The study of symmetry: groups, subgroups, homomorphisms, group actions.

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4
votes
1answer
76 views

Embedding $G$ into $G/N \times A$ for some small group $A$

Let $G$ be a finite group and let $N \unlhd G$ be a normal subgroup of $G$. I would like to embed $G$ in $(G/N) \times A$ for some small group $A$. I require the embedding to map $g \in G$ to $(gN, ...
10
votes
4answers
1k views

Finite groups with exactly one maximal subgroup

I was recently reading a proof in which the following property is used (and left as an exercise that I could not prove so far). Here is exactly how it is stated. Let $G$ be a finite group. Suppose ...
6
votes
2answers
721 views

Central elements in the Dihedral Group of order 2n

Continuing my independent journey through "Abstract Algebra" (see this previous question for context and notation), I am working on: If $n = 2k$ is even and $n \geq 4$, show that (a) $z = r^k$ ...
41
votes
3answers
3k views

The direct sum $\oplus$ versus the cartesian product $\times$

In the case of abelian groups, I have been treating these two set operations as more or less indistinguishable. In early mathematics courses, one normally defines $A^n := A\times A\times\ldots\times ...
3
votes
1answer
241 views

$G\otimes \mathbb Q=0$ implies torsion group

let $G$ be an abelian group. suppose that $G\otimes \mathbb Q=0$. Does this imply that $G$ is necessarily a torsion group?
7
votes
2answers
309 views

For $G$ a group and $H\unlhd G$, then $G$ is solvable iff $H$ and $G/H$ are solvable?

I recently read the well known theorem that for a group $G$ and $H$ a normal subgroup of $G$, then $G$ is solvable if and only if $H$ and $G/H$ are solvable. In my book, only the fact that $G$ is ...
1
vote
2answers
162 views

Results for elements in Dihedral Groups: $x \notin \langle r\rangle \Rightarrow rx = xr^{-1}$

As a follow-up to my previous question, the next two exercises state: Use the given generators and relations to show that if $x$ is any element of $D_{2n}$ which is not a power of $r$, then: ...
3
votes
1answer
2k views

Computing the order of elements in Dihedral Groups

I am working through "Abstract Algebra" by Dummit & Foote. Exercise 1.2.1 states: Compute the order of each of the elements in ... $D_6$, $D_8$, and $D_{12}$. I have found that: ...
7
votes
1answer
265 views

Explanation why an abelian tower admits a cyclic refinement

Now that school is wrapping up, I'm trying to crack down and get better at algebra. This proposition from Lang's Algebra loses me at the end. Here is my understanding so far: (Please excuse me if a ...
3
votes
1answer
73 views

$\vert G \vert < \infty$, $A,B,C \leq G$, $B \leq A \Rightarrow \vert A:B \vert \geq \vert C \cap A : C \cap B \vert$

I am stuck with the following problem. I am sure it cannot be that hard since it is intuitively true, but I can't find a way to prove it. Let $A,B,C \leq G$ where $G$ is a finite group. Suppose ...
4
votes
2answers
124 views

Probability and the Symmetric Group

Let $p_{n,k}$ be the probability that a random permutation from $S_n$, the symmetric group of order $n!$, has exactly $k$ fixed points. I am trying to compute $\lim _{n\to \infty}p_{n,k}$. After ...
5
votes
3answers
170 views

What are good ways of understandng a permutation group from a set of generators?

I'm trying to understand the structure of a Rubik's Cube-style puzzle I'm playing with; I have an expression of the solutions as the permutation group generated by four elements of $S_{16}$, each a ...
0
votes
1answer
250 views

Orbit of a subgroup

G is a group acting on S. H is a subgroup of G and let s be anything in S. Is it true that the order of s under action of G is divisible by the order of s under action of H?
3
votes
1answer
147 views

Proof of sylow theorem (1)

Let $G$ be a group of order $n\cdot p^m$. Let $K$ be the set of all subsets of $G$ of size $p^m$, and let $G_j$ be the group stabilizing $j$ in $K$. Why is $\rm{ord}(G_j) \leq |j|$?
5
votes
1answer
174 views

Centralizer of a $p$-element modulo the $p'$-core and conjugacy class sizes in quotient groups

Does $[ G : C_G(x) ] = [ G/K : C_{G/K}(x) ] [ K : C_K(x) ]$ hold for all finite groups $G$ and $p$-elements $x$, where $K = O_{p'}(G)$ is the largest normal subgroup of $G$ with order coprime ...
1
vote
2answers
397 views

The commutator subgroup of a quotient in terms of the commutator subgroup and the kernel

Let $H$ be the normal subgroup of $G$. Is it true that $(H[G,G])/H$ is isomorphic to $[G/H,G/H]$? If so, I want to make a surjective homomorphism $\phi\colon H[G,G]\to [G/H,G/H]$ with Kernel $H$ to ...
5
votes
4answers
245 views

Embedding of finite groups in small $\operatorname{GL}(n,F)$

How does we obtain a small (not necessarily smallest) value of $n$ such that a $p$-group of order $p^m$ can be embedded in $\operatorname{GL}(n,\mathbb{F}_p)$? [For embedding of a finite group $G$ in ...
6
votes
2answers
4k views

What is the meaning of an “irreducible representation”?

