p\to E\to (\mathbb{Z}/p)^n$ in cohomological terms
Let $d>1$ be an odd integer. Given a simplicial set $X$ and $[\gamma]\in H^2(X,\mathbb{Z}/d)$, there exists a fibration $N\mathbb{Z}/d\to E\to X$, with $$E= X_\gamma:=N\mathbb{Z}/d\times_{\gamma} X....
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Diameter of the "Masterball-puzzle" permutation groups by a kind of Cartier-Foata enumeration?
There is an wonderful blog post by Jordan S. Ellenberg SHOULD YOU BE SURPRISED BY THE DIAMETER OF THE NXNXN RUBIK’S GROUP?. Which explains how one can come to $N^2log(N)$ estimate of the diameter of ...
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Sylow subgroups of the free product of profinite groups
I am interested in the Sylow subgroups of the profinite completion of a free product of finite groups.
Is the following naive expectation true ? I assume things like this should be well-known, and am ...
5
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Number of $k$-tuples of elements generating a cyclic group
Let $k$, $m$ be natural numbers, and $C_m:=\mathbb{Z}/ m \mathbb{Z}$ be the cyclic group of order $m$.
Let $N_{k, \, m}$ be the cardinality of the following set: $$\{(a_1, \ldots, a_k) \in (C_m)^k \; ...
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Distance distribution for Cayley graphs of the fintie Heisenberg groups H3(Z/nZ) approaches Gaussian for large "n"?
I wonder several questions about Cayley graphs of finite Heisenberg groups H3(Z/nZ).
Question 1: do we know the diameter dependence on "n", at least for the standard choice of generators ? ...
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Proof of CFSG assuming every simple group is two-generated
It is well-known that one of the corollaries of the classification of finite simple groups (CFSG) is that every finite simple group can be generated by two elements. In a comment on an answer to an ...
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Random walk on N-Rubik cube group is going like sqrt(number of moves) or linear (number of moves) or? "commutative" vs. "free"(like) group pattern?
Consider higher (NxNxN) Rubik's cube group, with specific set of generators described below.
What is important - that there are huge COMMUTING subsets of generators.
Question: Consider a random walk ...
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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 ...
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N(H)/H and the Weyl group
Let $ H $ be a connected subgroup of $ G=\mathrm{SU}(n) $ such that $ N_G(H)/H $ is finite. Is $ N_G(H)/H $ always a subgroup of the symmetric group $ \mathrm{S}_n $?
I just noticed this from the ...
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Can modular representation theory be used to prove Sylow's existence theorem?
Edit 20/12: I added a more precise question at the bottom of the post.
Given a finite group $G$ and a prime $p$, we want to prove that $G$ has a $p$-subgroup $P$ such that $|G:P|$ is not divisible by $...
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Conditions of $N \triangleleft G$ such that the quotient $G/N$ has trivial centre
Let $G$ be a finite group whose centre is trivial and let $N$ be its normal subgroup. What does $N$ need to satisfy such that the quotient $G/N$ has a trivial centre?
References to any literature ...
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988
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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 ...
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Almost simple groups and their involutions without CFSG
Suppose $A$ is a finite almost simple group (meaning, by definition, that there exists a finite simple group $P$ such that $P \leq A \leq \mathrm{Aut}(P)$). Suppose furthermore that $A$ acts $2$-...
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Finitely generated G, such that x^3 = 1 for all x, is finite? [closed]
x^3 = e for any element x in finitely-generated group G. How to prove that G is finite?
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Minimal irrep of $\mathrm{SL}(2,2^r) $
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}$
This is in some ways a follow up post to Minimal irrep of $\mathrm{PSL}(2,p) $.
The ...
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Explicit character tables of non-existent finite simple groups
In connection with the historical development of the classification of finite simple groups, I am interested in a particular aspect that seems to be less well-documented than the main narrative of ...