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

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2answers
27 views

let $G$ be an infinite group of the form $G_1 \oplus G_2 \oplus \dots \oplus G_n$ , $n>1$. Prove that $G$ is not cyclic

let $G$ be an infinite group of the form $G_1 \oplus G_2 \oplus \dots \oplus G_n$ where each $G_i$ is a non trivial group and $n>1$. Prove that $G$ is not cyclic. Attempt : Let $G = G_1 \oplus G_2 ...
0
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2answers
30 views

Proof of a basic isomorphism.

How do you prove that $D_2 \cong V \cong \mathbb{Z}_2 \oplus \mathbb{Z}_2$? Where $V$ is the Klein-4 Group and $D_2$ is the dihedral group with cardinality 4. We have that $D_2 := \{1,r,s,sr \}$ ...
0
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0answers
10 views

tables of cyclic subgroups and conjugates

$G = S_5$, I need to construct tables for $H$ and $aHa^{-1}$ ($H =$ cyclic subgroup $(142)(35),$ and $a = (2354) \in G$) and see what can be inferred. In my attempt $H$ = $\{(142)(35), (124)(35), ...
-1
votes
1answer
20 views

Group Theory proving [on hold]

can someone help me with this question? 1) Given a natural number n≥1, let $G_n$ be the set of complex n-th roots of $1$, i.e. $G_{n} = \{z \in \mathbb{C} :z^n = 1\}$ Prove that $G_n$ is a group ...
0
votes
1answer
15 views

If $a\equiv b [p^k]$ then $a^p \equiv b^p [p^{k+1}]$

Can anyone explain the steps to this proof? I'm really lost/ If $k\geq 1$ and $a\equiv b[p^k]$ then $a^p \equiv b^p [p^{k+1}]$ Proof: Since $a= b + qp^k$ for some $q\in \mathbb{Z}$ we have $a^p = ...
1
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1answer
32 views

Prove that $E(\mathbb{C})^{\text{tor}} \cong \mathbb{Q}/\mathbb{Z} \times \mathbb{Q}/\mathbb{Z}$.

Let $E$ be an elliptic curve over $\mathbb{C}$. We know that $E(\mathbb{C}) \cong \mathbb{C}/L$ (this is a group isomorphism) for some lattice $L \subset \mathbb{C}$. Using this fact prove that ...
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1answer
31 views

A characterization for subgroups.

Let $G$ be a group and $a_0,a_1,...,a_n\in G$ and $$A=\{a_0,a_1,...,a_n\}$$ and $$(\forall m\le n)(\forall i\le m)(a_{i}a_{m-i}\in A)$$ Is $A$ a subgroup of $G$? How if $G$ is abelian?
4
votes
1answer
31 views

Image of the Brauer group under a field extension

For $k$ a field, let $Br(k)$ - the Brauer group of $k$ - denote the group of finite-dimensional central simple algebras over $k$, modulo Morita equivalence $(A\equiv B\iff \exists m, n(A\otimes_k ...
0
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2answers
27 views

Normalizer of a subgroup of $GL_2(\mathbb{R})$

I have following subgroup of $GL_2(\mathbb{R}):$ $$A=\Bigg\{\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right),\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} ...
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vote
2answers
22 views

Shown that this matrix is a representative of a group.

I have to show that this matrix (for which $ad - bc = 1$) \begin{pmatrix} a & b & 0\\ c & d & 0 \\ 0&0&1 \end{pmatrix} is a representative of a group. The same for this one: ...
1
vote
1answer
22 views

Conjugacy classes and centralizers of a SmallGroup

What is the complete lists of conjugacy classes and centralizers of SmallGroup(64,138)? Would someone be willing to provide the complete lists of conjugacy classes and centralizers of ...
1
vote
2answers
25 views

Difference between the definition of monoid action and group action?

The question is essentially in the title. From what I read in the wikipedia article about monoids it seems to me that we can define a monoid action in the exact same way we define a group action. Is ...
0
votes
1answer
29 views

$\exists a\in G-H$ such that $aHa^{-1}=H$

Let $G$ be a $p$-group with proper subgroup $H$. Show that there exists an element $a\in G -H$ such that $a^{-1} Ha = H$ Can you check my proof? Since $G$ and $H$ are $p$-groups their centers ...
1
vote
1answer
20 views

Notation question: Group generated by two elements?

