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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Prove there generally is no isomorphism between $R[x]/(x^2-a)$ and $R^2$

I have a ring $\mathbf R =(R, +, -, ., 0, 1)$ (note that there is no invers for multiplication, $R$ is not $\mathbb R$, it is any set for the given algebra). How do you prove that the following does ...
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Question about working in modulo?

This question is in essence asking for understanding of a step in Fermats theorem done Group style. For any field the nonzero elements form a group under field multiplication. So let us take the ...
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1answer
62 views

Matrices $P$ such that $A$ is symmetric $\Longrightarrow $ $PAP^{-1}$ is symmetric

Let $M_n(\mathbb{R})$ be the (vector) space of all $n\times n$ matrices over $\mathbb{R}$. Let $Sym_n(\mathbb{R})$ denote the subspace of symmetric $n\times n$ matrices. $GL(n,\mathbb{R})$ acts on ...
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2answers
61 views

Image of a normal subgroup under automorphism is the normal subgroup

Let $G$ be a finite group. Let $H$ be a normal subgroup of $G$ such that the order of H and the index of $H$ in $G$ are relatively prime. Let $f$ be an automorphism of G and let $J = f(H)$. Prove ...
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2answers
159 views

Books about braid theory

I'm looking for books that talk about braid theory, in the sense of braid groups mostly, and not too advanced, if possible. With material understandable for an undergraduate. Thanks for any ...
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34 views

If $G_{ab}$ is cyclic then $G$ is cyclic

In my notes I have the following theorem: Let $G$ be a (nilpotent?) group. Suppose that $G_{ab}$ is cyclic. Then $G$ is cyclic. Actually I don't know if the hypothesis that $G$ is nilpotent is ...
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2answers
33 views

Show that $\overline \varphi (a Z (D_4)) = Id$

Consider $$\begin{align}\overline \varphi : \frac{D_4}{Z(D_4)} &\to \frac{D_4}{Z(D_4)} \\aZ(D_4) &\mapsto xax^{-1}Z(D_4)\end{align}$$ where $$D_4 = \{id, \alpha, ...
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85 views

Group Acting on a Ring

What would be the definition for a group action on a ring? I could not find one online. Would this be acceptable? A group action of a group G on a ring R is a map from G x R to R defined by g(r)=g.r ...
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68 views

How to determine $\left|\operatorname{Aut}(\mathbb{Z}_2\times\mathbb{Z}_4)\right|$

I know that the group $\mathbb{Z}_2\times\mathbb{Z}_4$ has: 1 element of order 1 (AKA the identity) 3 elements of order 2 4 elements of order 4 I'm considering the set of all automorphisms on this ...
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Why is ${\bf N}\otimes\bar{\bf N} \cong{\bf 1}\oplus\text{(the adjoint representation)}$?

I just watched this lecture and there Susskind says that $${\bf N}\otimes\bar{\bf N} ~\cong~{\bf 1}\oplus\text{(the adjoint representation)}$$ for the Lie group $G= SU(N)$. Unfortunately, he does ...
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58 views

Show that $\det \rho(g)=1$ for elements g of odd order and $\det \rho(g)=-1$ for elements g of even order.

$G$ is a finite group. $\rho$ is a representation of $G$, then $\rho \mapsto \rho(g)$ is a $1$-dimensional representation. Show that if $\det \rho(g)=-1$ for some $g \in G$ then $G$ has a ...
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36 views

Prove that the inner product is the number of orbits of $G$ on $X \times Y$.

Let $X,Y$ be $G$-sets and $\mathbb{C}[X], \mathbb{C}[Y]$ the corresponding permutation representations. Prove that the inner product is the number of orbits of $G$ on $X \times Y$. Ive tried: ...
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93 views

When is $G\cong\operatorname{End}(G)$?

$\newcommand\End{\operatorname{End}}$Let $G$ be an Abelian group. Are there sufficient conditions for the existence of an isomorphism $G\cong\End(G)$, where $\End(G)$ is considered a group under ...
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59 views

What it means SO* (2N)?

I'm puzzled about the $"*"$ in the following notation for Lie groups: $SO^* (2N)$ or $SU^* (2N)$. I don't understand what is the meaning of this notation. It is introduced for example in Gilmore ...
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1answer
131 views

About Cayley graphs on finite fields.

If one is given $n$ vectors of length $n$ $\in \mathbb{F}_{p^k}^n$ for some prime number $p$ and $k \in \mathbb{Z}^+$ then how can one check if they are linearly independent? (the issue is if there ...
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Prove that $H_n=\{\sigma \in S_n \mid \sigma (i) \equiv i \pmod 3 \}$ with $n≥2$ is a subgroup of $S_n$.

Prove that $H_n=\{\sigma \in S_n \mid \sigma (i) \equiv i \pmod 3 \}$ with $n≥2$ is a subgroup of $S_n$. I'm doing this problem and I don't know if my approach is correctly done. First of ...
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138 views

If we have exactly 1 eight Sylow 7 subgroups, Show that there exits a normal subgroup $N$ of $G$ s.t. the index $[G:N]$ is divisible by 56 but not 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exits a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. Now this is my ...
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68 views

Showing that stabilizer of group action is conjugate to stabilizer.

