5
votes
1answer
46 views

Is S a group under matrix addition

Another matrix question! Let $$S=\{A \in M_2(\mathbb{R}):f(A)=0\}\text{ and }f\left(\begin{bmatrix}a&b\\c&d \end{bmatrix}\right)=b$$ Is S a group under matrix addition. Either prove that ...
2
votes
1answer
34 views

Please check my proof on: $\sim$ is an equivalence relation $\Leftrightarrow S<G$

Problem: Let $\emptyset\ne S\subset G$, where $G$ is a group, and define a relation on $G$ by $a\sim b\Leftrightarrow ab^{-1}\in S$. Show that $\sim$ is an equivalence relation if and only if $S$ is a ...
1
vote
2answers
65 views

Free action by cyclic group.

Let $G$ be a group acting on a set $X$. If $g\in G$ has no fixed points, prove or disprove the cyclic group $\left \langle g \right \rangle$ acts freely on $X$. edit: Can also assume $g$ has finite ...
0
votes
0answers
25 views

What is M-bar in factor groups? [duplicate]

If G is a group and N $\triangleleft$ G, show that if $\bar{M}$ is a subgroup of G/N and M = {a $\in$ G | Na $\in$ $\bar{M}$}, then M is a subgroup of G, and M $\supset$ N. If $\bar{M}$ is normal in ...
1
vote
0answers
24 views

For a discrete abelian group $G$, is the Gelfand Representation of $\ell^1(G)$ injective?

Given a discrete group $G$, we can consider the Banach $*$-algebra $\ell^1(G)$, with convolution product $(\xi*\eta)(g)=\sum_{h\in G}\xi(h)\eta(h^{-1}g)$ and involution ...
6
votes
2answers
83 views

Existence of an element in a group of certain order if an element of other order exists

Show that if a group $G$ of order $1089=3^2\cdot 11^2$ contains an element of order $9$ then it also contains an element of order $33$. I tried to see what would Sylow theorems tell for this problem ...
0
votes
2answers
194 views

Total number of non isomorphic groups of order 122.

Let $G$ be group of order $122 = 61 \cdot 2 = p \cdot q$ , where $p < q$ are primes. I know that there exists a unique non abelian group of order $pq$ and one abelian non isomorphic group of order ...
5
votes
3answers
50 views

Show that: $\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$

Show that: $$\frac{D_n}{\langle a\rangle}\simeq\mathbb{Z_2}$$ where $D_n$ is dihedral group and $a$ is generator of order $n$.
2
votes
0answers
52 views

isomorphism of algebric torus

I'm trying to prove the following: Let $D_n=(\mathbb{C}^{\times})^n$ (an algebric torus of rank $n$). Assuming $D_k$ is isomorphic to $D_n$ as an algebric group. Prove that $k=n$. So far, I managed ...
0
votes
2answers
28 views

Automorphism problem

For arbitrary group $(G,\cdot)$ let $Aut(G) = $ {$f: G \to G | f $ is a isomorphism} be set of all automorphisms of group $G$. We assume that $(Aut(G),\circ)$ where $\circ$ is addition of mappings is ...
2
votes
1answer
40 views

Subgroups of $(\mathbb Z_n,+)$

The problem is to define all subgroups of $(\mathbb Z_n,+), n \in \mathbb N$. My guess is if n is prime number, then there is only trivial subgroups. If n is not prime, then I can factorize it, and ...
3
votes
1answer
53 views

$eH \in G/H$ is the only element with finite order

$\newcommand{\ord}{\text{ord}}$This is a question in my book: $G$ is an infinite Abelian group and $H=\{ a \in G \mid \ord(a) < \infty \}$ is a normal subgroup of $G$, show that $eH$ is the ...
0
votes
2answers
61 views

Minimal non solvable group is simple

I suppose to prove in my homework that Every minimal non solvable group is simple. I can't find the way. Thank you.
0
votes
0answers
22 views

On modular group and quadratic forms

Let $\Gamma$ be the modular group, is the group of linear transformations of the upper half of the complex plane. Let $\mathbb Q_{N^2{d_K}}/\Gamma$ (the group of positive definite primitive quadratic ...
2
votes
1answer
89 views

How many normal subgroups does a non-abelian group $G$ of order $ 21$

How many normal subgroups does a non-abelian group $G$ of order $21$ have other than the identity subgroup $\{e\}$ and $G$? 1) 0 2) 1 3) 3 4) 7 I think option 1 is incorrect because every group ...
0
votes
2answers
35 views

Particular nontrivial group has a nonidentity automorphism [duplicate]

If $G$ is a nontrivial group that is not cyclic of order 2, then $G$ has a nonidentity automorphism. This is the exercise of hungerford algebra in the chapter $IV$ MODULES. Can you help me please?
1
vote
0answers
38 views

Calculate all the local automorphisms

The Kohn - Nirenberg domain $\Omega_{KN}$ defined by $$\Omega_{KN}=\left\{(z,w)\in \Bbb C^2:\text{Re}\ w+|zw|^2+|z|^8+\dfrac{15}{7}|z|^2\text{Re}\ z^6<0\right\}$$ How to compute all the ...
3
votes
2answers
66 views

How can there exist an isomorphism between this group and the cyclic group $(\mathbb Z,+)$?

