Questions tagged [finite-groups]
Questions on group theory which concern finite groups.
367 questions
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What are the rank 3 boolean intervals [H,G], with G simple group?
The rank $n$ boolean lattice $B_{n}$ is the subset lattice of $\{1,2, \dots , n\}$.
The lattice $B_{3}$ is the following:
Question: What are the rank $3$ boolean intervals of the form $[H,G]$, with $...
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1
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On $(2,3)$-generation of finite simple classical groups
A group $G$ is called $(a,b)$-generated if $G=\langle x,y\rangle$ for some $x,y\in G$ with $|x|=a$ and $|y|=b$.
I know some of the histories on this problem. For example, in this early paper in 1996 ...
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Finite simple groups with the same numbers of elements of orders p and q
Let $G$ be a nonabelian finite simple group, and let $p$ and $q$ be
distinct prime divisors of the order of $G$. Is it true that the
number of elements of $G$ of order $p$ never equals the number of ...
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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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A question about finite groups (a weak version of the converse of Lagrange theorem) [closed]
Let $G$ be a group of order $n$ and $d$ a positive divisor of $n$. Is it true that
there exists a subgroup of $G$ with order $d$ or $n/d$?
Of course this does not hold in full generality. -- In ...
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Kernel of modular representation of a finite group
Let $G$ be a finite group; let $F$ be a field of characteristic $p > 0$.
If I have an irreducible modular representation $\rho: G \to GL_n(F)$, does $\ker \rho$ contain all the normal $p$-...
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A problem with pointwise stabilizer subgroups of fixed-point subspaces II
Definitions: Let $W$ be a representation of a group $G$, $K$ a subgroup of $G$, and $X$ a subspace of $W$.
Let the fixed-point subspace $W^{K}:=\{w \in W \ \vert \ kw=w \ , \forall k \in K \}$.
Let ...
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How to find quotients of infinite triangle groups or von Dyck groups?
I need the following information about the quotients of infinite triangle (or von Dyck) groups.
(1) Let $G(l,m,n)$ defined as $S^l$ =$T^m$ = $(ST)^n$ = $E$ is the hyperbolic ($1/l+1/m+1/n<1$) ...
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Decomposition into irreducible of a representation of the wreath product $S_d \wr S_m$ (2)
This is a question following Decomposition into irreducible of a representation of the wreath product $S_d\wr S_n$
Let $F$ be the trivial and $S$ be the standard representations of $S_d$ (of ...
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relatively free groups in $Var(S_3)$
Suppose $S_3$ is the symmetric group of order 6. Which elements of the variety $Var(S_3)$ are relatively free?
This question is related to my previous question
Relatively free algebras in a variety ...
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An upper bound for the maximal subgroups at fixed index?
Let us call a subgroup an injective homomorphism between groups.
I warn the reader that a subgroup designates here an inclusion $(H \subset G)$, not $H$ alone.
A subgroup $H \subset G$ is ...
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Does $[H_i , G_i]$ distributive imply $[H_1 \times H_2, G_1 \times G_2]$ modular?
Let $L(G)$ be the subgroup lattice of $G$ and $[H, G]$ an interval in $L(G)$.
A lattice $(L, \wedge, \vee)$ is distributive if $a∨(b∧c) = (a∨b) ∧ (a∨c)$, $\forall a,b,c \in L $, and is modular if ...
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Projective characters with corresponding factor set
The following is just a follow up to my previous question. I have a finite group $H$ with 14 ordinary characters. The Schur multiplier $M(H)\cong 2^2$. Hence the group $H$ will have 3 sets of ...
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Collecting proofs that finite multiplicative subgroups of fields are cyclic
I teach elementary number theory and discrete mathematics to students who come with no abstract algebra. I have found proving the key theorem that finite multiplicative subgroups of fields are cyclic ...
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How do I verify the Coq proof of Feit-Thompson?
I probably don't have the appropriate background to even ask this question. I know next to nothing about formal or computer-aided proof, and very little even about group theory. And this question is ...
