The study of symmetry: groups, subgroups, homomorphisms, group actions.
45
votes
0answers
806 views
+50
Is there a characterization of groups with the property $\exists N \unlhd G : \not \exists H \leq G \,\,\,s.t.\,\,\,H\cong G/N$?
A common mistake for beginning group theory students is the belief that a quotient of a group $G$ is necessarily isomorphic to a subgroup of $G$. Is there a characterization for groups for which this ...
27
votes
0answers
1k views
Does a four-variable analog of the Hall-Witt identity exist?
Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125):
An amazing commutator formula is the Hall-Witt identity: ...
25
votes
0answers
484 views
Automorphisms inducing automorphisms of quotient groups
Let $G$ be a group, with $N$ characteristic in $G$. As $N$ is characteristic, every automorphism of $G$ induces an automorphism of $G/N$. Thus, $\operatorname{Aut}(G)\rightarrow ...
13
votes
0answers
193 views
What is the algebraic structure of functions with fixed points?
So I just noticed that the set of functions with a fixed point
$$f(x_0)=x_0,$$
are closed under composition
$$(f*g)(x):=g(f(x)),$$
and with $e(x)=x$, the inverible functions even seem to form a ...
12
votes
0answers
121 views
What are the symmetries of a colored rubiks cube?
Technically the symmetry group of the rubiks cube is the symmetry group of the cube with all its label peeled off. The normal rubiks cube with all its faces painted different colors has trivial ...
12
votes
0answers
291 views
The normalizer of $\mathrm{GL}(n,\mathbf Z)$ in $\mathrm{GL}(n,\mathbf Q)$
It seems that the normalizer of $H=\mathrm{GL}(n,\mathbf Z)$ in $G=\mathrm{GL}(n,\mathbf Q)$ is "almost" equal to itself, that is,
$$
N_G(\mathrm{GL}(n,\mathbf Z))=Z(G) \cdot \mathrm{GL}(n,\mathbf ...
11
votes
0answers
80 views
References on the theory of $2$-groups.
Many theorems about odd order $p$-groups fail miserably for $2$-groups. These can range from simple $2$-group exceptions (e.g. Frobenius complements can be either cyclic or generalized quaternion) to ...
11
votes
0answers
159 views
Does the Zariski closure of a maximal subgroup remain maximal?
Let $k$ be an algebraically closed field and let $G\leq\rm{GL}_n(k)$. Assume that $M<G$ is a maximal subgroup (in the abstract group sense). Denote by $\bar{G}^Z$ the Zariski closure of $G$ in ...
11
votes
0answers
188 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 ...
10
votes
0answers
226 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 ...
10
votes
0answers
117 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:
(a) the profinite completion of free (discrete) groups;
(b) the cartesian ...
10
votes
0answers
224 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 ...
9
votes
0answers
90 views
Existence of a certain functor $F:Grpd\rightarrow Grp$
Let $Grpd$ denote the category of all groupoids. Let $Grp$ denote the category of all groups. Are there functors $F:Grpd\rightarrow Grp, G:Grp\rightarrow Grpd$ such that $GF=1_{Grpd}$.
Dear all, I ...
9
votes
0answers
141 views
“Efficient version” of Cayley's Theorem in Group Theory
I'm considering finite groups only.
Cayley's theorem says the a group $G$ is isomorphic to a subgroup of $S_{|G|}$.
I think it's interesting to ask for smaller values of $n$ for which $G$ is a ...
9
votes
0answers
125 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 ...
9
votes
0answers
124 views
Group of isometries of $\mathbb{R}^n$
Determine the group of isometries of $\mathbb{R}^n$ equipped with the
sup metric.
My wild guess is $(a_1,\ldots,a_n)\mapsto(\pm a_{\sigma(1)},\ldots,\pm a_{\sigma(n)})+(c_1,\ldots,c_n)$ where ...
8
votes
0answers
68 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 ...
8
votes
0answers
199 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 ...
8
votes
0answers
392 views
How many associative binary operations there are on a finite set?
