Questions tagged [group-theory]

A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory is the study of groups.

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816 views

Geometric way to view the truncated braid groups?

This is perhaps a vague question, but hopefully there exists literature on the subject. The question is motivated by an answer I gave to this question. I also asked a related question on MO, although ...
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942 views

A question about Sylow subgroups and $C_G(x)$

Let $G=PQ$ where $P$ and $Q$ are $p$- and $q$-Sylow subgroups of $G$ respectively. In addition, suppose that $P\unlhd G$, $Q\ntrianglelefteq G$, $C_G(P)=Z(G)$ and $C_G(Q)\neq Z(G)$, where $Z(G)$ is ...
45
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1k views

Subgroups as isotropy subgroups and regular orbits on tuples

Is there some natural or character-theoretic description of the minimum value of d such that G has a regular orbit on Ωd, where G is a finite group acting faithfully on a set Ω? Motivation: In some ...
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477 views

Can Erdős-Turán $\frac{5}{8}$ theorem be generalised that way?

Suppose for an arbitrary group word $w$ ower the alphabet of $n$ symbols $\mathfrak{U_w}$ is a variety of all groups $G$, that satisfy an identity $\forall a_1, … , a_n \in G$ $w(a_1, … , a_n) = e$. ...
32
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444 views

What is $\tau(A_n)$?

Suppose G is a finite group. Define $\tau(G)$ as the minimal number, such that $\forall X \subset G$ if $|X| > \tau(G)$, then $XXX = \langle X \rangle$. What is $\tau(A_n)$? Similar problems for ...
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988 views

Is there a “ping-pong lemma proof” that $\langle x \mapsto x+1,x \mapsto x^3 \rangle$ is a free group of rank 2?

Let $f,g\colon \mathbb R \to \mathbb R$ be the permutations defined by $f\colon x \mapsto x+1$ and $g\colon x \mapsto x^3$, or maybe even have $g\colon x \mapsto x^p$, $p$ an odd prime. In the book, ...
21
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142 views

Iwahori versus Bruhat decompositions

I am faced with the following issue that I do not understand but seems contradictory, coming from the book of Roberts and Schmidt about $GSp(4)$. Consider a local non-archimedean field $F$, let $p$ be ...
19
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364 views

Proving there is no non-abelian finite simple group of order a Fibonacci number

"Prove there does not exist a finite simple non-abelian group of order of a Fibonacci number" I would like to answer the above question, but I currently have few ideas of where to begin. I ...
18
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210 views

Can you make the triviality of $\langle a,b,c \mid aba^{-1} = b^2, bcb^{-1} = c^2, cac^{-1} = a^2 \rangle$ more trivial?

I recently learned the following pleasant fact. (It was in the proof of Proposition 3.1 of this paper - but don't worry, there's no model theory in this question.) Let $G$ be a group, and let $a,b,c\...
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521 views

Description of Levi factors and unipotent radicals of parabolic subgroups in classical groups

For an algebraic group $G$ over an algebraically closed field $k$, a parabolic subgroup $P$ has factorization $P = Q \rtimes L$, where $Q$ is the unipotent radical of $P$ and $L$ is some Levi factor ...
18
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316 views

$f\colon I\rightarrow G$ and Gromov $\delta$-hyperbolicity

Please recall that $\left|\int_0^1 f(t)\,dt -w\right|\leq \int_0^1|f(t)-w|\,dt$. In general, let $(X,d)$ be a metric space. Given a function $f:I\to X$ let $m_f\in X$ be such that $d(m_f,w)\leq \int_0^...
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221 views

$G$ is a group of odd order, show that $a^2=b^2 \Rightarrow a=b$

I've come across this question and I tried to prove it, but my solution seems a little stealthy to me, is it correct? Let $|G|=2k+1$, then we have: $a=ae=aa^{|G|}=a^{|G|+1}=a^{2k+2}=(a^2)^{k+1}=(b^2)...
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232 views

Can we find a bound so that we can conclude $G$ is a $p$-group?

Let $n_p$ be number of the elements of order $p$ in a group $G$. My motivation is that if $n_2\ge\dfrac 34 |G|$ then $G$ is $2$-group. You can check it from this. Is there such general bound for $...
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318 views

How big must the union of a group's Sylow p-subgroups be?

For various orders $n$ it's a common exercise to prove that a finite group $G$ of order $n$ can't be simple by using the Sylow theorems to show that there is some prime $p \mid n$ such that the number ...
15
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372 views

Maximal subgroups that force solvability.

