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 studies groups.

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359 views

Projective profinite groups

I'm reading the first chapter of Serre's Galois Cohomology. On p. 58, He gives two examples of projective profinite groups: the profinite completion of free (discrete) groups; the cartesian product ...
23
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293 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 ...
21
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364 views

On distributivity of lattice of group topologies

Let $\frak L$ be the set of all topologies $\mathcal T$ on $\Bbb Q$ (the additive group of all rational numbers) such that $(\Bbb Q,\mathcal T)$ is a topological group. Then $(\frak L,\subseteq)$ is a ...
21
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600 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 ...
19
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365 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 ...
19
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554 views

Normalizers of automorphism groups

In abstract groups $\Gamma$ the normalizer $N_\Gamma(S)$ of a subset $S\subseteq\Gamma$ is the subgroup of all $x \in \Gamma$ that commute with $S$, i.e. $xS = Sx$, i.e. $x\ y\ x^{-1} \in S $ for all ...
17
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377 views

Kernels in $\mathbf{Top}$

There is a following well-known theorem for abelian categories (at least the ones I know, Ab, $R$-mod and so on... not so familiar with categorical language to be honest) which states the following : ...
16
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347 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 $f,g: \mathbb R \to \mathbb R$ be the permutations defined by $f: x \mapsto x+1$ and $g: x \mapsto x^3$, or maybe even have $g:x \mapsto x^p$, $p$ an odd prime. In the book, by Pierre de la ...
14
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259 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 ...
12
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116 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 ...
12
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135 views

$k[x_1, \dots, x_n]$ free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism.

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...
12
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398 views

$G$ is finite, $A \leq G$ and all double cosets $AxA$ have the same cardinality, show that $A \triangleleft G$

If $G$ is a finite group and $A$ is a subgroup of $G$ such that all double cosets $AxA$ have the same number of elements, show that $gAg^{-1}=A$ for all $g \in G$. Here is my attempt, I guess it's ...
11
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91 views

Commutator Identity and Commutator Subgroup

It is well known that if $G$ is any group in which $[G,G]$ is abelian, then for any $a,b,c\in G$, $$[[a,b],c].[[b,c],a].[[c,a].b]=1$$ Conversely, if this identity holds for all $a,b,c\in G$ then is it ...
11
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104 views

The order of subgroup generated by two distinct elements of order 2

$G$ is a group of order $2$6. If $x$ and $y$ are two distinct elements of order $2$, what could the order of $\langle x,y\rangle$ be? By lagranges theorem, $\langle x\rangle$ and $\langle ...
11
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184 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
11
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114 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 ...
11
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0answers
592 views

What do linearly ordered abelian groups look like?

Recently a post on MathOverflow connected a theorem about ordered abelian groups to the axiom of choice, and it made me want to try and play with these objects a bit. But it appears to me that I ...
11
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566 views

Why is the Monster group the largest sporadic finite simple group?

I do know that there is a "long proof" (I have read that it is +1000pages) that the Monster group is the largest sporadic simple group. My question is: Why can we be sure that there is no other ...
11
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195 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 ...
11
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484 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 ...
11
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167 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 ...
11
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301 views

“Semidirect product” of graphs?

The first subquestion is "has a standard notion of semidirect product been defined in graph theory"? If yes, i'd like to know if the definition i'm gonna give is equivalent to the standard one. I'd ...
11
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1k 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 ...
10
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351 views

Distributing groups of objects into boxes

How can I enumerate the number of ways of distributing distinct groups of identical objects (but various cardinality) into $k$ boxes such that at most one box is empty $(1)$ and no combination of ...
10
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173 views

A group acting on functions of functions of functions

Given a group acting on a set $X$, there is a standard way to define an action of the group on the set of functions of $X$. This can be extended to the set of functions of functions of $X$ as I show ...
10
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162 views

How useful are geometric aspects when studying finite groups?

My newbie impression when studying finite group theory is that geometric aspects are not very prevalent. Cayley graphs play quite some role for visualising finite groups, but compared to the study of ...
10
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168 views

Central Quotients of Finite Groups

There are more than 50 groups of order 48, and among them 16 groups have center of order 2, let $G$ be among such groups. Then $G/Z(G)$ is a group of order 24. What is this group of order 24? There ...
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170 views

Conditions for a group to admit the structure of a Lie group

This question is motivated by a previous one: Conditions for a smooth manifold to admit the structure of a Lie group and wants to be a sort of "converse". Here I am taking an abstract group $G$ and ...
10
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415 views

On group automorphism of subgroup a group $G$

Let $G$ be a group and $H$ be a subgroup of $G$. When is $\rm{Aut}(H)$ a subgroup of $\rm{Aut}(G)$?
9
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171 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 ...
9
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893 views

How do the infinitesimal generators generate the whole compact Lie group?

