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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A question about irreducible representation of symmetric group (permutation group) in tensor space and tensor contraction

In chapter 13 of the textbook of Group Theory in Physics by Wu-Ki Tung, Lemma 2 discusses the equivalence of two irreducible representations of GL(m) on ${T^i}_j$. In its proof, it simply mentioned ...
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7answers
513 views

Cayley table group visualization

how can I make graphics like this? random colors. I got a script in GAP that prints rows of numbers but I want it colored just random colors ...
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1answer
23 views

Folner sets in a quotient of a f.g. amenable group

Let $G$ be a finitely generated amenable group. I know that it's a basic result that every quotient of $G$ is amenable. Is it also true that every Folner sequence of $G$ projects onto a Folner ...
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1answer
27 views

Direct decompositions and quotients of abelian groups

Let $G = \langle a \rangle_{27} \oplus \langle b \rangle_{81}$. Find a direct decomposition $G = \langle 10a + 60b \rangle \oplus ?$. Find the elementary divisors of $G/ \langle 3a + 18b \rangle$. ...
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2answers
51 views

irreducible characters of a group

I am currently attempting a past exam paper and am stuck on the following question for part a) $\mu$ is an irreducible character iff it is equal to the character of an irreducible representation, ...
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1answer
48 views

Creating the Cayley table for $\mathbb{Z}_2 \times S_3$

Create the Cayley table for $\mathbb{Z}_2 \times S_3$ I know that the $\mathbb{Z}_2$ is: \begin{array}{c|cc} + & 0 & 1 \\\hline 0 & 0 & 1 \\ 1 & 1 & 0 \\ ...
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2answers
46 views

Looking for examples of finite loops and monoids

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms. Does anyone know some ressources for this? I ...
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1answer
72 views

character tables and solubility

I am currently going through a past exam paper for a group theory module and am unable to answer the following section of a question. The copy of my lecture notes doesn't seem to have a section on ...
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4answers
64 views

Topics in Algebra I.N.Herstein Problem 7

Given that if $A$ and $B$ are cyclic of orders m and n and $\gcd(m,n)=1$ then $A\times B$ is cyclic. Using this prove that if $u,v\in \mathbb Z$ then $\exists x$ such that $x\equiv u(\mod m);x\equiv ...
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1answer
27 views

How do I show that the identity element e is contained in the subset H of a group G?

Suppose the set $G$ is an additive group of integers $(G,+)$. For a subset $H$ of the set $G$ to be a subgroup, the subset $H$ must contain the identity element the subset $H$ must be closed under ...
2
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0answers
55 views

Completing Cayley table for a group

The task is to complete the following Cayley table for a given group. $e$ is of course the identity element. Together with group axioms and the fact that every Cayley table of a group must be a ...
4
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1answer
2k views

Cayley table help

I have a Cayley table with four elements and a binary structure $*$. I know that if I have the same element along the main diagonal (from top right corner to bottom left corner), then the set is ...
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1answer
50 views

Cayley Table Question

How can I figure out the order of the elements in a Cayley table?
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0answers
43 views

On definition and usefulness of “Cayley table”

Dummit-foote p.21 Let $G=\{g_1,\cdots,g_n\}$ be a finite group with $g_1=1$. The Cayley table of $G$ is the $n\times n$ matrix whose (i,j)-entry is $g_ig_j$. This definition is based on an ...
0
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2answers
62 views

Group in which one subgroup is contained in every subgroup

let $G$ be a finite abelian group such that it contains a subgroup $H_0\neq \{e\}$ which is contained in every subgroup $H\neq \{e\}$ of $G$ Prove that $G$ is cyclic.Find $o(G)$ How should I start? ...
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5answers
4k views

Group of order 15 is abelian

How do I prove group of order 15 is abelian? Is there any general strategy to prove that a group of particular order(composite order) is abelian?
2
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0answers
57 views

Showing the existence of $O^{\pi}(G)$

Assume G is a finite group. I am trying to show the existence of $O^{\pi}(G)$, the unique normal subgroup of G minimal such that $G/O^{\pi}(G)$ is a $\pi$-group, i.e. a group whose order is divisible ...
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2answers
544 views

Can I represent groups geometrically?

