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

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2
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0answers
121 views

Group with presentation $\langle x,y \ | \ x^2, y^2 \rangle$ is generated by $2$ elements of order $2$

Could you tell me how to prove that a group with presentation $\langle x,y \ | \ x^2, y^2 \rangle$ is generated by $2$ elements of order $2$? I know it's infinite, because we will have infinitely ...
0
votes
3answers
726 views

About stabilizer in group action

Let $X$ be a finite set and $x$ is an element of $X$. Let $G_x$, the stabilizer subgroup, be the subset of $S_X$ consisting of permutations that fix $x$. The question is Is stabilizer always a ...
0
votes
1answer
118 views

Basic Lie Algebra Question

Essentially, I'm trying to prove that when computing the tangent space for a group that there's nothing special about considering it at only the identity. Namely, there is an isomorphism of vector ...
1
vote
1answer
46 views

stab(S) is isomorphic to $ S_k \times S_{n-k} $

Show that the group $S_n$ (right) acts on the set of subsets of $\{1,2,\ldots n\}$ by $\rho(S,\sigma)=S\sigma$. Show that there are $n+1$ orbits, one for each possible value of $|S|$. Show ...
0
votes
1answer
40 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 $\pi:G\rightarrow G/N$ ...
2
votes
1answer
82 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 ...
-1
votes
0answers
21 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 $\sigma$.
0
votes
2answers
57 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
31
votes
9answers
6k views

Examples of nonabelian groups.

Can anybody provide some examples of finite nonabelian groups which are not symmetric groups or dihedral groups?
1
vote
1answer
32 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} ...
6
votes
3answers
3k views

Group theory applications along with a solved example

As I asked in previous question, I am very curious about applying Group theory. Still I have doubts about how I can apply group theory. I know about formal definitions and I can able to solve and ...
0
votes
1answer
42 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 ...
4
votes
2answers
75 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
0answers
52 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 ...
0
votes
1answer
29 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 $k,m \in Z_5$
2
votes
1answer
58 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
vote
2answers
55 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
290 views

Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.

it seems like it... should be? In that I can't think of any counterexamples off the top of my head. I was looking up these http://en.wikipedia.org/wiki/Hamiltonian_group and saw the quaternion group, ...
14
votes
10answers
6k views

Example of infinite groups, such that all its elements are of finite order

I am in need of: Example of infinite groups, such that all its elements are of finite order
1
vote
1answer
159 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
votes
2answers
43 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 ...
1
vote
2answers
21 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 ...
1
vote
2answers
56 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, ...
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 ...
2
votes
2answers
37 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 ...
1
vote
3answers
50 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) ...
15
votes
1answer
126 views

Decomposing $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles, formula for all $n$?

$``$Let $G = \text{SU}(2)$, and let $V_n$ be the space of homogeneous degree $n$ polynomials in $\mathbb{C}[x, y]$. Decompose $V_1^{\otimes n}$, $\text{Sym}^2V_n$ into irreducibles.$"$ For ...
2
votes
2answers
66 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
votes
0answers
59 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
vote
2answers
43 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 ...
2
votes
3answers
55 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
votes
0answers
31 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 ...
5
votes
0answers
158 views
+100

Factorizations in the symmetric group

Notation notes: The cycle $(i,i+1,\dots,j)\in S_n$ is the permutation $(1)(2)...(i,i+1,\dots,j)\dots(n)$ in cycle notation. Motivation Given a factorization of a permutation into certain cycles ...
1
vote
2answers
49 views

Proving Fermat's Little Theorem in general and use that to prove Euler's Generalization of Fermat's Little Theorem

Can anyone help me with this? I know there are many different ways to do this and threads explaining this question. However I can't seem to find one that uses only group/ring theory. I haven't ...
1
vote
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.
5
votes
4answers
249 views

Understanding homomorphism and kernels

Let $G$ be a group and $\phi$ a Homomorphism $$ \phi:G\to G' $$ Now I know that the size of the kernel tells you how many elements in $G$ map to the same element in $G'$ I couldn't find this in my ...
1
vote
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
votes
1answer
39 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 ...
0
votes
1answer
64 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 $mats$.
2
votes
2answers
263 views

Show that H is a subset of the normalizer

My Abstract Algebra professer assigned us this homework problem which I'm assuming he created himself: Suppose $H$ is a subgroup of $G$. Show that $H$ is a subset of the normalizer of $H$. The ...
1
vote
2answers
44 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
vote
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
votes
2answers
46 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
vote
1answer
42 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
votes
1answer
21 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
vote
8answers
363 views

Every infinite abelian group has at least one element of infinite order?

Is the statement true? Every infinite abelian group has at least one element of infinite order. I am searching for an infinite abelian group with all elements having finite order. Please help me ...
3
votes
2answers
51 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 ...
1
vote
1answer
45 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 ...
0
votes
2answers
85 views

Choosing “Conjugating element” from a subgroup

Let $H\leq G$ and $N\unlhd G$ such that $G=HN$ and $H\cap N= (1)$. My question is Prove that if two elements of $H$ are conjugate in $G$, then they are conjugate in $H$. What i have done so far ...
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 ...