Suppose that $G$ is a connected compact Lie group, $g(a)\in G$ is its element, and $a=(a_1,a_2,...,a_r)$ is the parameter of the group element $g(a)$. Then the infinitesimal generator of the group can ...
9
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282 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 ...
9
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75 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 \{ ...
9
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301 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 ...
9
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534 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 ...
8
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185 views

Inverse (finite group) isomorphism of a certain form exists

I have been working something in group theory for a long time and I have everything worked out but this one problem. I have reduced that problem to a conjecture. It takes some work to set it up, but I ...
7
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56 views

There exists a polynomial $f \in \mathbb{C}[V]^G$ such that $f|_{O_1} = 0$ and $f|_{O_2} = 1$.

What is this following theorem? Does it have a name? Can somebody provide a reference? It arose in my algebra class today in vague passing. Let $G \to GL(V)$ be a finite-dimensional continuous ...
7
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61 views

$|G:H|=p^n$ means $O_p(H)\leq O_p(G)$?

Let $H\leq G$ (finite group) and $|G:H|=p^n$, ($p$ is a prime number) prove that: $$O_p(H)\leq O_p(G)$$ note: $O_p(G)$ defined as the intersection of all Sylow-$p$ groups in $G$ I try to prove ...
7
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0answers
38 views

Number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$

For $p$ a prime, what is the number of elements of the group $SL_6(\mathbb{Z}/p^k\mathbb{Z})$, $k \ge 1$? I can answer the $k=1$ case. For each element of $SL_6(\mathbb{Z}/p\mathbb{Z})$, there are ...
7
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127 views

index 2 subgroups of the infinite product of Z/2Z

Is it possible to describe all the index 2 subgroups of the group $G = \prod_{i\in \mathbb{N}}\; \mathbb{Z}/2\mathbb{Z}$? For example, one can take the kernel of the $i$-th projection map ...
7
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189 views

Classify groups of order 171

This is a problem from Stanford Algebra Qualifying Exam, Fall 1998. I know the standard way is to use Sylow theorems and semidirect product. $171 = 9\cdot 19$. By Sylow theorems, $n_3|19$ and ...
7
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268 views

Cardinality of set of groups

After analyzing this question I started wondering. Thoughts. Everyone can give a simple example of a countable group $G, |G| = \aleph_0$ which has uncountable number $2^{\aleph_0}$ subgroups, for ...
7
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358 views

A question about the article 'You can't hear the shape of a drum'

I have read the article http://www.math.upenn.edu/~kazdan/425S11/Drum-Gordon-Webb.pdf, , where Gordon and Webb describe in a simple a way the contruction of a pair of isospectral but non isometric ...
7
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111 views

Wreath product (Rotman, p.173)

I have a question about the explanation of wreath product in Rotman's "An introduction to the theory of groups (4th ed.)." Let $D$ and $Q$ be groups, $\Lambda$ be a $D$-set, $\Omega$ be a finite ...
7
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99 views

Automorphisms of $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$

Let $p$ a prime and $n$ a positive integer, What are the outer automorphisms of the finite linear group $GL_2(\mathbb{Z}/p^{n}\mathbb{Z})$? do we know a complete list of them? Is there any thing on ...
7
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79 views

I would like to show that all reflections in a finite reflection group $W :=\langle t_1, \ldots , t_n\rangle$ are of the form $wt_iw^{-1}.$

I would like to show that all reflections in a finite reflection group $W := \langle t_1, \ldots , t_n\rangle$ are of the form $wt_iw^{-1}$ for some $i$ and some $w \in W$ Clearly all such elements ...
7
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123 views

A connection between nonplanar complete graphs and the alternating group?

I went to an undergrad's senior honors thesis presentation a few days ago. She was discussing crossing numbers and mentioned that complete graphs $K_n$ are nonplanar iff $n \geq 5$. ?Coincidentally? ...
7
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272 views

Is there some sort of correspondence between groups and partitions of a set?

Every group action on a set $S$ partitions the set into orbits. Conversely, for every partition of $S$ is there a group action such that the set of orbits of the group action equals the partition? ...
7
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272 views

Combinatorial total space for finitely generated torsion-free groups?

Motivation: I'm an operator algebraist and I'm looking for an answer to the main question in order to build non-trivial spectral triples for a class (as large as possible) of discrete groups. $\to$ ...
7
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304 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 ...