Skip to main content

Questions tagged [quotient-group]

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

Filter by
Sorted by
Tagged with
-2 votes
1 answer
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 $x\in\mathbb{R^*}$, ...
Mathew's user avatar
  • 17
3 votes
1 answer
140 views

If $G/Z(G)$ is isomorphic to a subgroup of $\mathbb Q$ then $G$ is abelian.

I want to show the following two statements: a) Let $G$ be a group such that $G/Z(G)$ is isomorphic to a subgroup of $\mathbb Q$ then $G$ is abelian. For part a) I have the following facts "If $G/...
Fuat Ray's user avatar
  • 1,150
0 votes
0 answers
30 views

Quotients of triangle groups

I want to find all quotients of ordinary hyperbolic triangle group (2,3,8). In general the "Triangle type" indicates the three positive integers p, q and r in the defining presentation ...
Zahid Malik's user avatar
1 vote
1 answer
45 views

Very basic Question regarding quotients of free abelian groups [closed]

I recently stumbled upon this something that I found a little bit confusing. So say we have some finite free abelian group B of rank m, so we have an expression of B as the direct sum $B = \mathbb{Z}...
froitmi's user avatar
  • 87
0 votes
0 answers
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 $|H \backslash G / K|$, $|H:H'|$...
J. S.'s user avatar
  • 414
0 votes
1 answer
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 ...
JAG131's user avatar
  • 917
0 votes
0 answers
78 views

Prove that $U(40)/U_8(40) $ is cyclic but $U(40)/U_5(40)$ is not cyclic.

I was reading the chapter about Factor groups in Joseph A. Gallian's Contemporary abstract algebra and I ran into the following problem. $$\text{Prove that }U(40)/U_8(40) \text{ is cyclic but } U(40)/...
Selim Bamri's user avatar
2 votes
1 answer
50 views

Is there a name for the arc $\mathbb{S}^1 / (x \sim360 - x)$

I was playing with some ideas in a vague way and I have encountered this structure that arises from taking the space of angles $\mathbb{S}^1$ and quotienting it by the relation $(x, 360-x)$ (here $360$...
Sidharth Ghoshal's user avatar
0 votes
1 answer
50 views

Every abelian quotient group is a quotient group of $G/G'$ [duplicate]

A group theory book (not in English) I'm reading states the following: $G/G'$ is the largest abelian quotient group. In fact, every other abelian quotient group is also a quotient group of $G/G'$, ...
M_N1's user avatar
  • 149
1 vote
0 answers
41 views

The free product has the direct product as a factor group. What's the corresponding normal subgroup?

Let $G$ and $H$ be groups. Consider the free product $G * H$ and the direct product $G \times H$. There is a particular way of identifying $G \times H$ as a factor group of $G * H$. Namely, the ...
Dannyu NDos's user avatar
  • 2,049
0 votes
0 answers
18 views

Sequence involving plane curves is exact

I'm reading through Andreas Gathmann's Algebraic Curves and there is this affirmative on Proposition 2.10 at page 14: Let $P\in\mathbb{A}^2$, and $F, G, H$ curves (or polynomials). If $F$ and $G$ have ...
Mand's user avatar
  • 303
0 votes
1 answer
37 views

Cohomology ring of $S^2 \times S^2$ and $P2\#\bar{P2}$ are not isomorphic.

I want to see an algebraic proof for the fact that cohomology rings $H^*(S^2\times S^2; R)$ and $H^*(\mathrm{P}2\#\bar{\mathrm{P}2}; R)$ are not isomorphic if $2^{-1}\notin R$. The algebras are : $H^*...
Mukilraj K's user avatar
0 votes
1 answer
57 views

Closed subgroups of the additive group of adeles and their "modulo 2" quotients

I was wondering if in the adeles ring over $\mathbb{Q}$, viewed as an additive group, any closed subgroup H is such that $((1/2)H)/H$ is finite or compact, where $1/2H$ is the group of adeles $x$ such ...
Eric Chopin's user avatar
0 votes
0 answers
57 views

Quotient group of G with Commutator [G,A] is equal to Centraliser $G / [G,A] = C_{G / [G,A]} (A)$

Edit: $G$ is a $\pi$-group and $A$ a $\pi'$-group acting on $G$. Suppose either $G$ or $A$ is solvable. I have a hard time understanding this statement. For $G,A$ it is $$G / [G,A] = C_{G / [G,A]} (A)$...
Stippinator's user avatar
1 vote
1 answer
46 views