Let there be $H$ subgroup of symmetric group $S_4$, so that $H= \langle (12)(34),(234) \rangle$. What does the notation $\langle (12)(34),(234) \rangle$ mean? I know that if there's one elements, then ...
2
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1answer
22 views

determine if $G = \{ f : \mathbb{R_+} \to \mathbb{R_+} \}$ is a group

I'm confused about how the identity was formed - if $e(x) = x$, then how does one get from $f(x)e(x) = f(x)\cdot 1$
0
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1answer
19 views

A Criterion for being Sylow p-group

Show that if $H$ is a $p$-group of finite group $G$ and $N_G(H)=H$ then $H$ is a Sylow $p$-group of $G$? Or prove the following more general property,$$[G:H]\equiv1\ (\mod\ p)$$
3
votes
0answers
28 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
2
votes
1answer
25 views

$Z$ is cyclic and has generators 1 and -1

I know that group G is cyclic if there exist $ g \in G$ such that $ G = \{g^k : k \in Z\}$. However I don't understand how Z has generators $1$ and $-1$. Does $g^k$, so in this case, $1^k$ mean ...
0
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1answer
21 views

Finding explicit form of group homomorphism

Let there be $f: \mathbb Z_{50} \rightarrow \mathbb Z_{15}$ group homomorphism so that $f(7)=6$. Find explicit form of $f$. What's the approach to this type of questions? Is it possible that the ...
2
votes
1answer
18 views

Determining the structure of the abelian group, integral matrix

I am revising for my upcoming university exams and I have a past exam question that I am finding particularly challenging... a) Consider the integral matrix $$R=\begin{bmatrix} 2 & 2 & ...
3
votes
2answers
36 views

Group having an element $x$ of order $p$ where $p$ is the smallest prime dividing |G|

Let $G$ be a finite group and $p$ be the smallest prime dividing $|G|$ and $x\in G$ be an element of order $p$. Let $h\in G,$ and $hxh^{-1}=x^{10}$. Then prove that $p=3$. If $H=<h>, ...
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1answer
15 views

Question about proof about index and subgroups

Let $G$ be a group so that $H\lhd G$. There is an element $g \in G$ so that $g$ isn't in $H$ but $g^2$ is in $H$. Show the index is even. Can't I just say that the cosets of $H$ are $H$ and $Hg$ ...
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3answers
22 views

A question about normal subgroups and index

Let $G$ be a group, and $H$ be a normal subgroup of $G$. $|H|=11$ and $[G:H]=24$. Let there be $x \in G$ and $x^{11}=e$. Show $x \in H$. Would like hints etc' on how to solve this. Is proving that ...
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0answers
29 views

What Is The Permutation Isomorphic? [on hold]

What is the definition of permutation isomorphic of two groups in their actions on a group?
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1answer
24 views

Groups with single conjugacy class of subgroups. (modified)

I am going to modify my previous question. What are those finite non abelian groups in which non normal subgroups of same order are conjugate. e.g. Dihedral groups of order $4n+2$.
4
votes
6answers
60 views

Why is $\mathbb{R}/\mathbb{Z}$ isomorphic to the complex numbers of length one?

Wikipedia states that the quotient group $\mathbb{R}/\mathbb{Z}$ is isomorphic to all complex numbers of length $1$. I have a hard time making sense out of this, and in particular, how complex numbers ...
1
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1answer
32 views

Let $G$ be a group. Then verify the statements with justification:

Let $G$ be a group. Then verify the statements with justification: $\bullet$ If $G$ has nontrivial centre $C$, then $G/C$ has trivial centre. $\bullet$ If $G$ does not equal $1$, there exists a ...
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1answer
39 views

Question about group theory and order in $\mathbb Z_n$

This is a only theoritical. Why is the order $o( \bar x )$ of $\bar x∈\mathbb Z_n$ the smallest non-negative integer $k$ such that $kx \equiv 0$ (mod $n$)? I don't understand how it follows from ...
3
votes
3answers
56 views

Only $1$ Nontrivial Subgroup $\Longrightarrow |G| = p^2$ [duplicate]

I am pretty new to this site , so I am not sure how things work, but I am in desperate help with a question that I don't know where to start or finish with. It is for a test I have to study for. Here ...
1
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1answer
36 views

Can we give of the fact that a group of order $9$ is abelian without using an argument involving the product of two cyclic groups of order $3$?

A group of order $9$ is always abelian. I've seen proofs of this result, but I would like to prove it the following way: Let $G$ be a group of order $9$. If $G$ has an element $a$ of order $9$, then ...
2
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2answers
52 views

What is the most elementary proof of the following: a non-abelian group of order $6$ is isomorphic to $S_3$

We know that, a non-abelian group of order $6$ is isomorphic to $S_3$. While I've been able to locate different proofs of this result, I would like to have one that is as elementary as it can be. So ...
1
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1answer
32 views

An assumption used in defining a group of order $mn$

I want to show that the defining relations $a^m=b^n=e, ab=ba^k$ define a group of order $mn$ with a normal subgroup of order $m$, if $k^n \equiv 1 \pmod m$. Consider the set of symbols of the form ...
3
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1answer
38 views

How do I effectively solve this kinds of problems?