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Fun applications of representations of finite groups
Are there some fun applications of the theory of representations of finite groups? I would like to have some examples that could be explained to a student who knows what is a finite group but does not ...
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Are there $n$ groups of order $n$ for some $n>1$?
Given a positive integer $n$, let $N(n)$ denote the number of groups of order $n$, up to isomorphism.
Question: Does $N(n)=n$ hold for some $n>1$?
I checked the OEIS-sequence https://oeis.org/...
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How do you *state* the Classification of finite simple groups?
From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite ...
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How much of the ATLAS of finite groups is independently checked and/or computer verified?
In a recent talk Finite groups, yesterday and today Serre made some comments about proofs that rely on the classification of finite simple groups (CFSG) and on the ATLAS of Finite Groups. Namely, he ...
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Why are the sporadic simple groups HUGE?
I'm merely a grad student right now, but I don't think an exploration of the sporadic groups is standard fare for graduate algebra, so I'd like to ask the experts on MO. I did a little reading on them ...
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What is Atiyah's topological formulation of the odd order theorem?
Here is a quote from an article by Daniel Gorenstein on the history of the classification of finite simple groups (available here).
During that year in Harvard, Thompson began his monumental ...
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Is each squared finite group trivial?
A semigroup $S$ is defined to be squared if there exists a subset $A\subseteq S$ such that the function $A\times A\to S$, $(x,y)\mapsto xy$, is bijective.
Problem: Is each squared finite group ...
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Which small finite simple groups are not yet known to be Galois groups over Q?
The subject line pretty much says it all. To expand just a little bit:
1) What is the smallest simple group that is not yet known to occur as a Galois group over $\mathbb{Q}$? (Variants: not known ...
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Do all exact $1 \to A \to A \times B \to B \to 1$ split for finite groups?
Let $A$, $B$ be finite groups. Is it true that all short exact sequences $1 \rightarrow A \rightarrow A \times B \rightarrow B \rightarrow 1$ split on the right?
In other words, do there exist ...
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Known and fixed gaps in the proof of the CFSG
As the "second-generation" proof of the Classification of Finite Simple Groups is being written up in the volumes by Gorenstein, Lyons, Aschbacher, Smith, Solomon, and others (see e.g. this ...
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Why are Schur multipliers of finite simple groups so small?
Given a finite simple group $G$, we can consider the quasisimple extensions $\tilde G$ of $G$, that is to say central extensions which remain perfect. Some basic group cohomology (based on the ...
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For which $n$ is there only one group of order $n$?
Let $f(n)$ denote the number of (isomorphism classes of) groups of order $n$. A couple easy facts:
If $n$ is not squarefree, then there are multiple abelian groups of order $n$.
If $n \geq 4$ is even,...
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Why does the monster group exist?
Recently, I was watching an interview that John Conway did with Numberphile. By the end of the video, Brady Haran asked John:
If you were to come back a hundred years after your death, what problem ...
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A group-theoretic perspective on Frankl's union closed problem
Here is a group theoretic phrasing of a special case of the union closed conjecture:
Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...
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Applications of Frobenius theorem and conjecture
A theorem of Frobenius states that if $n$ divides the order of a finite group $G$, then the number of solutions to $x^n = 1$ in $G$ is a multiple of $n$. Frobenius conjectured that if the number of ...
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Being a subgroup: proof by character theory
Let me first cite a theorem due to Frobenius:
Let $G$ be a finite group, with $H$ a proper subgroup ($H\ne (1)$ and $G$). Suppose that for every $g\not\in H$, we have $H\cap gHg^{-1}=(1)$. Then
$...
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Lecture notes on representations of finite groups
Next term I am supposed to teach a course on representation of finite groups. This is a third year course for undegrads. I was thinking to use the book of Grodon James and Martin Liebeck "...
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Generating a finite group from elements in each conjugacy class
Is there a finite group such that, if you pick one element from each conjugacy class, these don't necessarily generate the entire group?