I have a group over $\mathbb Z$, defined by the binary operation $*$, such that $a*b:=a+b+2$. From the previous exercise, I have deduced that the identity-element is $-2$ and that it is an abelian ...
2
votes
1answer
32 views

Chain condition (ACC and DCC ) for a group

If $G$ is a group that satisfies both chain conditions (ACC & DCC ) and there exists a group $H$ with $G×G≅H×H$, we can say $G≅H$? If G satisfies both chain conditions and K and H groups satisfies ...
4
votes
1answer
58 views

Subgroups of $F^*$ are cyclic

Q: If $F$ a field then every finite subgroup of $F^*$ is cyclic. Solution: Suppose $d\mid |G|$ for $G$ subgroup of $F^*$ and $G$ not cyclic. Suppose $A,B$ subgroups of $G$ of order $d$. Then ...
0
votes
1answer
72 views

What is the orbit of a set?

I have the following question: Let $ G = S_4 $. What is $\textrm{Orb}(H)$ (under $G$) when $H =V_4$, $H = \textrm{Sym} \{1,2,3\}, H = <(1234)>$ ($G$ is acting by conjugation)? I don't quite ...
0
votes
3answers
45 views

Algebra I: Cyclic Generators

The direct product $\mathbb{Z}_{45} \times \mathbb{Z}_{98}$ is cyclic and isomorphic to $\mathbb{Z}_{4410}$ because $gcd(45,98)=1$; furthermore the element $n=([1]_{45},[1]_{98})$ is a cyclic ...
0
votes
1answer
33 views

All permutations from $S_6$ and $S_7$ by which $(1,2)(3,4,5)$ is conjugate to itself

The task is to find all permutation $\tau$ from $S_6$ and $S_7$ such that: $$\tau^{-1}(12)(345)\tau=(12)(345)$$ I think the answer is: $\{id \in S_6 , id \in S_7 , (67) \in S_7\}$ I would just like ...
1
vote
0answers
60 views

A group theory problem

Having trouble with a group theory question. By 'trouble' I mean that I, nor my peers, haven't any idea where to start or go. Let $a, b \in \mathbb{Z}$ with $0 < a, b < n$. Define $S_{a,b}=\{ ...
1
vote
2answers
18 views

Suposse that U is a finite non empty subset of G and that it is closed under multiplication. Prove that U is a group.

Suppose that $U$ is a finite non empty subset of $G$ and that it is closed under multiplication. Prove that $U$ is a group. So if it is non empty and closed under multiplication all that is needed to ...
1
vote
1answer
49 views

Example of an extension group

If you have short exact sequence $$K \;\overset{f}{\rightarrow}\; G \;\overset{g}{\rightarrow}\; Q$$ (where $f$ is monomorphism and $g$ is epimorphism) then $G$ is said to be an extension of $K$ by ...
1
vote
2answers
40 views

Proof of group isomorphism

Text of problem: "You have group (K, ·), and it have two normal subgroups: G and H. $ G⋂H $= {1} and group generated by $G∪H = K. $ Write a proof that mapping alpha : G × H → K defined by: alpha((g, ...
0
votes
1answer
35 views

D_4 cannot be written as direct product of groups A x B.

Show that none of the following groups is a direct product of groups (A x B). (a) D_4 , with cardinality 8 (b) D_5, with cardinality 10. Attempt: We need to show for (a) that D_4 is not isomorphic ...
0
votes
1answer
33 views

Cyclic and abelian groups

Just looking for the criteria which I would use to say if these groups are cyclic. Like a short proof? for (i), (ii), (iii), (iv) (v) Thank you.
1
vote
2answers
39 views

Prove this wreath product is a group [Homework]

I'm not usually one to post unworked problems here... I usually try to at least have an attempt, but unfortunately in this case I'm unable to even get an intuitive sense of what's going on here - and ...
0
votes
1answer
43 views

Direct product, homomorphism

Prove (G1 x G2)/(K1 x K2) is isomorphic to G1/K1 x G2/K2. Attempt: First, we need to show a homomorphism exists, such that f : G1 x G2 → G1/K1 x G2/K2. Second, we need to verify Imf = G1/K1 x ...
1
vote
0answers
49 views

showing the group is abelian [duplicate]

I need someone to guide me to solve the following problem Let $a, b \in G$. a. f $(a *b)^i = a^i * b^i$ for three consecutive natural numbers, then show that $G$ is Abelian. b. If the above ...
2
votes
1answer
25 views

Factor groups and Burnside's lemma

I'm supposed to find the number of orbits in $\{1,2,3,4,5,6,7,8\}$ under the cyclic subgroup generated by $(1,3,5,6)$ of $S_8$. I would have very much appreciated an explanation to this exercise since ...
4
votes
1answer
135 views

Abelian group generators and relations

(a) Define what it means for an abelian group to be finitely generated. Explain the terms elementary divisors and rank of $G$ and describe the structure theorem for finitely generated abelian ...
1
vote
1answer
43 views

Prove that the center of group G is a subgroup of G.

