Questions tagged [quotient-group]

This tag is for questions relating to "Quotient Group".

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3
votes
0answers
42 views

Quotient group of $\mathbb{C^*}$ by $n$th roots of unity

I need to find image and kernel of the following homomorphism of groups ($\mathbb{C^*}$ is multiplicative group of complex numbers) $ f: \mathbb{C^*} \rightarrow \mathbb{C^*}$ given by $f(z)=z^n$. I ...
3
votes
1answer
57 views

Groups of order $16$ with a cyclic quotient of order $4$

Question: I am interested in (a) listing the groups $G$ of order $16$ which have a cyclic quotient of order $4$; (b) in each case knowing in how many essentially different ways this occurs (...
1
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0answers
28 views

Tangent space of $G/H$ at the identity

Let $G$ be a compact Lie group and let $H$ be Lie subgroup of $G$ with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ respectively. 1)- How do we prove that : $$T_{[e]} (G/H) \simeq \mathfrak{g}/\...
1
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0answers
22 views

Is a group formed by a partition always a quotient group? [duplicate]

Let $G$ be a group. If $N$ is a normal subgroup of $G$, the set of cosets of $N$ in $G$ form a group under set multiplication - this is the quotient group $G/ N$. I'm trying to improve my intuition ...
0
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1answer
69 views

Question about "Quotient Group of Cyclic Group is Cyclic"

I found a proof of the fact that if $G$ is a cyclic group and $H$ is a subgroup of $G$, then $G/H$ is a cyclic subgroup. They don't mention that $H$ is a normal subgroup. But to define the quotient ...
0
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2answers
65 views

Is there a way to prove other than using a counterexample that $\sum_4/V_4$ is not abelian?

Suppose $\sum_4/V_4$ abelian, then if $a,b\in \sum_4$, then $aV_4bV_4=bV_4aV_4$, so $abV_4=baV_4$, which is only possible if there exists $ a^{-1}b^{-1}ab\in V_4$. If $a=(1,2),b=(1,2,3,4)$ then $a^{-1}...
-5
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1answer
45 views

Order of quotient group [closed]

What is order of quotient group G/G and {e}/{e} ? Please someone help me with proper explanation.. Thanks in advance 🙏
1
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0answers
44 views

Element of a quotient group that equals to 0

I may be a bit confused but somehow I do not see why the following holds: Let $R$ be a ring and $I$ an ideal. Then we have the quotient $R/I$. Let $a+I \in R/I$. Why does it hold that if $a+I = 0 \...
2
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0answers
73 views

Centers and Quotients of Groups

This question occured to me when I was working with upper central series: Let $K\le N$ be normal subgroups of $G$. Then there is some normal subgroup $M$ of $G$ satisfying $M/N = Z(G/N)$. I claim that ...
2
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0answers
29 views

All $p$-groups have a normal subgroup of each possible order. [duplicate]

I was trying to prove that a group $G$ of order $p^n$, where $p$ is prime, has a normal subgroup of each order $p^i$, $1\leq i\leq n$. This has been asked before (for example here) but I want to know ...
0
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1answer
50 views

Number of subgroup $G<\Bbb Z^3$ such that $\Bbb Z^3/G\simeq \Bbb Z/3\Bbb Z\oplus\Bbb Z/3\Bbb Z$ [closed]

Compute the number of subgroups $G<\Bbb Z^3$ such that $\Bbb Z^3/G\simeq \Bbb Z/3\Bbb Z\oplus\Bbb Z/3\Bbb Z$. Possible forms of $G$ are $G = \Bbb Z\oplus 3\Bbb Z\oplus 3\Bbb Z$, $3\Bbb Z\oplus 3\...
2
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1answer
43 views

Finding quotient of additive abelian group in Sage

I have $G=\mathbb{Z}^3$, $H=\langle(3,2,4),(6,1,7),(2,3,6)\rangle \leq G$. My task was to find the quotient $G/H$ as the direct product of infinite cyclic groups and prime-power-order finite groups $(\...
0
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1answer
97 views

Understanding a proof of: If $N\unlhd G$ s.t. $N$ and $G/N$ are solvable, then $G$ is solvable.