I am studying Scott's book Group Theory. In the Exercise $1.1.17$ he asks us to show that if $S$ is a set and $|S|=n$, then there are $n^{\frac{n^{2}+n}{2}}$ commutative binary operations on $S$. But ...
7
votes
0answers
109 views
Covering a group with the conjugates of two subgroups related by an automorphism
Let $G$ be a finite group and $H$ a proper subgroup. Then $G$ is not the union of the conjugates of $H$. This is a standard homework problem; Arturo gives a nice solution here.
It is also not ...
7
votes
0answers
62 views
Way to Tietze's Transformation Theorem
during our knot-theory lecture we have talking about the following theorem:
Given two finite presentations of the same group, one can be obtained from the other by a finite sequence of Tietze ...
7
votes
0answers
76 views
If $H$ is a subgroup of $G$ and $x,y\in G$, what is $xHy$ called?
For a group $G$, its subgroup $H$ and $x,y\in G,$ we call $xH$ a left coset of $H,$ and we call $Hy$ a right coset of $H.$ Is there a special name for sets of the form $xHy$? Is there a name or ...
7
votes
0answers
96 views
Generic properties of $p$-groups
I have the impression that most people in algebra believe the following statement to be true, but I have no reference for it.
Fix a natural number $n$. Consider for each prime $p$ the set of all ...
7
votes
0answers
137 views
Exercise 2.13, I. Martin Isaacs' Character Theory
I am trying to solve the exercise 2.13 in Isaacs' Character Theory Book. However I met some difficulties, let me sketch out what I am thinking so that you may tell me a hint.
The problem 2.13 is ...
7
votes
0answers
95 views
Any affine algebraic group is linear.
It is a well-known result that any affine algebraic group is a closed subgroup of some $\mathrm{Gl}_n(\Bbbk)$. However, I would like to see a proof for that, so I looked it up in various books, more ...
7
votes
0answers
382 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 ...
7
votes
0answers
134 views
When metabelian?
I'm interested in knowing whether certain groups $G$ are metabelian.
In general, my groups $G$ have the following form:
there is an exact sequence $1\to N\to G\to Q\to 1$ where
$N$ is abelian, and ...
6
votes
0answers
94 views
How confident can we be about the validity of the classification of finite simple groups?
The completion of the classification of finite simple proofs was first announced in 1983. However, as late as 2008 minor gaps were still found and closed.
How certain can we be that
The proof of ...
6
votes
0answers
56 views
On the Amalgamated Product
Let $F_2=\langle x,y\rangle$ be Free group of rank 2, and $\mathbb{Z}/2=\langle z : z^2\rangle$ be the cyclic group of order 2. Consider an injection $1\mapsto y$ from $\mathbb{Z}$ to $F_2$ and a ...
6
votes
0answers
67 views
Information-theoretic aspects of mathematical systems?
It occured to me that when you perform division in some algebraic system, such as $\frac a b = c$ in $\mathbb R$, the division itself represents a relation of sorts between $a$ and $b$, and once you ...
6
votes
0answers
73 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 ...
6
votes
0answers
246 views
Resource: Group Theory
There is a website providing recent thesis in finite geometry. Is there any website with collection of recent thesis on (finite) group theory.
I want to see the Ph.D. thesis of Raymond T. Shepherd, ...
6
votes
0answers
419 views
Galois closure of a $p$-extension is also a $p$-extension
I'm working on a problem in Dummit & Foote and I'm quite stumped. The problem reads:
Let $p$ be a prime and let $F$ be a field. Let $K$ be a Galois extension of $F$ whose
Galois group is a ...
6
votes
0answers
205 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 ...
6
votes
0answers
124 views
Virtually cyclic groups
Let $G$ be a group with finite generating set $A$, define the distance $$d(g,h)=\mathrm{min}\{n:gh^{-1}=a_1^{\varepsilon_1}\dots a_n^{\varepsilon_n}, a_i\in A,\varepsilon_i=\pm1\}.$$ Define the ball ...
6
votes
0answers
156 views
Conditions for the Finitely Generated Intersection Property
While going through Gratzer's "General Lattice Theory", I was surprised to learn (via some exercise) that the intersection of two finitely generated subgroups is not necessarily finitely generated. ...