Let $G\rightarrow X$. Show that $stab(g\cdot x)$ is conjugate to $stab(x)$. To make $G$ act on itself by conjugation, take $X = G$ and let $ g \times x = gxg^{-1}$ Here $g \in G$ and $x \in G$ Since ...
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Abstract algebra question (1) order problem (2) ring problem

Let G be an abelian group and let a, b ∈ G have orders m and n respectively. Suppose furthermore that gcd(m, n) = 1. Show that ab has order mn. I noticed that if gcd(m,n) is not 1, then order of ab ...
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65 views

Can closure of quaternions under multiplication be shown with a cayley table?

Unsure about my understanding of groups and quaternions. I'm trying to figure out if just using a cayley table (specifically this one) can show closure of quaternions under multiplication, is ...
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76 views

Decomposition of a group whose Cayley graph is a tree

This is an exercise taken from Chapter 9 of a French book, Géométrie et Théorie des Groupes. It says, roughly, the following: Show that a finitely generated hyperbolic group, whose Cayley graph is a ...
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Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup.

I'm stuck on this exercise: Let $G$ be a group, with $|G|=pqr$, $p,q,r$ different primes, $q<r$, $r \not\equiv 1$ (mod $q$), $qr<p$. Show $G$ has an unique Sylow $p$-subgroup $P$. What ...
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1answer
48 views

Order of elements in group cohomology

OK, I'm just learning about cohomology of groups, and I want to make sure I'm not missing anything in this question: Let $n$ be an abelian group, let $G$ be a group, let $\tau$ be an action of ...
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1answer
37 views

Prove that $|H|$ is odd and $|G/H|$ = $2^n$ for some positive integer n

The problem assumes G is a finite abelian group, and $H$={$a\in G$|$|a|$ is odd} is a subgroup of G. Prove that $|H|$ is odd and $|G/H|$ = $2^n$ for some positive integer n. As a Hint, it says to ...
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1answer
117 views

About first Sylow Theorem proof

I've been struggling with the first Sylow theorem proof we were given in class. This is how my professor introduced us the theorem: First Sylow Theorem: Let $G$ be a finite group. Let $\rvert G ...
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1answer
50 views

Product of chracter

From Isaac's character theory book; $3.12$ Let $x\in Irr(G)$ and $g,h\in G$. Show that $$\chi(g)\chi(h)=\dfrac{\chi(1)}{|G|}\sum_{z\in G}\chi(gh^z)$$ I had thought that it was related to $3.9)$; ...
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Question about torsion group

In my book, it says that torsion group is a group all of whose elements have finite order. It seems to me that then every subgroup of finite group must be torsion group since if group is finite, then ...
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1answer
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Inductive Step when working with Quotient Groups, Proof about normal subgroups of special groups

I have a question on the following proof of the characterisation of normal subgroups of groups which are direct products of non-abelian simple groups: Theorem: Let $G = G_1 \times \cdots \times ...
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2answers
95 views

if G is generated by {a,b} and ab=ba, then prove G is Abelian

if G is generated by {a,b} and ab=ba, then prove G is Abelian All elements of G will be of the form $a^kb^j,\,\, k,j \in \mathbb{Z}^+$ so I need to get $a^kb^j = b^ja^k$
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Showing two groups are not isomorphic using the order of their elements.

I am trying to solve this question: "$\text{Prove that no two of the groups } C_2 \times C_2 \times C_2 , C_2 \times C_4 \text{ and } C_8 \text{ are isomorphic.} $" I understand that to show they ...
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1answer
97 views

An example subset of group such that its not subset of normalizer

I read book of Dummit and Foot Abstract algebra. I need some help with the following question Let $H$ be a subgroup $G.$ Show that $H$ is subgroup of $N_G(H).$ Give an example to show that this ...
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1answer
20 views

Finite subgroups which are normal in a matrix Lie group

I have the following question: Let $G$ be a closed subgroup of $GL(n,\mathbb{C})$. Denote $Z(G)$ by the center of $G$. ${\bf Question}:$ Is it true that every finite normal subgroup of $G$ are ...
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89 views

Number of left cosets of the special linear group in the general linear group

Let $F$ be a field of cardinality $q$. I need to prove that $\frac{\left |{GL_n(F)}\right |}{\left |{SL_n(F)}\right |}=q-1$. I try to find a bijection between the left cosets of ${\left ...
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1answer
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The order of a $k$-cycle in $S_n$ is $k$.

Here's what I have right now: The order of a $k$-cycle in $S_n$ is $k$. Proof. Let $\sigma$ represent the $k$-cycle $$\sigma=(x_1 \ x_2 \ \cdots \ x_k)$$with distinct elements $x_i$. Note that the ...
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1answer
44 views

Is it possible to define a binary operation $\mu$ on $(0,1]$ such that ($(0,1], \mu$) is an abelian group?