What does it mean to talk about the "irreducible representatives of SO(3)"? I'm struggling to understand the concept of irreducible representations. Could someone give a concrete example for someone ...
0
votes
2answers
83 views

order of stabilizer

$G$ is a group which acts transitively on a set $S$. $G_s$ is the stabilizer of $s$. How to see the map from $G/G_s$ to $S$ given by $g*G_s \to g(s)$ is one to one?
21
votes
2answers
729 views

Surprising but simple group theory result on conjugacy classes

I have read that for any group $G$ of order $2m+1$ (odd) with $n$ conjugacy classes, it is always the case that $16$ divides the value $(2m+1)-n = |G|-n$. This seems to me like an astonishing ...
14
votes
3answers
468 views

Finding groups $G$ such that $G \cong \mathscr{S}(G)$

Let $G$ be a group and let $\mathscr{S}(G)$ denote the group of Inner-Automorphisms of $G$. The only isomorphism theorem I know, that connects a group to its inner-automorphism is: $$G/Z(G) ...
7
votes
3answers
664 views

Classifying the irreducible representations of $\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}$

I need help with the following problem: Suppose $n$ is some positive integer, and $n|p-1$. Classify all irreducible representations of $$\mathbb{Z}/p\mathbb{Z}\rtimes \mathbb{Z}/n \mathbb{Z}.$$ ...
2
votes
1answer
360 views

questions about unipotent radical

If G is a linear group, then G has the largest normal subgroup consisting of unipotent matrices. This is called the unipotent radical of G. If G is brought by conjugation in Mn(K) to a block diagonal ...
0
votes
2answers
270 views

Maximal subgroups of a finite group

Let $G$ be a finite group acting on a finite set $X$. If the action is primitive then the stabilizers are maximal subgroups of $G$ (converse also true). Is there any criteria to get maximal subgroups ...
3
votes
3answers
144 views

$G\rtimes_\phi H \cong G\rtimes_\psi H$ when certain automorphisms exist

I'm reviewing for exams and came across this problem from an older exam: Let $G$ and $H$ be groups and let $\phi,\psi : H\to \operatorname{Aut}(G)$ be homomorphisms from $H$ into the group of ...
4
votes
1answer
98 views

Group elements with small centralizer

For any element $x$ of a group $G$, $x$ is centralized by any element of the subgroup $\langle x \rangle$ generated by $x$. Is there a name for, or are there any equivalent descriptions for elements ...
17
votes
2answers
558 views

Categorical description of algebraic structures

There is a well-known description of a group as "a category with one object in which all morphisms are invertible." As I understand it, the Yoneda Lemma applied to such a category is simply a ...
2
votes
1answer
175 views

Finite non-abelian groups such that $Soc(G)=G$

Let $G$ be a finite group. It is possible that $Soc(G)=G$. For example: if $G=\prod_{i=1}^n \mathbb{Z}_{p_i}$ for some primes $p_1,\dotsc,p_n$ (not necessarily distinct), then $Soc(G)=G$. Can this ...
1
vote
2answers
629 views

S4 decompose into D4 and cosets?

Can $S_4$ (symmetry group of $4$) be represented by the union of $D_4$ (dihedral group of $4$) and the cosets (in $S_4$) thereof? If not why not?
2
votes
2answers
509 views

generators of the symplectic group

In Masoud Kamgarpour's paper "Weil Representations" he uses a set of generators for the symplectic group, referring to a book by R. Steinberg which I do not have access to. If it matters at all, I am ...
4
votes
1answer
203 views

Finite groups $G$ such that $Soc(G)$ is abelian and contains normal subgroups only

This question is a generalization of a previous one: Groups such that any nontrivial normal subgroup intersects the center nontrivially (see Jack Schmidt's answer): What finite groups $G$ have the ...
3
votes
1answer
166 views

How to determine whether or not a group is a free group?

How do i show whether or not a group could be free. For example the Reals.
3
votes
1answer
101 views

Is this statement true for divisible Groups?