6
votes
0answers
79 views
Can the infinite von Dyck groups be subgroups of $SU(n)$?
I know by constructing some particular cases that I can find unitary matrices $X$, $Y$ and $Z$ such that
$X^m = Y^n = Z^p = XYZ = 1$
with
$$
\frac{1}{m} + \frac{1}{n}+\frac{1}{p} < 1
$$
...
6
votes
0answers
114 views
Connecting cells by line and column permutations in a finite grid
I'd like to know whether the following simple problem has been studied before and if any solution is known.
Let G be a finite (MxN) grid, S a subset of G's cells (the "crumbs"). Two crumbs are said ...
6
votes
0answers
143 views
A problem in transfer theory
This is about problem 5B.1 page 157 in Isaacs' "Finite Group Theory" book. This chapter is definitely giving me trouble. The problem reads:
Let $G$ be a finite group, $P \in \operatorname{Syl}_p(G)$ ...
6
votes
0answers
143 views
Is there a standard category-theoretic way to express a loop or quasigroup?
The standard way to encode a group as a category is as a "category with one object and all arrows invertible". All of the arrows are group elements, and composition of arrows is the group operation.
...
5
votes
0answers
44 views
+100
Amenable group rings embeddable in skew fields
I'm looking for a reference of the following fact:
given a (countable?) amenable group $G$ and a (skew) field $K$, the following are equivalent:
(1) the group ring $K[G]$ is a domain;
(2) $K[G]$ is ...
5
votes
0answers
48 views
Why is $PGL_2(5)\cong S_5$?
Why is $PGL_2(5)\cong S_5$? And is there a set of 5 elements on which $PGL_2(5)$ acts?
5
votes
0answers
46 views
Binary tetrahedral group and $\rm{SL}_2(\mathbb F_3)$
The binary tetrahedral group $\mathbb T$ is an interesting 24-element group. For instance it can be expressed as the subgroup
$$ \mathbb T = \left\{ \pm 1, \pm i, \pm j, \pm k, \dfrac{\pm 1 \pm i \pm ...
5
votes
0answers
174 views
What is the class of languages accepted by DFAs whose transition monoids are transitive permutation groups?
In the Wiki page
A permutation automaton, or pure-group automaton, is a deterministic finite automaton such that each input symbol permutes the set of states. ..... A formal language is p-regular ...
5
votes
0answers
50 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 ...
5
votes
0answers
45 views
Order of profinite groups
So as we know, if $G$ is a profinite group, then the order of $G$ is defined to be
$|G| = \textrm{lcm}(\{G/N\})$
where $N$ runs over all normal subgroups of finite index of $G$. (see here)
...
5
votes
0answers
87 views
How is $\mathrm{PGL}(V)$ a subgroup of $\mathrm{P\Gamma L}(V)$?
I've stumbled upon a strange exercise while reading "Notes on Infinite Permutation Groups" by Bhattacharjee, Möller, Macpherson and Neumann. If you have the book, the exercise is 7(ix) on page 66.
...
5
votes
0answers
69 views
Universal property of $N\rtimes K$
Given groups $N$ and $K$, if $K$ acts on $N$ by \begin{equation}
K\xrightarrow{\theta}\operatorname{Aut_{Grp}}(N),
\end{equation} then we can define a group $N\rtimes_{\theta}K$ whose elements are ...
5
votes
0answers
51 views
Subgroup generated by two noncommuting plane isometries
In the group of orientation preserving isometries of the plane, can we characterize the subgroups generated by two non commuting elements ?
Denote an isometry by
$$ \varphi(z) = e^{i\theta}z +b.$$
In ...
5
votes
0answers
72 views
$H=\langle a,b| a=bab, b=aba\rangle $ and $\frac{H}{A}\cong\ Q_8$
Here is my problem:
Let $$H=\langle a,b| a=bab, b=aba\rangle $$ and $\frac{H}{A}\cong\ Q_8$ wherein $A\leq Z(H)\cap H'$. Show that $H\cong Q_8$.
Working on the elements, I could see that ...