Is it possible to define a binary operation $\mu$ on $(0,1]$ such that ($(0,1], \mu$) is an abelian group? I tried to define $\mu(a,b) = a+b \pmod 1$. But $0$ is not in $(0,1]$. Thank you very much. ...
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1answer
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A confusing statement from “Berkeley Problems in Mathematics”

This is from pg. 402 of "Berkeley Problems in Mathematics". The statement below forms part of the solution of problem 6.4.12 Let $\phi:G\to S_n$ be a homomorphism between group $G$ and symmetric ...
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1answer
43 views

Looking for example(s) of infinite abelian group $G$ , other than $\mathbb Q / \mathbb Z$, such that $\mathbb Z^+=\{o(g):g \in G\}$

Give example of an infinite abelian group $G$ (if exists) , other than $\mathbb Q / \mathbb Z$ , such that $\mathbb Z^+=\{o(g):g \in G\}$ . Please help
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Young tableaux of $8\otimes 8$ in $SU(3)$

In Georgi's Lie Algebras in Particle Physics, one finds the following Young tableaux for $8\otimes 8$ in $SU(3)$: I am unsure of all the cancellations. Let us number the canceled tableaus increasing ...
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52 views

Isomorphic relation between dihedral groups

Theorem Let $G,H$ be abelian groups such that $Dih(G)\cong Dih(H)$ If $G$ is finitely generated, then $G\cong H$. I'm curious whether "finitely generated" hypothesis can be removed. If ...
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Determine if $\mathbb{Z}/12\mathbb{Z}\setminus\{0\}$ is a group under usual product.

I'm doing an exercise about determining if some sets with binary operations have group structures. I'm struggling with this one: The set $\mathbb{Z}/12\mathbb{Z} \setminus \{0\}$, with the usual ...
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2answers
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Prove: $aH = bH \iff Ha^{-1} = Hb^{-1}$

I have to prove this exercise for my math-study: Let $G$ be a group and $H \subset G$ a subgroup. Prove that for every $a,b \in G$ holds: $$aH = bH \iff Ha^{-1} = Hb^{-1}$$ I tried this, but I'm ...
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1answer
51 views

Number of cyclic subgroups order $p^2$ in $\mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_{p^2}$

Let $$G={ {<a>}_{p} \times {<b>}_{p} \times {<c>}_{p^2}} \cong \mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_{p^2} \text{, $p$ is prime}$$ There are $p^3-1$ elements with order ...
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1answer
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Sufficient conditions such that $A+xB= \mathbb{Z}/n\mathbb{Z} $

I'm working in an exercise and I need some results about additive theory of numbers, I encountered this problem: Given an element $x\in \mathbb{Z}/n\mathbb{Z}$ and two subsets $A$ and $B$ of $ ...
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Does the general linear group equals the automorphism group of the additive group of a K-vector space?

I am wondering given a $K$-vector space $E$ of dimension $n$ if $GL_n(K)=Aut(E,+)$. For a finite field it seems to be true. For example $Aut(\mathbb Z/2\mathbb Z^3,+)=GL_3(\mathbb Z/2\mathbb Z)$. If ...
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1answer
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$G=H_1 \cup H_2 \cup H_3$ index of $H_i$

Let $G=H_1 \cup H_2 \cup H_3$ be a finite group, where each $H_i$ is proper subgroup of G. (I can show) $H_i \neq H_j$ where $i\neq j$ Show that each $H_i$ has index two in G Any suggestion?
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71 views

Group under composition

Which proper subsets of $S_3$ for a group under composition? I'm not really sure how to approach this. I know the four requirements of groups - identity, closure, inverse and associativity. And I ...
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2answers
122 views

Centre of a group and normalizers

Let $G$ be a group and let $A \subset G$ be a non empty subset of $G$.Define the following subsets of $G$ $$Z(G) = \{z \in G \space | \space zx =xz \space \space \forall x \in G \}$$ $$N_G(A) = \{h ...
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1answer
42 views

Is this extension of $Sp(4,2)$ a semidirect product?

Somebody I trust has been insisting to me that a certain extension of $Sp(4,2)$ is actually a semidirect product, and I'm inclined to believe him, but I haven't been able to convince myself he's ...
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2answers
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Do there exist pro-$p$ groups with finite quotients of non $p$ power order?

We define a pro-$p$ group to be a projective (i.e. inverse) limit of $p$-groups. My question is exactly as stated in the title: If a subgroup $H$ of a pro-$p$ group $G$ has finite quotient, must ...
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1answer
61 views

A normal intermediate subgroup in $B_3$ lattice with an additional index condition?

This post is a sequel of: A normal intermediate subgroup in $B_3$ lattice? Let $G$ be a finite group and $H$ a subgroup. Let $\mathcal{L}(H \subset G )$ be the lattice of intermediate subgroups ...