For which finite groups $M$ is it the case that every finite group $G$ with $M$ as a maximal subgroup solvable? If $M$ satisfies this condition then $M$ is solvable. Also, if $M$ is abelian then $M$ ...
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411 views

Groups of order $180$, $540$, $1080$ are not simple.

Here's how I solve the problems. Thanks for pointing out what might be the weakness of my solutions. Actually, what I want are other ways of solving this kind of problems, appart from counting the ...
14
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156 views

If $g$ is commutator then so is $g^m$ for $(m,o(g))=1$

There are certain theorems in finite group theory whose proofs involve character theory and for which there are still no character-free proofs. Among such is Frobenius theorem on transitive ...
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280 views

Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-...
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651 views

Character theory of $2$-Frobenius groups.

Edit Summary: I've posted this on MO and received a partial answer there. Can anybody help me expand on this? Definition. Let $G$ be a finite group and $F_1=\text{Fit}\,G$ and $F_2/F=\text{Fit}\...
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2k views

Fundamental group of a compact manifold

In an article I am currently reading, the author tells us that for compact (finite dimensional topological) manifolds X and finite groups $\Gamma$, the set $$\mathrm{Hom}(\pi_1X,\Gamma)/\Gamma$$ where ...
12
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128 views

Have “groupy” numbers been studied before?

In number theory, a positive integer $n$ is called highly composite if it has more divisors than any smaller positive integer. This notion has been studied by several notable mathematicians; for ...
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241 views

Are all verbal automorphisms inner power automorphisms?

Suppose $G$ is a group. $\DeclareMathOperator{\Wa}{Wa}\DeclareMathOperator{\Tame}{Tame}\DeclareMathOperator{\Aut}{Aut}$ Lets call $\phi \in \Aut(G)$ verbal automorphism iff $\exists n \in \mathbb{N}, ...
12
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592 views

Lowest dimensional faithful representation of a finite group

How does one compute the lowest dimensional faithful representation of a finite group? This question originated in the context of given a finite group $G$: trying to find the lowest dimensional shape ...
12
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2k views

The multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$

Let $G$ be the multiplicative group of all complex $2^n$-th roots of unity, where $n = 0 , 1, 2 , \ldots$. Then assess the following claims: Every proper subgroup of $G$ is finite. $G$ has a ...
12
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284 views

The division algebras arising in the Wedderburn decomposition of a finite group modulo its radical in characteristic $p$

The following question is probably straightforward for those who know. However, I am used to working either over splitting fields or in characteristic zero. Question. Let $G$ be a finite ...
12
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498 views

Higman group with 3 generators is trivial

The group $G$ generated by $x,y,z$ subject to the relations $[x,y]=y$, $[y,z]=z$, $[z,x]=x$ is trivial. This isn't the case for the corresponding group with 4 generators, which is the famous Higman ...
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186 views

Is almost any group generated by two generators?

What is the asymptotic probability that a randomly chosen finite group can be presented with $2$ generators? More precisely, what is $$ \lim _{n \to \infty} \frac{\text{number of 2-generated groups of ...
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240 views

Subgroup structure of $\mathrm{SL}(2, p^2)$ and its irreducible characters

I am taking a course in representation theory of finite groups, and somehow I ended up getting assigned to write a report on subgroup structure and irreducible characters of $\mathrm{SL}(2,7)$ and $\...
12
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200 views

Official name(s) for a certain type of p-group

I'm implementing a class of groups into Sage (sagemath.org), a computer algebra system, and I'm wondering if it has any official names. I found it in Gorenstein's "Finite Groups." It is there called $...
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566 views

What groups are semidirect products of simple groups?

The question I want to ask is actually slightly broader than that in the title: what is the smallest class of finite groups which contains all finite simple groups groups, and is closed under ...
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2k views

Was Atiyah's proof of the odd order (Feit-Thompson) theorem false?

I read last year that Atiyah thought he had found a proof of the odd order theorem of only 12 pages, using $K$-theory, and that people were trying to figure out if it was correct or not. But I never ...
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154 views

$\text{SL}(2, \mathbb{F}_q)$, for which characters is the $G$-representation irreducible?

Followup to here. Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = \text{SL}_2(\mathbb{F})$. The group $G$ acts linearly on the $2$-dimensional vector space $\mathbb{F}^2$ and fixes ...
11
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1k views

Induced representation, Ind(Res(U))

I am reading a book of Fulton and Harris "Representation theory, a first course". Now it's all about representation theory of finite groups, and there is one exercise, which I can't solve: If $U$ is ...
11
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739 views

How strong is the statement that Thompson F is amenable?