Let $N$ be a normal subgroup of $G$ s.t. $N$ and $G/N$ are solvable, then $G$ is solvable. Proof: Because $N$, $G/N$ are solvable $\Rightarrow$ $N^{(s)}=\{e\}$, $(G/N)^{(t)}=\{e\}$ for $s,t\in \mathbb{...
-1
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0answers
41 views

Show the isomorophism between a factor group and a cyclic group

Suppose that $G$ is a finite abelian group and $N$ is a normal subgroup it. Suppose that, $G\simeq \mathbb{Z}_a\times \mathbb{Z}_b$ and $N\simeq \mathbb{Z}_a\times \mathbb{Z}_d$ where $d$ is a ...
2
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0answers
37 views

Mistake in my Second Isomorphism Theorem example

Suppose that $G = \mathbb{Z}$, $N = 6\mathbb{Z}$, and $H = 4\mathbb{Z}$. Is $HN = 24\mathbb{Z}$? Is $H \cap N = 12\mathbb{Z}$? Is $HN/N \cong H/(H \cap N)$? I am having trouble with why this ...
1
vote
1answer
79 views

Justify $\frac{\mathbb{Z}_a\times \mathbb{Z}_b}{\langle a/c\rangle \times \langle b/d\rangle}$ is isomorphic to $\mathbb{Z}_c\times \mathbb{Z}_d$.

Let $c\mid a, d\mid b$ in $\mathbb{N}$. Then the external direct product $\mathbb{Z}_a\times \mathbb{Z}_b$ has a subgroup $\langle a/c\rangle \times \langle b/d\rangle$ of order $cd$. To examine ...
0
votes
1answer
134 views

When is $G/N \cong H/K$?

Suppose the following: $G \cong H$ $N \trianglelefteq G$ $K \trianglelefteq H$ $N \cong K$ What is the requirement that $\alpha: G/N \to H/K$ is an isomorphism? I found a counterexample like for $G =...
2
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0answers
93 views

Under what conditions can we say that $G/N \cong S \implies G \cong N \times S$?

For a recent project (which I have since completed) I needed to derive the automorphism group of the cube graph, and I wanted to do so with some reasonable degree of rigor. I defined a group action of ...
1
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0answers
20 views

Quotients of automorphism groups of finite field extensions

If $k\subset \ell\subset m$ are finite fields, then $\mathrm{Aut}(\ell /k)$ is the quotient $\mathrm{Aut}(m/k)/\mathrm{Aut}(m/\ell)$ (correct me if I'm wrong). Does this work in a more general setting?...
2
votes
1answer
71 views

Let $H$ be the subgroup of $\Bbb Z^3$ generated by elements $(5,−4, 3), (7, 2, 3)$ and $(21, 8, 9)$. Classify the factor group $\Bbb Z^{3}/H$.

I am studying for an abstract algebra exam and am working through practice problems provided by the professor. The problem statement is: Let $H$ be the subgroup of $\Bbb Z^{3}$ generated by elements $(...
1
vote
1answer
50 views

Cyclicness of a quotient of subgroups of infinite cyclic group

Let $A$ be an abelian group and let $G(A) = A \times S_3$ be the direct product of $A$ and the symmetric group of $3$ elements $S_3$. We define the following operation on $G(A)$: For each $(x,\sigma),(...
1
vote
1answer
121 views

Discuss $\mathbb R[X]/(aX^2 +bX + c)$ in terms of $\Delta = b^2-4ac$

Discuss $\mathbb R[X]/(aX^2 +bX + c)$ in terms of $\Delta = b^2-4ac$. I've already found that $\mathbb R[X]/(aX^2 +bX + c) \simeq \mathbb{R} \times \mathbb{R}$ if $\Delta > 0$ and $\mathbb R[X]/(...
0
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0answers
183 views

If $G$ is finitely generated with $\mathcal{Z}(G)$ finite, $N \triangleleft G$ finite, then $\mathcal{Z}(G/N)$ is finite.

I'm looking for a reference in the literature for the fact stated in the title, i.e. If $G$ is a finitely generated group with $\mathcal{Z}(G)$ finite and $N$ is a finite normal subgroup of $G$, then $...
0
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0answers
60 views

Quotient $G/N$ of solvable group $G$ is solvable. Is my (trivial) proof correct?

I'd like some feedback from experts in group theory. Let $N \trianglelefteq G$, $G$ solvable. Then $G/N$ is solvable. My proof: Take canonical homomorphism $G \rightarrow G/N$ ($x \rightarrow xN$) and ...
1
vote
0answers
58 views

Abelianization of quotient group

Let $G$ be a group. Let $N$ $\trianglelefteq$ $G$. We define $\phi$ : $G$ $\rightarrow$ $G^{ab}$ a homomorphism, where $G^{ab} : = G/[G,G]$ is the abelianization of group $G$. Is it true that $G^{ab}/\...
0
votes
2answers
84 views

Is it true that for a Group $G$ with Normal Group $N: G/N = GN/N$?