I have just taken up abstract algebra for my college and my professor was giving me an introduction to groups, but since I like geometric definitions or ways of looking at stuff, I kept thinking, "How ...
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2answers
41 views

what does this notation means in group theory?

Quoting from my text, "Observe that if we let $$\mathbb{R}\times \mathbb{R}$$ denote all ordered pairs of real numbers,....." What does the notation "$\times$" means?
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3answers
40 views

Automorphisms of a direct product of cyclic groups

Let $G=A\times A$; $A$ be cyclic group of order $p$ where $p$ is a prime .How many automorphisms does $G$ have? My thoughts: If we have a cyclic group $G$ of order $n$ then I know that ...
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0answers
30 views

Harmonic Analysis of Finite Groups

If I understand correctly, the basic goal of harmonic analysis on finite groups is to find isotypical subspaces of a given set. Why is it important to do so? What are the advantages of decomposing a ...
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1answer
27 views

Standard definition for $a$ being congurent to $b$ mod $n$

My text puts the definition for $$a\equiv b \bmod n$$ as $$n\mid(a-b).$$ On the other hand, certain sources puts the definition as $$n\mid(b-a).$$ Which exactly is the standard notation or is there a ...
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2answers
73 views

Help to prove that $ U_{p} $ is a cyclic group.

As part of my study of Abstract Algebra, I’m trying to prove that $ U_{p} $ is cyclic for $ p $ a prime number. It’s a classical result, but I’m trying to prove it following four steps stated as ...
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0answers
53 views

Characterization of conjugacy classes of $A_n$: intuition

Note the following theorem (quoted after handouts by Keith Conrad (UoCT) found online): Let $\pi \in A_n$. Its conjugacy class (cc) in $S_n$ remains the same in $A_n$, or it breaks into two cc's of ...
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0answers
34 views

Idemptent of Young Tableaux

I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. I understood that different Young Diagrams corresponds to inequivalent irreducible representations of ...
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3answers
923 views

$(\mathbb{Q},+)$ has no maximal subgroup

I have a problem that I don't have any idea. show that group $(\mathbb{Q},+)$ has no maximal subgroup.
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2answers
83 views

Is every subgroup of a group normal?

Is there a simple example that can be used to show that not every subgroup of a group is normal? thanks,
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2answers
369 views

Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$

Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element?
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2answers
38 views

Theory for describing how mathematical entities will behave [closed]

I've realized that operations like adding and multiplication can mean different things under different situations. For example, the multiplication of real numbers is the same as scaling while ...
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5answers
94 views

Prove every group of order less or equal to five is abelian [closed]

Is it possible to prove that every group of order less or equal to five is abelian? thanks,
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1answer
20 views

Group Actions Question

Suppose that $G$ is a group with $|G| = 32$, which acts on some set $S$. Prove that if $|S|$ is odd, then there exists an element $s \in S$ such that $g(s) = s$ for all $g \in G$. I was going to ...
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0answers
37 views

Sofic groups alternative definition

I am trying to solve exercise two from the following document : http://mtm.ufsc.br/~daemi/soficworkshop/Course%20notes/Lupini%20Lecture%202.pdf I suspect there is an error in the exercise, but I'm ...
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1answer
36 views

$\tilde H_0 \oplus \mathbb Z =\mathbb Z\oplus \mathbb Z$

Generaly $\tilde H_0 \oplus \mathbb Z =H_0$. (reduced homology and homology) I'm interested in the specific case $H_0 =\mathbb Z \oplus \mathbb Z$ or a little more generally $H_0 =\bigoplus_{1\leq ...
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1answer
33 views

Test for a $G$-torsor to be trivial?