I'm preparing for a test on Monday. This is an exercise in the past homework. Find all subgroups of $\mathbb{Z}_2\times \mathbb{Z}_2\times\mathbb{Z}_4$ isomorphic to $Z_2\times Z_2$. Is there a ...
1
vote
1answer
22 views

Non simplicity of a group of order $p^{100}q$ given some conditions.

Let $G$ be a group $p$, $q$ primes $p\neq q$, $|G|=p^{100}q$. 1.- Assume that $G$ has two different $p$-Sylow subgroups and intersection for each pair of $p$-Sylow subgroups is trivial. Show that ...
2
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1answer
46 views

Which non-Abelian finite groups contain the two specific centralizers? - part II

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers isomorphic to both of these two groups (but may contain other ...
2
votes
0answers
44 views

Which finite groups contain the two specific centralizers?

This is a question requiring the good knowledge of group theory: (Q1) Which finite groups $G$ contains some specific centralizers both of these two groups: i. the elementary group $Z_2^4$, ...
0
votes
0answers
28 views

Burnside's Lemma and Stirling Numbers of the First Kind

I've seen that $n!=\displaystyle\sum_{p=0}^n s(n, p)n^p$, where $s(n, p)$ are the signed Stirling Numbers of the First Kind, whose absolute values count the number of permutations in $S_n$ which have ...
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2answers
45 views

The Cayley Representation Theorem.

This theorem states that "Any group is isomorphic to a subgroup of a group permutations." I only ask if someone could provide a simple example so that i can fully understand this theorem.
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0answers
36 views

Ring Theory in Group Theory [on hold]

how can we check that Complex, Real and Rational numbers are ring, division ring and field? please give as many examples as anyone can.
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2answers
54 views

Subgroup of group of order $44$

Pick the correct statement(s) below: $(a)$ There exists a group of order $44$ with a subgroup isomorphic to $ Z_2 + Z_2 $. $(b)$ There exists a group of order $44$ with a subgroup isomorphic to $ ...
4
votes
1answer
28 views

Closed conjugacy classes in $M_n(k)$

Let $k$ be an algebraically closed field, $n$ a positive integer, and consider the action of $\mathrm{GL}_n(k)$ on $M_n(k)$ by conjugation. My professor tells me that semisimple conjugacy classes are ...
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0answers
26 views

To Find a subgroup of order $6$ in $U(700).$

Find a subgroup of order $6$ in $U(700).$ Attempt: $U(700)=U(2^2.5^2.7)\thickapprox U(2^2) \oplus U(5^2)\oplus U(7)$ $\thickapprox \mathbb Z_2 \oplus \mathbb Z_{5^2-5} \oplus \mathbb Z_{7-7^0}$ ...
2
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0answers
17 views

normal group $(\text{ker }g)(\text{ker } f)$ as a kernel of some group using $f$ and $g$

For group homomorphism $f: A\to B$ and $g: A\to C$ we know $\text{ker }f\cap \text{ker } g$ is kernel of $(f,g): A\to B\times C$. $(\text{ker }g)(\text{ker } f)$ is trivially normal in $A$, can we ...
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0answers
8 views

About representations and transformations under an $SU(n)$ Lie Group

I think my problem is that I misunderstand what "transforms under" really means. Let's take $SU(3)$, for the $\mathbf{3}$ with Dynkin indices $(1,0)$, a state transforms like : $ψ→gψ$. For the ...
6
votes
1answer
39 views

Order of groups and group elements? [duplicate]

Let G be a group and let p be a prime. Let g and h be elements of G with order p. I am wondering how I can use group theory to find the possible orders of the intersection between ...
1
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1answer
31 views

Can one conjugate any element in $S_3$ to any other element?

I've come across the following problem from herstein: Let $\varphi$ be an automorphism of $S_3$. Show that there is an element $\sigma \in S_3$ such that $\varphi(\tau) = \sigma^{-1}\tau\sigma$ ...
2
votes
1answer
41 views

Equivalent axioms for a group

Claim: Assume $(G,\times)$ is a set with an associative binary operation such that $\forall a,b\in G$ the equations $ax=b$ and $ya=b$ have a unique solution in $G$. Then $G$ is a group. Thought: Let ...
1
vote
1answer
49 views

Is this type of a subgroup always normal? [duplicate]

Let $G$ be a finite group, and let $H$ be a subgroup of $G$ such that the index $(G\colon H)$ of $H$ in $G$ is the smallest prime that divides the order of $G$. Can we say anything about whether or ...
0
votes
1answer
20 views

Basic question about dimensionality of Euclidean group

I have a basic question about the dimensionality of the Euclidean group. Why are degrees of freedom greater than the dimension? I thought that a degree of freedom is the same as a dimension, as in, ...
0
votes
1answer
49 views

If G is a finite group and $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$

Let G be a finite group. Show that, given $x \in G$, there is an integer n $\geq 1$ such that $x^n = e$. I'm trying to use the info that is finite, but I can't find a way. For instance, if G is a ...