Ordering of group elements in MultiplicationTable vs CosetTable in GAP

I construct a finitely presented discrete group $G$ in GAP, a normal subgroup $H\triangleleft G$ of finite index $N$, and the factor group $G/H$ of order $N$. Assume for simplicity that $G$ has two ...
MathPhysGeek's user avatar
-1 votes
1 answer
83 views

Decomposition of $z^q-z$ with $q=p^n$ in $\mathbb{K}[x]$ with $|\mathbb{K}|=q$. [duplicate]

Let $q=p^n$ be a prime power, and $\mathbb{K}$ a field of $q$ elements. We need to show that $z^q-z$ decomposes into linear polynomials in $\mathbb{K}[z]$. Have been reading notes all over and suspect ...
Alex A.G.'s user avatar
  • 177
1 vote
0 answers
40 views

Subgroups and Quotient Groups of Infinite bounded Abelian p-groups

I am looking to a reference to a solution of Exercise 13 (page 91) of L. Fuchs, Infinite Abelian Groups, Vol. 1, Academic Press, 1970. There is no solution to this exercise in a collection Exercises ...
George's user avatar
  • 11
1 vote
0 answers
86 views

Prove that $\mathbb{R}[x]/(x^2+1)^2$ is isomorphic to $\mathbb{C}[y]/(y^2)$ as a $\mathbb{C}$-algebra.

How could I prove that $A=\mathbb{R}[x]/(x^2+1)^2$ is isomorphic to $\mathbb{C}[y]/(y^2)$ as a $\mathbb{C}$ algebra? Are there any known isomorphisms or is there a trivial way to do so? I have been ...
Alex A.G.'s user avatar
  • 177
1 vote
0 answers
87 views

Let $R$ be a P.I.D. and let $a \neq 0$ be an element in $R$. Prove that for a prime element $p$, $p(R/(a)) = ((p) + (a))/(a)$.

Let $R$ be a P.I.D. and let $a \neq 0$ be an element in $R$. Prove that for a prime element $p$, $p(R/(a)) = ((p) + (a))/(a)$. This is something that came up while reading Dummit and Foote's textbook ...
Squirrel-Power's user avatar
0 votes
0 answers
37 views

Rules of Products of Quotient Groups [duplicate]

Suppose that you have 2 groups, $G$ and $H$, such that $H\trianglelefteq G$. I was wondering if we can say that $$ (G/H) \times H \cong G $$ I know some rules about product and quotient groups work &...
Joshua G-F's user avatar
4 votes
2 answers
64 views

Example for $[G : G\cap H] \neq [H:G\cap H]$ with isomorphic subgroups $G \cong H$?

For isomorphic finite subgroups $G,H$ of a group $A$ it holds $[G : G\cap H] = [H: G\cap H]$, since $[G : G\cap H] = \frac{|G|}{|G\cap H|} = \frac{|H|}{|G\cap H|} =[H: G\cap H]$. Does this also hold ...
psl2Z's user avatar
  • 2,813
1 vote
2 answers
56 views

Let $K\le H\le G$ with $[G:H]<\infty$. Is $[G:K]<\infty$ necessarily?

While reading this proof for the Tower Law for Subgroups, the data for the theorem only stipulates that H is a subgroup of G with finite index, and that K is a subgroup of H. Later in the proof ...
giorgio's user avatar
  • 583
1 vote
1 answer
75 views

If $G \cong G'$ and $H \cong H'$ where $H \trianglelefteq G$ and $H' \trianglelefteq G'$, is then also $G/H \cong G'/H'$?

This seems pretty trivial, but I used it in an assignment and want to double-, triple check. If $G \cong G'$ and $H \cong H'$ where $H \trianglelefteq G$ and $H' \trianglelefteq G'$ are normal ...
soggycornflakes's user avatar
2 votes
1 answer
72 views

Showing that a normal subgroup is equal to the kernel of a homomorphism.