I just have a very short question, why is a $G$-torsor trivial precisely when it has a section?
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1answer
38 views

Sum of Inverses of the elements in $\mathbb Z_p^*$

If $p $ is an odd prime and if $1+\frac{1}{2}+\cdots +\frac{1}{p-1}=\frac{a}{b}$ where $a,b $ are integers prove that $p|a$. If $p>3\implies p^2|a$ My Try: Can the problem be interpreted as a ...
1
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1answer
44 views

Is a finite group which is generated by two characteristic abelian subgroup always abelian?

Let $G$ be a finite group. If there exist two characteristic subgroups $H,K$ of $G$ such that $H$ and $K$ are abelian and generate the whole group $G$. Then can we conclude that $G$ is abelian? All ...
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2answers
38 views

Show that if $G$ is a finite cyclic group then $G^*$ is cyclic and $o(G)=o(G^*)$

Let $G$ be a group and $G^*$ be the group of all homomorphisms from a group $G$ to the set $\mathbb C^*$ i.e the group of all non-zero complex numbers. Show that if $G$ is a finite cyclic group then ...
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0answers
17 views

Algebraic determination of asymmetric unit (aka irreducible wedge) in Brillouin zone of lattice

In Solid State physics the reciprocal space is of utmost importance to predict the band structure and thus most of the electrical transport parameters like effective mass, etc. The First Brillouin ...
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1answer
27 views

Groups of order 56 with a normal Sylow 7-subgroup

I am trying to classify the groups of order 56 containing a normal Sylow 7-subgroup. It is easy enough to see that any such group $G$ must be a semidirect product $Z_7 \rtimes_\varphi P_2$ where $P_2 ...
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0answers
22 views

Class preserving Autpmorphism

This time I am having interest in theory of Class preserving Automorphism and central preserving Automorphism and some topics related to Automorphism of groups. So Could you please tell me some ...
5
votes
5answers
231 views

Mapping from $\{1,\ldots,n!\}$ to the symmetric group $S_n$

Is there an easy known bijective mapping formula between the set $\{1,\ldots,n!\}$ and the symmetric group $S_n$? I want to pick a number $k \in \{1,\ldots ,n!\}$ and assign a unique permutation of ...
35
votes
5answers
1k views

What structure does the alternating group preserve?

A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other ...
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1answer
31 views

How can we show the torsion subgroup of a group is pure?

I found a definition of pure subgroup: Let $G$ be an abelian group and $H\leq G$. $H$ is a pure subgoup of $G$ if $\forall h \in H$, if $h$ is divisible by $n$ in $G$, then it is divisible by $n$ in ...
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2answers
28 views

Group Theory: how to find subgroups

I am trying to address my weak points with group theory, and thought I could learn this through an example: Let $G = (\mathbb{Z}_4 \times\mathbb{Z}_6, +)$. Find $3$ subgroups of $G$ of size $12$. ...
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1answer
47 views

On a classification of all the characteristic subgroups of a finite abelian $p$-group.

For any finite abelian group $G$, any $n\mid\exp G$ and any $m\mid\frac{\exp G}{n}$, let $nG[m]:=\{g\in nG\mid mg=0\}$. I wonder if every characteristic subgroup of a finite abelian $p$-group $P$ is ...
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2answers
36 views

Finitely generated modules, related to the Gaussian integers

Let $M = \mathbb{Z}[i] = \{a + bi|a, b \in \mathbb{Z}\}$ be the additive group of the Gaussian integers. Consider the $\mathbb{Z}$-submodules $N_1 = 2M$, $N_2 = (1 + i)M$. I now want to figure out ...
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3answers
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Count the number group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$ ?

I have to count the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$ ? 1 2 3 6 I aware of the formula for calculating group homomorphisms defined on cyclic groups..here ...
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1answer
37 views

Product of a character with an irreducible character a non-negative integer

Why is the inner product of a character with an irreducible character a non-negative integer? I can see that by properties of the inner product it will be non-negative but I cannot see why it would ...
17
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1answer
197 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 ...
5
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2answers
50 views

Must a non-simple group have a normal Sylow subgroup?

In class, one way we're taught to prove a group is not simple is to exhibit a normal Sylow subgroup. I'm wondering if the converse is true, i.e. if a group is not simple, must it have a normal Sylow ...