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Navigating $\mathbb{Z}/p\mathbb{Z}$
$\newcommand{\Z}{\mathbb{Z}}$Let's consider a silly-looking question first. Consider $\Z/p\Z$. Say I am allowed the two operations $x\mapsto x+1$ and $x\mapsto 2x$. Then, starting from $0$, I can ...
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Guess that group via product queries
Suppose someone (person B) knows a finite group $G$ of order $n$.
You (person A) know only the order $n$,
and that $1$ is the name of the identity element.
The group elements are named $1,2,\ldots,n$ ...
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Are there always more conjugacy classes in the kernel of a morphism to $Z_2$ than not?
Let $G$ be a finite group and let $\phi:G\to Z_2$ be a homomorphism to the group with two elements. Is it always the case that there are more conjugacy classes in the kernel of $\phi$ than conjugacy ...
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Highly transitive groups (without assuming the classification of finite simple groups)
What is known about the classification of n-transitive group actions for n large without using the classification of finite simple groups? With the classification of finite simple groups a complete ...
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Mathieu group $M_{23}$ as an algebraic group via additive polynomials
An elegant description of the Mathieu group $M_{23}$ is the following: Let $C$ be the multiplicative subgroup of order $23$ in the field $F=\mathbb F_{2^{11}}$ with $2^{11}$ elements. Then $M_{23}$ is ...
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Has anyone catalogued the "first generation" proof of the classification of finite simple groups?
It has been estimated that the original proof of the CFSG spans around 15,000 journal pages written by hundreds of authors over most of the 20th century. The GLS project attempted to simplify this ...
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Monstrous Moonshine for Thompson group $Th$?
I. As a background, in Traces of Singular Moduli (p.2), Zagier defines the modular form of weight 3/2,
$$g(\tau) = \frac{\eta^2(\tau)}{\eta(2\tau)}\frac{E_4(4\tau)}{\eta^6(4\tau)}=\vartheta_4(\tau)\, ...
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What is this subgroup of $\mathfrak S_{12}$?
On some occasion I was gifted a calendar. It displays a math quizz every day of the year. Not really exciting in general, but at least one of them let me raise a group-theoretic question.
The quizz: ...
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Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group
Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$?
I have no real ...
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In what sense is the classification of all finite groups "impossible"?
I think there is a general belief that the classification of all finite groups is "impossible". I would like to know if this claim can be made more precise in any way. For instance, if there is a ...
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What is the standard 2-generating set of the symmetric group good for?
I apologize for this question which is obviously not research-level. I've been teaching to master students the standard generating sets of the symmetric and alternating groups and I wasn't able to ...
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What's the maximum probability of associativity for triples in a nonassociative loop?
In a finite nonabelian group, the probability that two randomly chosen elements commute cannot exceed 5/8. One easy proof also makes it easy to find the smallest groups that attain this bound, namely ...
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Has any attempt been made to classify finite groupoids?
I recently stumbled upon the Mathieu groupoid and I found them fascinating.
It appears as a subset of $S_{13}$ which is not closed under multiplication, but it turns out to be a groupoid with 13 ...
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Geodesics in finite groups
It seems that I can generalize a result from compact, connected Lie groups to finite groups, but in order to do so, I need to have some kind of geodesics on finite groups.
Below is a proposition for ...
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Realizing groups as commutator subgroups
What are the groups $X$ for which there exists a group $G$ such that $G' \cong X$?
My considerations:
$\bullet$ If $X$ is perfect we are happy with $G=X$.
$\bullet$ If $X$ is abelian then $G := X \...
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What finite simple groups we can obtain using octonions?
Rearranged on 2017-05-31
What I am missing is a uniform definition of finite simple groups. Especially sporadic groups are difficult to define.
Many of the finite groups are defined using machinery ...
user21230
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A new combinatorial property for the character table of a finite group?
Let $G$ be a finite group and $\Lambda = (\lambda_{i,j})$ its character table with $\lambda_{i,1}$ the degree of the ith character.
Consider the following combinatorial property of $\Lambda$: for ...