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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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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0answers
29 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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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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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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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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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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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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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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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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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
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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5answers
91 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
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
27 views

What is the Lie group that leaves this matrix invariant?

What is the group that leaves \begin{equation} Y = \bigoplus_{j=1}^{k} \alpha_j \mathbb{I}_{2n_j\times 2n_j} \end{equation} invariant under congruence ($Y = XYX^T$) where $\alpha_j \in \mathbb{R}$ ...
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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 ...
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136 views

graph product that commutes with automorphism, and semi direct

Is there a way to construct a "product" of graphs $G\rtimes H$ such that $Aut(G \rtimes H ) \simeq Aut(G) \rtimes Aut(H) $? A related topic is "Semidirect product" of graphs? but not quite ...
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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
16 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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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 ...
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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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2answers
28 views

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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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
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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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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23 views

Show $\mathbb Z_{p^\infty}$ is locally cyclic and locally finite [closed]

Show $\mathbb Z_{p^\infty}$ is locally cyclic and locally finite.
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39 views

Cyclic groups and other groups

Ok So I know that a cyclic group is a group that is generated by a single element like $\large{(Z_n,+)}$. Now I was wondering that if every group has a generator , and I found that the answer is yes ...
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1answer
59 views

How to show $1+\sqrt 2$ generate an infinite cyclic group of units in $\mathbb Z[\sqrt 2]$?

The answers given here seem very convoluted: The units of $\mathbb Z[\sqrt{2}]$. Is it possible to provide a more explanatory proof?
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1answer
41 views

What does the category of $G$-set look like when $G = C_p$?

Let $G$ be a finite group. The category of $G$-set consists of finite $G$-sets as objects and $G$-equivariant maps as morphisms. Each finite $G$-set is isomorphic to a disjoint union of $G/H$'s, where ...
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$\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$ possible?

Is it possible to have $\operatorname{Gal}(\overline{K}/K)=\mathbb{Z}$? My question comes from the link beetween covering and field extensions. For covering the simplest example is ...
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2answers
35 views

Arrangements of sets of k positions in a n-competitors race

Let $E(n)$ be the set of all possible ending arrangements of a race of $n$ competitors. Obviously, because it's a race, each one of the $n$ competitors wants to win. Hence, the order of the ...
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2answers
41 views

infinite order of element with element in an infinite group

If $G$ is a infinite group, then $G$ must have an element of infinite order. Is this true? I know that if $G$ is infinite cyclic, then it's isomorphic to $\mathbb Z$. (I guess fact is irrelevant ...
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2answers
52 views

Order of all elements $(\mathbb{Z}/7161\mathbb{Z}^*)$ divisor of 30

How do I show that for all $x \in (\mathbb{Z}/7161\mathbb{Z})^*$ the order of $x$ is a divisor of $30$? I thought about using the Chinese Rest Theorem, but I don't know how. Alsois there an element ...
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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 ...
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2answers
23 views

Locally graded group with all proper subgroups abelian

A group $G$ is said to be locally graded if every finitely generated nontrivial subgroup of $G$ contains a proper subgroup of finite index. I have to prove that a locally graded group with all proper ...
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1answer
47 views

Group Properties - “$a$” commutes “$b$”?

Dr. Pinter's "A Book of Abstract Algebra" presents this problem from Chapter 4: If $a$ and $b$ are in $G$ and $ab=ba$, we say that $a$ and $b$ commute. Assuming that $a$ and $b$ commute, prove the ...
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4answers
48 views

How do I show that $b^8=a^3ba^{-3}$?

Suppose $G$ is a group. I am trying to show that for $a,b\in G$, if $aba^{-1}=b^2$, then $b^8=a^3ba^{-3}$. I am not even sure if this is true but I found this in Artin's Algebra. My work: ...
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1answer
22 views

substraction of groups in direct sum

Assume I have a sub group $G\leq \mathbb Z^n$ and I have $\mathbb Z^n = G\oplus \mathbb Z^m $ for some $m\leq n$. I want to deduce $G\cong\mathbb Z^{n-m}$. Is that true? how can I do it?
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1answer
14 views

The concept of continuity in a topological group

I am now learning the Lie group theory. People talk about the fundamental group of a topological group. The problem is, how is the continuity defined in a topological group? In other words, in which ...
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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
65 views

Is there a cyclic subgroup C of S8 such that the interval lattice [C,S8] is distributive?

I've checked by hand that for any $n \le 7$, there is a cyclic subgroup $C$ of $S_n$ such that the intermediate subgroups lattice $\mathcal{L}(C \subset S_n)$ is distributive. Question: Is it the ...
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1answer
42 views

normal subgroup in $S_3$?

Is $\{(1),(1,3)\}$ a normal subgroup in $S_3$? I know that a normal subgroup means that the left cosets are equal to the right cosets.
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1answer
24 views

Short exact sequence of abelian groups implies long exact sequnce of cohomologies

I am trying to compute cohomologies $H^i(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}, \mathbb{Z})$. Actually it is not a big deal, because I have already computed $H^i(\mathbb{Z}/n\mathbb{Z}, ...
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1answer
35 views

Prove that $A$ is a free abelian group

Suppose $a_1, \dots, a_n$ generate an abelian group $A$, and for any abelian group $B$, and any $b_1, \dots, b_n \in B$ we can find a homomorphism $\varphi: A \to B$ given by $\varphi(a_i) = b_i ~ ...
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1answer
35 views

Right group axioms from left group axioms

I was working a question in group theory where we are given that the left axioms hold for a set $G$ together with a binary operation $*$. We would then like to prove that $(G, *)$ is a group. I read a ...
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80 views

Group of Order $5$

Let $G$ be a group of order $5$ with elements $a, b, c, d, 1$ where $1$ is the identity element. This is the definition of the group. We all know that this can't be a group because any group of ...
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1answer
36 views

Consequences of $G$ having exactly $3$ irreducible characters

Suppose a finite group $G$ has exactly irreducible characters $\chi_1 = \mathbb{I}, \chi_2,\chi_3$. i) Show that G is soluble and deduce it has a non-trivial one-dimensional character $\chi_2$. ii) ...
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Semi-linear transformations form a group.

Let $K$ be a field and $a,b,c,d\in K$ such that $ad-bc\neq 0$ and $\sigma \in Aut(K)$, we define a semilinear transformation $f: K \cup \lbrace \infty \rbrace \to K \cup \lbrace \infty \rbrace$ ...
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1answer
32 views

Proof that elements of a free generating set have infinite order.

I'm trying to show that elements of a free generating set $S$ have infinite order straight from the definition of a free group being generated by $S$. The definition I'm using is that a group $F$ is ...