# Questions tagged [quotient-group]

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

879 questions
Filter by
Sorted by
Tagged with
56 views

### Let $\mathbb{R^*}$ be the multiplicative group of nonzero real numbers.Which of the following statements are true? [closed]

Let $\mathbb{R^*}$ be the multiplicative group of nonzero real numbers.Which of the following is true? 1. If $H\subseteq\mathbb{R^*}$ is a subgroup such that $x^2\in H$ for all $x\in\mathbb{R^*}$, ...
• 17
140 views

• 87
32 views

### Counting the number of double cosets for subgroup

Let $H,K \le G$ and consider the set of double cosets $H \backslash G / K$. If we have $H' \le H$, is there a formula expressing $| H' \backslash G / K|$ in terms of $|H \backslash G / K|$, $|H:H'|$...
• 414
69 views

### Finitely Generated Group and Isomorphic Copy of Quotient Groups [duplicate]

Given finitely generated (not necessarily abelian) group $G$ and normal subgroup $N\triangleleft G$, then $G/N\cong H\leq G$. Is this statement correct? The reason why I am suspicious this to be true ...
• 917
78 views

572 views

### Does $\mathbb{R}/\mathbb{Q}$ contain a subgroup isomorphic to $\mathbb{Q}$?

I was wondering how to prove that $\mathbb{R}/\mathbb{Q}$ contains a subgroup isomorphic to $\mathbb{Q}$. I know that $\mathbb{R}/\mathbb{Q}$ is torsion-free, but I don't know if using that would help ...
94 views

### In $(G_1\times G_2)/G_2$, I am confused since $G_2$ is clearly not a subgroup of $G_1\times G_2$

I have seen the following expression in the text book of algebra chapter$0$. $(G_1\times G_2)/G_2$. I am confused since $G_2$ is clearly not a subgroup of $G_1\times G_2$, and hence not a normal ...
• 21
217 views

### First Isomorphism Theorem not concerned with Injectivity?

I have a question regarding the application of the First Isomorphism Theorem for Groups in proofs; why are the proofs not concerned with whether the respective map is injective? To clarify my question,...
• 53
1 vote
181 views

### Classify $(\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z})/\left<(1,1,2)\right>$

I am following a solution to classify $G/H$, $G=\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}$, $H=\langle (1,1,2)\rangle$, according to the theorem of finitely generated abelian groups. This is my first ...
1 vote
44 views

### How to represent elements from $\bar{E}=E/Z(E)$ in the form $(a|b)$

From https://arxiv.org/abs/quant-ph/9608006 Background The group $E$ of tensor products $\pm w_{1} \otimes \dots \otimes w_{n}$ and $\pm i w_{1} \otimes \dots \otimes w_{n}$, where each $w_{j}$ is one ...
• 329
113 views

### Order of $\mathbb{Q}_{p}^{\times }/ (\mathbb{Q}_{p}^{\times })^{2}$ [duplicate]

Let $p\neq 2$ be a prime. An element $x\in\mathbb{Q}_{p}$ is a square if and only if it can be written as $x=p^{2n}y^{2}$ with $n\in\mathbb{Z}$ and $y\in\mathbb{Z}_{p}^{\times }$ a $p$-adic unit. The ...
• 137
95 views

### Quotient group and classification of quotient groups $\mathbb{Z}^3/H$

I'm studying quotient groups (by myself) and having a hard time with them. I will reference Calculate the quotient groups and classify $\mathbb{Z^3}/(1, 1, 1)$ - Fraleigh p. 151 15.8 Classification ...
1 vote
45 views

### What is the relationship between between the quotient groups $G/\overline{H}$ and $N(H)/H$?

Given a group $G$ and a subgroup $H\subseteq G$, let $\overline{H}=\langle ghg^{-1}\mid h\in H\text{ and } g\in G\rangle$ be the normal closure of $H$ and $N(H)=\{g\in G\mid gHg^{-1}=H\}$ be the ...
• 11
1k views

### When can we "switch" isomorphic things

I'm finishing a course on basic abstract algebra, which covers groups, rings, modules, and finite group representations. However, up to this point, I am not very sure about the concept of isomorphism. ...
• 1,722
1 vote
69 views

### Prove by universal property that epimorphism onto quotient group kills the subgroup

Today I'm obsessed with universal properties and I'm trying to prove that the universal property of quotient groups implies that the subgroup is killed by the cannonical epimorphism. I hope this ...
• 1,490
106 views

### How do we know that the quotient group $\bar{E} = E/Z(E)$ is an elementary abelian group

My question is: How do we know that the quotient group $\bar{E}=E/Z(E)$ is an elementary abelian group? Please find below some background information on the different relevant groups involved from ...
• 329
482 views

1 vote
102 views

### True/false: If $N\unlhd G$, then for any homomorphism $f:G\to G/N$, the kernel of $f$ contains $N$. [closed]

I want to know whether the following statement is true. And if it is, how one can prove it. "If $N\trianglelefteq G$, then for any homomorphism $f:G\to G/N$, the kernel of $f$ contains $N$."
• 37
67 views

### Proving that $A_4/V_4$ can be generated by $\sigma V_4$ where $\sigma$ is a 3-cycle

Define $V_4 = \{ \text{id}, (1,2)(3,4), (1,3)(2,4), (1,4)(2,3) \}$ I want to prove that given a 3-cycle $\sigma$, $A_4/V_4 = \langle\sigma V_4\rangle$. So far I have proved that $V_4$ is a normal ...
• 342
146 views

### Pontryagin duality and quotient groups

I am studying the Pontryagin duality for LCA groups, and I came across two results in which I am finding some difficulty. Here I will denote by $G^*$ the dual of the group $G$, i.e. the group of all ...
63 views

### If every quotient group of G by non-trivial normal subgroups are finite, then G is finite

If every quotient group of $G$ by a non-trivial normal subgroup is finite, then $G$ is finite. This a statement that I'm supposed to prove if it's true or not. If $G = (\mathbb{Z}, +)$, then all non-...
• 343
1 vote
57 views

### If every quotient group of a group G by non-trivial normal subgroups is abelian, G is abelian

If every quotient group of a group G by non-trivial normal subgroups is abelian, G is abelian. I'm asked about the veracity of this statement. I thought about the group $\mathrm{D_3}$ and the only ...
• 343
65 views

### Quotient Ring example issue

I am currently learning about lattices and came across a Mr.Bill Buchanan video that explains the Ring-LWE problem. I understand how modulo works and quite understand the math behind it. However, I ...
• 1
### Quotient group isomophic to $\mathbb{Z}$
In a certain document I am asked to show that if, given a group $G$, there is a normal subgroup $N$ such that $G/N \simeq \mathbb{Z}$, $\forall n \in \mathbb{N}$ there exists a subgroup $H$ such that \$...