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

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19 views

Please show that $W$ is a normal subspace of $\sigma$. [on hold]

Suppose that $\sigma$ is a symmetry transformation, $V$ is a space, and $W$ is a subspace of $V$. Please show that $W$ is a normal subspace of $\sigma$.
2
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1answer
13 views

minimal number of relations in nonabelian dihedral groups of order a power of 2

In my recent minimal number of relations r question for the symmetric group of degree 3, it was demonstrated that r = 2. What is r in general for nonabelian dihedral groups of order a power of 2? ...
0
votes
2answers
49 views

To show that $\langle\mathbb Q,+\rangle$ is not isomorphic to $\langle\mathbb Q \setminus\{0\} , \,\cdot\,\rangle$

For this question is it enough to say that they both dont have same cardinality so they are not isomorphic.Can we exhibhit any structural property diference here? THANKS
1
vote
1answer
23 views

Show that $H$ is transitive on the set $G$.

Let $G$ be a group and let a be a fixed element of $G$. The map $\lambda_{a}: G \to G$, given by $\lambda_{a}(g) = ag$ for all $g \in G$, is a permutation of the set $G$. Note $H = \{\lambda_{a} ...
1
vote
1answer
30 views

Proof of the fact that the set of (p,q) shuffles is a cross section of the subgroup $S_p\times S_q$

Definition Let $G$ be a group and $H$ its subgroup. We name a subset $K$ of $G$ a cross section if it has exactly one element from each left coset of $G/H$. Definition Let $n=p+q$ for some ...
2
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1answer
71 views

Is my understanding for group algebra correct?

Let $G$ be a group, $k$ be a commutative unital ring. Consider $\mathbf{Alg}_k^{inv}$ the category of unital $k$-algebras whose multiplicative semigroup is a group. Then there is a forgetful functor ...
0
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1answer
35 views

Representing group by permutaions

Let the group is given by this relations $<a,b\ |\ a^5=b^4=e,bab^{-1}=a^2>$. I am asked to find the cycle index of this group. In order to find cycle index I need to represent the group with ...
3
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0answers
40 views

A name for the automorphisms induced by the normalizer by conjugation?

Let $N$ be a subgroup of a group $G$, $H := \operatorname{N}\left( N \right)$ the normalizer of $N$. There is a natural morphism from $H$ to $\operatorname{Aut}\left( N \right)$ given by $h \mapsto ...
2
votes
1answer
33 views

Constructing an irreducible representation for a finite group

This is not a homework. Recently, I have been starting to study the representation and character theory and I am doing some exercises in this field, but I have some problem with some of them, like: ...
2
votes
1answer
50 views

Show that a simple group of order 60 has no proper subgroup of order greater than 12

I am trying to show that any simple group of order $60$ has no proper subgroup of order greater than $12$. I know that $G$ is isomorphic to $A_5$, a non-abelian simple group of order $60$. I suppose ...
1
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2answers
20 views

Statement of Sylow's Fourth Theorem (single conjugacy class)

I am confused by the statement of Sylow's Fourth Theorem: Let $G$ be a finite group, $p$ a prime. The Sylow $p$-subgroups of $G$ form a single conjugacy class of subgroups. In particular, I do not ...
2
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2answers
36 views

Alternative proof of '$I$ is maximal iff $R/I$ is a field'

For any commutative ring $R$ and an ideal $I$ of $R$, $I \neq R$, show that $I$ is a maximal ideal iff $R/I$ is a field. I write my own proof and it checks with the 'traditional' proof which ...
2
votes
2answers
59 views

What can we say about the order of a group?

Let $G$ be a group and $a ∈ G$. If $a^{12}= e$, what can we say about the order of $a$? What can we say about the order of $G$? We know that $|a|$ divides $12$, but what can we say about the order of ...
3
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0answers
55 views

Groups with more than half elements of order 2

In Dihedral groups, at least half elements are of order two. Question: If a (non-abelian) finite group has at least half elements order two, then what can be said about the group?
1
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2answers
39 views

$K \unlhd H\unlhd G$, K is sylow in H, proof $K \unlhd G$

$K \unlhd H\unlhd G$, K is sylow in H, proof $K \unlhd G$ I know in general it's false, I wonder how should I use the condition that K is sylow? is it that the K is the unique sylow subgroup? and ...
0
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1answer
36 views

Quotient Objects in $\mathsf{Grp}$ II

This question is a sort of continuation of a previous one. In CWM, Maclane says ... every quotient object of a group $G$ in $\mathsf{Grp}$ is represented by the projection $\pi:G\rightarrow G/N$ ...
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3answers
37 views