If $G$ and $H$ are divisible groups each of which is isomorphic to a subgroup of the other, then $G$ is isomorphic to $H$. Here, $G$ and $H$ are abelian groups. Can we assume another adjective ...
1
vote
2answers
312 views

Finding generators of a group $G$; given a homomorphism $h:G \to G'$, and generators for $\text{Ker}(h)$, and $\text{Im}(h)$ are known

Everyone: I am trying to understand how to obtain a set of generators of a group $G$, given a homomorphism $h:G \to G'$ ($G'$ also a group); once we know the generators of $\ker(h)$ and ...
2
votes
2answers
200 views

Conjugation and the semidirect product

Where does the following argument break down? Let $G$ be a group, with $H \triangleleft G$, $K \leq G$ and $H \cap K = 1$. Taking the semidirect product, we have $G \cong H \rtimes K$. Identifying ...
3
votes
1answer
231 views

Group theory SU(2) and so on - a good reference

I studied algebra and group theory at the university about 20 years ago. Lately I've been reading the occasional maths book/article and they mention things like $\rm{SO}(n)$ and $\rm{SU}(2)$ as ...
5
votes
1answer
496 views

Does the quotient manifold inherit the Riemannian structure?

Let $(M, g)$ be a Riemannian manifold, and let $G$ be a group acting freely and properly on $M$. From differential geometry we know that the quotient set $M/G$, i.e., the set of the orbits, is a ...
1
vote
1answer
338 views

Breaking RSA in a special case

This is a part of homework assignment, and I am stuck. The RSA signature is being calculated using Chinese Remainder theorem technique. Find the detailed description here. Public and private keys are ...
4
votes
1answer
237 views

Wikipedia article on Sylow's theorems

(Edit: The wikipedia article is correct. I messed up the notions normalizer and normal closure.) Please take a look at the wikipedia article on Sylow's theorems here, more precisely, at the last ...
2
votes
1answer
262 views

Setting up Brauer character theory

My question relates to p. 147 of Serre's Linear Representations of Finite Groups, where he is setting up the definitions relevant to Brauer character theory. Having fixed an algebraically closed ...
4
votes
2answers
262 views

What does Virtually Z give you?

I have a group, and it has a subgroup of finite index which is isomorphic to $\mathbb{Z}$. My questions are these: -Is my group a semidirect product, $\mathbb{Z} \rtimes K$, or even $\mathbb{Z} ...
1
vote
2answers
564 views

Help on Subgroup and Normal Subgroup

We know that a subgroup N of an abelian group G must be normal. However, is the reverse necessarily true? An illustration would enlighten me. Any resource links or names of illustrative texts ...
3
votes
1answer
107 views

Isomorphism between multiplicative R and R x integers modulo 2

Got this question on a recent exam, and though it may seem trivial, I cannot seem to figure it out. Show that $\mathbb{R}^{*} \cong \mathbb{R} \times \mathbb{Z}/2\mathbb{Z}$. I had one of these ...
1
vote
1answer
522 views

Induced homomorphism by passing to the quotient

Let $G_1$ and $G_2$ be two abelian groups with respective subgroups $N_1$ and $N_2$. and let $f:G_1 \to G_2$ be a homomorphism. Under what conditions is the induced map $f':G_1/N_1 \to G_2/N_2$ ...
3
votes
2answers
309 views

Minimal embedding of a group into the group $S_n$

I am aware this has come up recently (Embedding of finite groups for example) but after searching I haven't found the particular answer I'm looking for. Suppose I know the character table and can ...
0
votes
1answer
83 views

Needed hint for dG

As it is defined; dG is a group generated by all the divisible subgroups of G. So, if the group G is reduced then dG=0. Does it mean that if G is a reduced group then G has no any non zero divisible ...
11
votes
1answer
1k views

The Sum of Perfect Squares

In Symmetry and the Monster, I ran across this interesting fact: Let $\displaystyle f(n) = 1^2 + 2^2 + 3^2 + \cdots + n^2 = \sum_{k=1}^{n} k^2$ Let $x$ be an integer Then $f(n) = x^2$ for only two ...
10
votes
2answers
229 views

Applications of the fact that a group is never the union of two of its proper subgroups

It is well-known that a group cannot be written as the union of two its proper subgroups. Has anybody come across some consequences from this fact? The small one I know is that if H is a proper ...
12
votes
2answers
465 views

Groups such that any nontrivial normal subgroup intersects the center nontrivially

It is well known that if $G$ is a finite p-group and $1 \neq H \unlhd G$ then $H \cap Z(G) \neq 1$. Are there other families of groups with this property?
4
votes
2answers
165 views

order-independent accumulator operations?

This is a followup to an answer I posted on stackoverflow about calculating a cumulative operation. Are there any other invertible operations on integers besides addition (+), subtraction (-) and XOR ...