I'm having trouble understanding part of the proof of the classical version of the Seifert-Van Kampen theorem in Munkres: Theorem 70.2 (Seifert-Van Kampen theorem, classical version). Let $X=U\cup V$,...
Ricky's user avatar
  • 3,165
2 votes
1 answer
100 views

Elements of $\mathbb{Z}/p\mathbb{Z}$ and prove it's simple group

I'm trying to understand properties of groups. Since $\mathbb{Z}/p\mathbb{Z}$ ($p$ is prime) is quotient group It should be set of all left cosets of $p\mathbb{Z}$ in $\mathbb{Z}$ which is, $ \{ m+pn ...
user avatar
2 votes
1 answer
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 ...
user avatar
2 votes
1 answer
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 ...
azheng's user avatar
  • 21
3 votes
2 answers
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,...
Spectral's user avatar
1 vote
1 answer
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 ...
hirdajarzu's user avatar
1 vote
1 answer
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 ...
am567's user avatar
  • 329
0 votes
0 answers
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 ...
Dungessio's user avatar
  • 137
0 votes
1 answer
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 ...
hirdajarzu's user avatar
1 vote
0 answers
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 ...
X.Huang's user avatar
  • 11
9 votes
3 answers
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. ...
wsz_fantasy's user avatar
  • 1,722
1 vote
1 answer
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 ...
Al.G.'s user avatar
  • 1,490
0 votes
1 answer
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 ...
am567's user avatar
  • 329
4 votes
2 answers
482 views

$H$ acts on $G/H$. Show that $\mathrm{Fix}(H) = G/H$

So $H$ acts on $G/H$. Where $H$ is a subgroup of $G$. We are also given that $|G| = p^2$ and $|H| = p$, where p is a prime. $$ H \times G/H \rightarrow G/H, \quad (h, gH) \rightarrow h * (gH) = (hg)H $...
GodelEscher's user avatar
1 vote
0 answers
55 views

Is there a way to relate the real numbers mod 1 to quotient groups?

My group theory textbook gives the following definition for the real numbers mod 1 Let $G = \{x \in \mathbb{R} \mid 0 \leq x < 1\}$ and for $x, y \in G$ let $x + y$ be the fractional part of $x + ...
Ethan Kharitonov's user avatar
1 vote
3 answers
80 views

Simple question concerning definition of the equivalence relation and the group operation in quotient group

I have simple, yet critical questions for my understanding on the quotient group $G/H$. Remember that $H$ is a sub group of $G$. So we say that, for any given element $x,y \in G$, $x$ and $y$ are ...
X0-user-0X's user avatar
1 vote
1 answer
65 views

Series of nested normal subgroup (composition series) induces a sequence of quotient groups

In Group theory in a nutshell by A. Zee on pg. 66 he introduces sequences of nested normal subgroups: $G \rhd H_1 \rhd H_2 \rhd \dots \rhd H_k \rhd I$. Then he says that this induces a sequence of ...
Jens Wagemaker's user avatar
0 votes
1 answer
90 views

Let $G$ be a group then does $\{e\}/G$ make sense?

For context, I was trying to show that the quotient of a solvable group is solvable. For a normal subgroup $N$ I was able to show that ${(G/N)}^{(k)}={G}^{(k)}/N$ and thus by using the fact that $G$ ...
Al-Amin Miah's user avatar
1 vote
0 answers
52 views

Index of the multiple of a group in itself

Let $m, n \geq 1$, prove that $[\mathbb{Z}/m \mathbb{Z} : n\left( \mathbb{Z}/m \mathbb{Z} \right)] = \text{gcd}(m, n)$ When writing out the index as the cardinality of the quotient space, this looks ...
strugglingStudent's user avatar
3 votes
2 answers
191 views

What are the elements of the group $15 \mathbb Z/ 15 \mathbb Z$?

My question is: What are the elements of the group $15 \mathbb Z/ 15 \mathbb Z$? I am confused between if it is the 0 coset or the 1 coset. I want to conclude that $$12 \mathbb Z/ 24 \mathbb Z \times ...
Idonotknow's user avatar
1 vote
1 answer
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 $N$."
FSY's user avatar
  • 37
0 votes
1 answer
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 ...
GodelEscher's user avatar
2 votes
1 answer
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 ...
Alice in Wonderland's user avatar
0 votes
1 answer
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-...
J P's user avatar
  • 343
1 vote
2 answers
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 ...
J P's user avatar
  • 343
-2 votes
1 answer
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 ...
A. H's user avatar
  • 1
2 votes
2 answers
59 views

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 $...
Emmy N.'s user avatar
  • 1,361

1
2 3 4 5
18