Is it true that for a Group $G$ with Normal Group $N: G/N = GN/N$? I think the statement is correct. But why do we have to write: $[G,G]N/N$ here instead of just $[G,G]/N$? Thanks!
5
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2answers
71 views

Let $H\leq G$. Prove $x^{-1}y^{-1}xy\in H\text{ }\forall x,y\in G \iff H\trianglelefteq G \text{ and } G/H \text{ is abelian}$.

Question: Let $H\leq G$. Prove $x^{-1}y^{-1}xy\in H\text{ }\forall x,y\in G \iff H\trianglelefteq G \text{ and } G/H \text{ is abelian}$. my thoughts: In the forward direction, if $x^{-1}y^{-1}xy\in ...
0
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0answers
62 views

Another version of correspondence theorem.

Let $X,Y$ be a sets and $\sim$ is an equivalence relation on $X$. $X/\sim$ is the quotient set or partition by $\sim$. Let us consider the sets $\{\tilde f:X/\sim\to Y\}$ and $\{f:X\to Y:f$ has the ...
0
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0answers
21 views

The cokernel of $f(x,y,z) = (2x-y,2y-z,2z)$ over $\mathbb Z$

Let $L$ denote the subgroup $$ \langle (2,0,0),\ (-1,2,0),\ (0,-1,2) \rangle $$ of $\mathbb Z^3 = \mathbb Z \oplus \mathbb Z \oplus \mathbb Z$. Then the quotient group $ G = \mathbb Z^3 / L $ is a ...
2
votes
1answer
75 views

Find the cosets in $G/K$

I'm trying to find the cosets in $G/K$ and write down the multiplication table of $G/K$ for $G = ⟨a⟩ × ⟨b⟩$, where $o(a) = 8$ and $o(b) = 2$, and $K = \langle (a^2, b) \rangle$. (In other words, $G=...
0
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1answer
58 views

What is the operation through which a Coset is created?

Most books initially explain Cosets in terms of $\mathbb{Z}$ & then use the subgroup $n\mathbb{Z}$ to partition into $n$ Cosets. Is the group $\mathbb{Z}$ used here the additive $\mathbb{Z}^+$ or ...
1
vote
1answer
53 views

How do I prove this equality with cardinalities?

I have the following problem and I am a bit stuck. Let $G$ be a finite group acting on $X$. We first consider the case that $X=G/H$ where $H$ is a subgroup of $G$. Here we consider the natural action $...
0
votes
1answer
130 views

Suppose that $H$ and $K$ are normal subgroups of $G$ such that $H \cap K=\{e\}$. Show that $G$ is isomorphic with some subgroup of $G/H \times G/K$. [duplicate]

Let $G$ be a group with identity $e$. Suppose that $H$ and $K$ are normal subgroups of $G$ such that $H \cap K=\{e\}$. Show that $G$ is isomorphic with some subgroup of $G/H \times G/K$. It seems ...
2
votes
1answer
110 views

To prove $O/aO$ is finitely generated

Let $R$ be one dimensional Noetherian integral domain and $K$ be it's fraction field. Let $L/K$ be a finite extension, and $O$ be the integral closure of $R$ in $L$. Then, why $O/aO$ ($a$ is an ...
2
votes
0answers
60 views

Correspondence theorem: Subgroups of $\mathbb Z^2$ that contain $2\mathbb Z \times 4\mathbb Z$

Determine, and identify, the subgroups of $G=\mathbb Z^2$ that contain the smaller subgroup $H=2\mathbb Z \times 4\mathbb Z$. Note: I would like to identify subgroups not just determine them like in ...
2
votes
1answer
75 views

finitely generated quotient group: From $G/H = \langle g_1H, \ldots, g_m H \rangle$, we have $G=\langle g_1, \ldots, g_m \rangle H$?

Please help me with my own proof for this: A group with finitely generated normal subgroup and finitely generated quotient is finitely generated itself Let $G$ be a group with $G \trianglerighteq H$ ...
1
vote
0answers
26 views

If $G / \gamma_2(G)$ is finitely generated, then $\gamma_i(G)/\gamma_{i+1}(G)$ is finitely generated. [duplicate]

I am having some difficulties proving that if $G/\gamma_2(G)$ is finitely generated, then $\gamma_i(G)/\gamma_{i+1}(G)$ is finitely generated. This is a well-known fact. Assume that $G/\gamma_2(G)=\...
0
votes
0answers
57 views

Application of correspondence theorem.