By definition, the center of G is the set: $Z(G)$ = {$g\in G|g^{-1}xg=x$ for all $x\in G$} We need to show that: The identity element exists It is closed under the operation For every element $g$, ...
1
vote
1answer
35 views

Prove that Z(G) which is the center of G is a subgroup of G

Question: Let G be a group.Prove that Z(G) is a subgroup of G. If i want to show that Z(G) is the subgroup of G that means i have to show that it is closed under group operation? Here is my attempt. ...
0
votes
0answers
62 views

Alternative proof only two groups of order 6

I need to prove that a group of order 6 is either $C_6$ or $S_3$. I'm aware that this questions has been answered several times, but I'm restricted in the methods I'm allowed to use. The first one is ...
0
votes
1answer
62 views

Showing that a group is simple.

Given a non-commutative group of order $\mathrm{ord}(G)=343$. Prove that $G$ is a simple group. So I have to show that the only normal subgroups are the trivial group and the group itself. But I ...
1
vote
1answer
55 views

Orbit-Stabilizer Theorem proofs?

I posted 3 full questions to give context. But my main problem is the second part of the questions and how would they be answered/proved.
0
votes
2answers
30 views

Set or Ring, and group of units?

I have a couple of questions. I understand the axioms needed for a ring. But am confused about a unitary ring? does this just mean its a ring but has to have the unit 1? Also I do not understand ...
0
votes
1answer
55 views

Proving that Aut($S_3$) is isomorphic to $S_3$

I'm doing an exercise were I had to first prove that all automorphisms of $S_3$ induce a permutation in $X= \{ \alpha \in S_3 \, / \, $order$(\alpha) = 2\}$, which was easy enough. Now I have to ...
0
votes
1answer
30 views

Symmetry Group Regular Tetrahedron

Looking for some help of how to do this, which could also be expanded to other shapes. Thanks.
0
votes
0answers
31 views

Test if a group is abelian or not? [duplicate]

$a$ and $b$ both being elements of a group $(G, \ast)$ and for all $a,b$ belonging to $G$, $$(a*b)^n = (a^n)*(b^n)$$ holds for three consecutive integers $n, n+1,n+2$. We have to show that its an ...
0
votes
0answers
25 views

About First Orthogonality theorem

Let $G$ be a finite group, $(U,\theta_1)$ and $(V,\theta_2)$ be irreducible $k$-representations, $m=\dim_k U$ and $n=\dim_kV$. By the way, $K$ is an algebraically closed field. Let ...
0
votes
1answer
49 views

Group Theory, Permutations.

Let $x := (175)(2436)$ and $y := (1234567)$ Compute the elements $x^{-1}yx$ and $y^{-1}xy$? And if $z := (1326745)$ for which $u$ is there an element $v$ such that $v^{-1}zv=u$. If $u$ exists find ...
2
votes
2answers
60 views

Number of homomorphisms from $Z_4 \times Z_4$ to $Z_4$

So, I have to count number of homomorphisms like: $ \beta:(\mathbb{Z}_4,+)\times (\mathbb{Z}_4,+) \to (\mathbb{Z}_4,+) . $ I count order of the $(\mathbb{Z}_4,+) $, it is $4$. So the elements ...
0
votes
1answer
45 views

group algebra $kS_2$ isomorphic to direct product of matrix algebras

if we let $k=\mathbb{C}$, I need to show that the group algebra $kS_2$ isomorphic to direct product of matrix algebras. Another question is if it is true for an arbitrary field $k$. If we let ...
-1
votes
1answer
35 views

Define product of cyclic groups isomorphic to other group [closed]

My task is: define product of cyclic groups $ (\mathbb Z_{n_1},+) \times (\mathbb Z_{n_2},+) \times \dots \times (\mathbb Z_{n_\ell},+)$ isomorphic to group $ (\mathbb Z^{\times}_{20},\cdot) $. How do ...
1
vote
2answers
22 views

if $f$ is of prime order, then can the orbit of $s$ under $f$ have one element?

if $f\in A(S)$ has order $p$, $p$ a prime, show that for every $s\in S$ the orbit of $s$ under $f$ has one or $p$ elements.* Since the cyclic group generated by $f$ has $p$ elements, therefore ...
0
votes
5answers
58 views

In a group $G$, if $aba^{-1}=b^i$, show that $a^rba^{-r}=b^{(i^r)}$ for all positive integers $r$.

In a group $G$, if $aba^{-1}=b^i$, show that $a^rba^{-r}=b^{(i^r)}$ for all positive integers $r$. If we interpret it as a rule then we will that $a(aba^{-1})a^{-1}={(b^{i})}^i=b^{i^2}$ and so ...