Justin Moore's proof turned out to have an error I just attended Justin Moore's talk on this today. Since I am neither a group theorist nor a combinatorist, and is not familiar with ultrafilters I ...
10
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0answers
151 views

Raising to the power of $i$

We all know the usual exponentiation $a^i$ in the complex setting; one can ask is there such a map in a given group; more specifically is there a criterion in the general sense such that given an ...
10
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111 views

Is there a theory of “almost symmetry” generalizing group theory?

Apologies for the inescapably soft question. Does there exist a theory that aims to develop tools analogous to those of group theory, except for the study of objects that are merely almost ...
10
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129 views

Subgroups of $\text{Spin}(7)$.

I am interested in subgroups of $\text{Spin}(7)$ and identifying certain elementary properties that they have. Specifically relating to commutativity. Let me apologise at this point for my ignorance ...
10
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71 views

Small group characterizing identity matrix

I am looking for a small (say, finite and of small cardinality) subgroup of the general linear group whose centralizer consists only of scalar matrices. I work over complex numbers. A more precise ...
10
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175 views

What is this group called?

Let $X$ denote a set. There's a corresponding group obtained by taking the group freely generated by $X^2$ and then quotienting out by the following families of relations: $(x,x) = 1$ $(x,y)(y,z) = (...
10
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260 views

Derived functors of abelianization

The abelianization functor from the category of groups to that of abelian groups is right exact in the sense that it takes a short exact sequence $$1 \to K \to G \to H \to 1$$ to a shorter exact ...
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106 views

Affine group, $[L : \mathbb{Q}] = n\varphi(n)$ or ${1\over2}n\varphi(n)$

Let $L$ be the Galois closure of $K = \mathbb{Q}(\sqrt[n]{a})$, where $a \in \mathbb{Q}$, $a > 0$ and suppose $[K : \mathbb{Q}] = n$. How do I see that $[L: \mathbb{Q}] = n\varphi(n)$ or ${1\over2}...
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327 views

Geometrical interpretation of a group action of $SU_2$ on $\mathbb S^3$

Background There're some nomenclatures from Michael Artin's Algebra to explain. 3-Sphere, or $\mathbb S^3$, is the locus of $x_0^2+x_1^2+x_2^2+x_3^2=1$, where $(x_0,x_1,x_2,x_3)\in\mathbb R^4$. $SU_2$...
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572 views

Examples of finite groups that are not a semidirect product

I'm looking for examples of (families of) finite groups that are not semidirect products. When first learning group theory, the first such group that one encounters is $Q_8$. In my search for other ...
10
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0answers
103 views

Cover and avoidance properties of chief factors: can we avoid exactly the ones we want?

If $G$ is a finite solvable group with chief series $1 = G_0 \leq \ldots \leq G_n = G$ (so each $G_i \unlhd G$ and if $G_{i-1} < H < G_i$ then $H$ is not normal in $G$) and $I \subseteq \{ 1,\...
9
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0answers
104 views

Is there a categorification of “(virtually) solvable”?

If this question doesn't make sense or is otherwise poor quality, then I'm sorry. Motivation: As part of my research, I study virtually solvable (1) groups. These are goups that have a solvable ...
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140 views

Is this specific group finite?

I have the following group presentation: $G=\left\langle a,b,c\ |\ a^2,b^{11},(ab)^{4},(ab^2)^6,ab^2abab^{-1}abab^{-2}ab^2ab^{-1},c^2,(ac)^3,(bc)^2\right\rangle$ Is $G$ finite? GAP's Size(G) runs ...
9
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0answers
115 views

Finite group of “linear substitutions”

From what I can tell, a linear substitution is an operation on a set of variables $x_1,\ldots,x_n$ which sends them to a new set of variables $y_1,\ldots, y_n$ via a linear transformation $$\vec{y} = ...
9
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0answers
215 views

Seeing symmetries

Preliminaries Let $[n] = \{0,\dots,n-1\}$ and $P([n])$ be the power set of $[n]$. Let the correlation between two subsets $x,y$ of $[n]$ be the number $\kappa(x,y) = 1 - \frac{2}{n}|x\triangle y|$ ...
9
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0answers
224 views

Developing Intuition for the Monster Group

Here is my context. I recently finished my undergraduate studies and am moving on to graduate work. I have learned up to and including the Sylow theorems, group actions, and conjugacy classes. Over ...
8
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0answers
158 views

Subgroups of $GL_n$ and group actions

For my abstract algebra class I have to do some exercises concerning group actions of $GL_n(K)$ on the set of flags $$F_n = \{0 \subseteq V_1 \subseteq V_2 \subseteq \ldots \subseteq V_r = K^n \,\vert\...