Let $q:G\to G/H$ be the quotient map where $H$ is a normal subgroup. Call the set of all subgroups containing $H$ by $\mathcal K$ and the set of all subgroups of $G/H$ by $\mathcal L$. Then define $\...
0
votes
0answers
16 views

Computing quotient groups arising from Baer sum

My question is a general one about computing quotient groups of (a subgroup of) a direct sum of $\mathbb{Z}$'s and $\mathbb{Z}/n\mathbb{Z}$'s, but it arises from computing the Baer sum of two ...
1
vote
1answer
53 views

If $N$ and $H$ are normal subgroups of $G$, then $N\cap H$ is also normal in $G$ and $G/(N\cap H)$ is isomorphic to a subgroup of $(G/N)\times(G/H)$?

I've already shown that $N\cap H$ is a normal subgroup of $G$, so I just need to determine an isomorphism. My problem is that I'm not entirely sure I understand what $(G/N)\times(G/H)$ looks like. I ...
1
vote
1answer
99 views

Classification of $\mathbb{Z}\times \mathbb{Z} \times \mathbb{Z} / \langle (n,n,n) \rangle$

I want to classify the abelian groups of the form \begin{align} \mathbb{Z}\times \mathbb{Z} \times \mathbb{Z} / \langle (n,n,n) \rangle \end{align} using fundamental theorem of finitely generated ...
1
vote
0answers
78 views

In $F=F^{ab}(A)$, define $f\sim f'$ if and only if $f-f'=2g$ for some $g\in F$. Show $F/\sim$ is finite if and only if $A$ is finite.

this is a question from Aluffi's Algebra: Chapter 0, Exercise II.5.10. The full question is the following: Given a free abelian group over a set $A$, denoted $F=F^{ab}(A)$, we define an equivalence ...
2
votes
2answers
59 views

If $(g,a) \mapsto g \cdot a$ is an action of $G$ on $A$ and $H \vartriangleleft G$, is $(\overline{g},a) \mapsto g \cdot a$ an action of $G/H$ on $A$?

I'm learning about group actions from Dummit and Foote, and I have a question about the following remark that the authors make in section 4.1: In particular, an action of [a group] $G$ on [a nonempty ...
2
votes
2answers
89 views

The structure of $\Bbb Z\times\Bbb Z/H$, where $H=\langle(3,-2)\rangle$

Consider the group $\DeclareMathOperator{\bZ}{\mathbb{Z}}\DeclareMathOperator{\bN}{\mathbb{N}}G=\bZ\times\bZ/H$ where $H=\langle(3,-2)\rangle$ (in general it looks like it works the same for subgroups ...
1
vote
0answers
51 views

Let $G$ be abelian s.t. $r_p(G)<\infty$ if $p=0$ or a prime. Prove $G/nG$ is finite for all $n>0$.

This is Exercise 4.3.9(b) of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. This question is referred to later on in the text. The ...
0
votes
1answer
65 views

Let $p$ be odd prime.Why $a,b∈\mathbb Z_p^×-{(\mathbb Z_p^×)}^2$ implies $a/b∈{(\mathbb Z_p^×)}^2$?

Let $p$ be odd prime.Why $a,b∈\mathbb Z_p^×-{(\mathbb Z_p^×)}^2$ implies $a/b∈{(\mathbb Z_p^×)}^2$ ? For example, $2, -1∈\mathbb Z_3^×-{(\mathbb Z_3^×)}^2$ and $-2∈{(\mathbb Z_3^×)}^2$. I heard the ...
0
votes
0answers
37 views

A homeomorphism to the open upper-half complex plane

I am reading a quick course on modular forms, and I do not understand a lemma: Let $\alpha = \begin{pmatrix}a&b\\c&d \end{pmatrix} \in SL_2(\mathbb{R}), z\in \mathbb{C}^{*}$ (the Riemann ...
1
vote
1answer
99 views

Let $G$ be abelian s.t. $r_p(G)<\infty$ if $p=0$ or prime. If $H\le G$, show $r_p(G/H)\le r_0(G)+r_p(G)$ for $p>0$.

This is Exercise 4.3.9(a) of Robinson's "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. It is an exercise noted as being referred to later on ...
0
votes
0answers
23 views

How do I identify the quotient groups in $D_{12}?$ [duplicate]

I am given $D_{12}=\{1,x,x^2,x^3,x^4,x^5,y,xy,x^2y,x^3y,x^4y,x^5y\}$. I know that the normal subgroups are as follows: $\langle x\rangle, \langle x^2\rangle,\langle x^3\rangle,\langle x^2,y\rangle,\...
0
votes
1answer
70 views

A natural way of thinking about quotient of groups

Given a ring $A$, the subsets by one could quotient it while maintaining the ring structure are precisely the ideals. This happens because when considering $A$ as a module over itself, the ideals as ...

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