Mapping Normalizer to Automorphism

I am squeezing my brain trying to understand this problem: Let H be a subgroup of $G$, and denoted by $\operatorname{Aut}(H)$ the group of all automorphisms of $H. $ Define $$\alpha : N_G(H) ...
0
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0answers
30 views

If $H<S_n$, $H$ is abelian and transitive on $\{1,2,…,n\}$, then the order of $H$ is $n$ [duplicate]

If $H<S_n$, if $H$ is Abelian and transitive on $\{1,2,...,n\}$, then the order of $H$ is $n$. So far I have: $H$ is transitive therefore the group orbit is the group itself. I know I should ...
1
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2answers
51 views

A question in finite group theory with proof using representation theory

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises in this field, but I have a problem with an exercise: I want to prove ...
0
votes
1answer
33 views

commuting subsets in a group

For a countable infinite discrete group $G$, consider the following three properties. (P1) $G$ is abelian. (P2) For any finite subset $K$ of $G$, there exists an element $s\in G$, such that the two ...
1
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2answers
37 views

order of group generated by two element with some relation.

The group defined by generators $a,b$ and relations $a^{8}=b^{2}a^{4}=ab^{-1}ab=e$ has order at most 16. How to prove that? I have no idea.
0
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0answers
50 views

The structure of the Sylow p-subgroups of $Sym(n)$

For the structure of the Sylow $p$-subgroups of $Sym(n)$, there is a standard proof by using the properties of the wreath product like as in the Passman's book. But I want to understand this proof. In ...
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1answer
51 views

If every proper subgroup of a group is finite, does it follow that the group is finite? [duplicate]

Suppose that every proper subgroup of a group is finite. Does it imply that the group is finite?
2
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1answer
37 views

A question in character theory of finite groups

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises from "A Course in the Theory of Groups by Robinson" and "character ...
1
vote
2answers
55 views

Groups such that inclusion on collection of all its subgroup is a total order

Characterize the Groups with the following property: Suppose G is any group such that for any two subgroups, H and K either H $\subseteq$K or K $\subseteq$ H. Now what can we tell about cardinality, ...
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2answers
43 views

Show that if $G$ is abelian then the set of elements in $G$ of finite order form a subgroup

Let G be a group. Show that if $G$ is abelian then the set of elements in $G$ of finite order form a subgroup. I have a proof for this question but I dont understand how the group has to be abelian ...
1
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1answer
41 views

Show that a group is soluble

I am wondering how I would go about showing that any group of order $p^2q$ or $pq^2$ is soluble, where $p,q$ are primes with $p<q$. Could you give me a hint or outline for how to get started ...
0
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2answers
43 views

Show $G=(\mathbb{Q},+)$ is not finitely generated

I have been reading a proof for this question and I do not understand the final contradiction that the proof arrives at. Show $G=(\mathbb{Q},+)$ is not finitely generated. (i.e. not generated by a ...
1
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1answer
39 views

Center of Group and Conjugacy Classes

I am trying to prove that the center of a group G is the union of the trivial conjugacy classes of G. So far what I have: We know the center Z($G$) of group $G$ is defined by {$b \in G $ | $ ba= ab$ ...
0
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1answer
27 views

Finding cycle index of matrix group

I need hint to find cyclic index if this matrix group \begin{pmatrix} k & 0 \\ m & 1 \\ \end{pmatrix} where $k,m \in Z_5$
0
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1answer
19 views

Normalizer and Centralizer coincide

I am working on the following question: Suppose $G$ is a finite group that has a cyclic 2-Sylow subgroup $H$. I want to show that the centralizer, $C_G(H)$, and $\text{normalizer,} \ N_G(H)$ coincide. ...
1
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1answer
43 views

Automorphisms in $Z_n$

I know an Automorphism is a group G that is an isomorphism where $h:G \rightarrow G$, (G is being mapped to itself) and that Aut(G) is the set of all Automorphisms in G. I was wondering how I would ...
4
votes
2answers
71 views

Automorphism that is an Involution of a finite group

I am studying for a final and am trying to solve this problem: Let $G$ be a finite group with an automorphism $\sigma:G\rightarrow G$ such that $\sigma \circ \sigma=1$ and whose only fixed point is ...
3
votes
2answers
50 views

Find the number of subgroups in $Z_p \times Z_p \times Z_p$

let $p$ be a prime number ; I want to find the number of subgroups in $G = Z_p \times Z_p \times Z_p$ $(Z_p = \mathbb{Z}/p\mathbb{Z})$. I know that there is $p^2 + p + 1$ copies of $Z_p$ in $G$ for ...
0
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2answers
42 views

Find all Quaternions Satisfying..

Let H be the skew field of quaternions. Find all quaternions x satisfying $(i + j)x(i + k) = 2$ I'm having trouble figuring out what to do with this question. I know the "i j k i j k" formula for ...
0
votes
1answer
61 views

Fixed Spaces for Group Elements

what is the GAP code for finding the fixed space? A list of row vectors that form a base of the vector space $V$ such that $v M = v$ for all $v$ in $V$ and all matrices $M$ in the list $mats$.
1
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3answers
37 views

orthogonal and special orthogonal group of dimension $2$, group of isometries of $S_1$, $\mathbb{R}^2$ [on hold]

In my abstract algebra class, my teacher gave us this problem as to help review for the final. Unfortunately, I am not very well versed with linear algebra so I don't understand all that well what ...
4
votes
1answer
51 views

If a nilpotent group $A$ act on a group $G$ then $G$ is solvable.

Theorem: If a nilpotent group $A$ act on $G$ by automorphism and $C_G(A)=e$ then $G$ is solvable. I hope that the statement of theorem is true. It should belong to Hartley, but when I googled it I ...
0
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1answer
43 views

Compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$

How do I compute the index of $\mathbf{Z}[\alpha]$ in $\mathbf{Z}[\alpha,\beta,\gamma]$? For example, $\alpha = \sqrt[3]{-19}$ and $\beta = (\alpha^2 - \alpha + 1)/3$ satisfy $(\alpha + 1)\beta = ...
2
votes
1answer
52 views

For $G$ an abelian group and $H$ a subgroup, is $[G : H]$ the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$?

Let $G$ be an abelian group and $H$ a subgroup. What is the smallest positive integer $n$ such that $ng \in H$ for all $g \in G$? Is it $[G : H]$, or can it be strictly smaller (a divisor of $[G : ...
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0answers
35 views

Algebraic Structures Books [duplicate]

I wanted to ask you guys if you know any books where I can learn basic stuff about Algebraic Structures and Groups. Thank you.
2
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3answers
54 views

Quotient Objects in $\mathsf{Grp}$

I don't know how to precisely formulate my question, but here goes: Subobjects and quotient objects are duals, so a quotient object in $\mathsf{Grp}^{\text{op}}$ is a subobject in $\mathsf{Grp}$. The ...
0
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2answers
29 views

S3 group action faithful?

I'm struggling with understanding the term "faithful". I read that a group action for example $S_3$ is faithful on {1,2,3}. Does that mean $S_3$ is not faithful on {1,2,3,4} because it never changes ...
0
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1answer
138 views

What is a Simple Group?

I am working on this problem from class note: Let $G'$ be a group and let $\phi$ be a homomorphism from $G$ to $G'$. Assume that $G$ is simple, that $|G| \neq 2$, and that $G'$ has a normal ...
0
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1answer
26 views

twists of unipotent algebraic groups

Let $U$ be a unipotent linear algebraic group over some field $k$ with char$k$=0. Let $U'$ be a linear algebraic group over $k$ such that $U'_{\bar{k}} = U_{\bar{k}}$ (ie $U'$ is a $\bar{k}/k$-twist ...
2
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3answers
39 views

a question about abstract algebra, prove that $HK\cong H\times K$

Let $H$ and $K$ be subgroups of a group $G$, $HK=KH$ and $H\cap K=\{1\}$. Prove that $$HK\cong H\times K.$$ Can some one tell me how to prove this question? I have spent too much time in it, ...
1
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4answers
125 views

To which group is $G/\ker \phi$ isomorphic to?

$G = \langle H, \odot_7\rangle$ where $H= \{ 1,2,3,4,5,6\}$ and $\odot_7$ denotes the operation, multiplication modulo $7$, the function $\phi: G \to G$ defined by $\phi(g)=g^2$ . List the elements ...
0
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1answer
17 views

Prove that for a general 3-cycle \sigma one can find a permutation …

I need to prove that for a general 3-cycle $\sigma$ one can find a permutation $\tau \in S_4$ such that $\tau \sigma\tau^{-1} = (123)$, and use this to show that 3-cycles in $S_4$ are even. How do i ...
1
vote
1answer
33 views

Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$

I am writing another exam in Algebra this week and this time the main topic is automorphism. I was again going through the example exercises and exams from previous years and this problem is giving me ...
4
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0answers
36 views

$n_p(GL_2(\mathbb{F}_p))=p+1$

I'm interested in the following problem from Dummit & Foote's Abstract Algebra text (Exercise 40 of Section 4.5): Prove that the number of Sylow p-subgroups of $GL_2(\mathbb{F}_p)$ is $p+1$. ...