Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [quotient-group]

The tag has no usage guidance, but it has a tag wiki.

2
votes
2answers
29 views

Suppose $H$ is a subgroup of a group $G$ and $aH$ is a left coset. Prove that there exists some $K$ (a subgroup of $G$) , which $aH$ is equal to $Ka$.

Suppose $H$ is a subgroup of a group $G$ and $aH$ is a left coset. Prove that there exists some $K$ (a subgroup of $G$) , which $aH$ is equal to $Ka$. I've tried to show this statement,but I cant ...
2
votes
1answer
41 views

$H$ is a subgroup of a finite group $G$ such that $|H|$ and $\big(|G:H|-1\big)!$ are relatively prime. Prove that $H$ is normal in $G$.

Suppose that $H$ is a subgroup of a finite group $G$ and that $|H|$ and $\big([G:H]-1\big)!$ are relatively prime. Prove that $H$ is normal in G Let $[G:H]=m$ Let $G$ act on set $A$ of left cosets ...
2
votes
1answer
50 views

Show that the quotient group $G/N$ contains a subgroup isomorphic to $H$. [closed]

Let $G$ be a finite group, $N\mathrel{\lhd}G$ a normal subgroup of $G$, and $H\leq G$ a subgroup of $G$. Suppose that $|H|$ and $|N|$ are relatively prime (i.e., $\gcd(|H|,|N|)=1$). Show that the ...
1
vote
2answers
24 views

What is $D_{16}/ Z(D_{16})$?

I was asked the following: Let $D_{16}$ be the dihedral group of order $16$. What is $D_{16} / Z(D_{16})$? I know that the center of $D_{16}$ har order $2$. So therefore, the quotient has order $16/2 ...
0
votes
0answers
17 views

Quotient Set of a Quotient Set

Can anyone help me with this problem? Given: $A = \{a,b,c\}$ $G=I_A \cup\{(a,b), (b,a),(b,c),(c,b)\}$ $H=I_A \cup\{(b,c), (c,b)\}$ (Note: H is a refinement of G) Then: $A|G = \{G_a, G_b,G_c\}$ ...
18
votes
4answers
1k views

Give an example of: A group with an element A of order 3, an element B with order 4, where order of AB is less than 12

I'm a mathematics major studying at University as an undergrad. This is a question on the study guide for the upcoming final in Math 344 - Group Theory: "Give an example of a group G with an element ...
0
votes
0answers
21 views

Quotient group as a manifold

Let $V_k(n) \subset \prod _{i=1} ^k S^n$ the real Stiefel space endowed with subspace topology and defined via $$V_k(n) := \{(v_1, v_2, ..., v_k) \vert \text{ } v_i \bot v_j \text{ for } i \neq j \...
3
votes
1answer
67 views

Is there a general way to calculate the fundamental group of a quotient space?

Suppose $X$ is a path-connected topological space, and $A$ is a path-connected subset of $X$. My question is, is there a way to calculate the fundamental group of the quotient space $X / A$ in terms ...
0
votes
0answers
44 views

Let $G$ be a group and $H, H'$ be subgroups of $G$ where $H$ is normal. Under which circumstances is $H \cap H'$ a normal subgroup of $H$?

To be more specific, what kind of assumptions do I have to make for $H'$ to obtain this assertion? Which one are necessary and which one are sufficient? For instance, we could say $H \subset H'$, but ...
0
votes
1answer
17 views

Question concerning isomorphism of quotient groups

I saw a video that tells that if $p, q$ are integers such that $p|q$, then $Z_q/Z_p$ is isomorphic to $Z_{q/p}$. If it is true, can you give me a hint about how can I prove this?
1
vote
2answers
47 views

Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H \times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$.

Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H \times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$. Also prove that if $G = HK$, then $G/(H∩K)$ is isomorphic to $G/H \...
1
vote
2answers
50 views

$\mathbb{F}_2[X]/(S) \cong \mathbb{F}_4 $

Let $S(X) = X^2 +X+1 \in \mathbb{F}_2[X]$ Prove that $\mathbb{F}_2[X]/(S) \cong \mathbb{F}_4 $ What I did: $\{1, X \}$ is a basis of $\mathbb{F}_2[X]/(S)$ and S is irreducible in $\mathbb{F}_2$ so $...
0
votes
0answers
43 views

General concept of 'quotient'

Working with group theory I've found multiple times the idea of quotient group as $G/H = \{gH\ |\ g\in G, H < G\}$. Nevertheless, you can find similar things in vectorial spaces as $\mathbb{R}^2/L =...
1
vote
1answer
54 views

If we divide a group and a proper subgroup of this group by the same normal subgroup, can the quotients be equal?

Let $G$ be a group and $H$ be a proper subgroup of $G$. Let $N$ be a normal subgroup of both $G$ and $H$. Question: Is it possible that $G/N = H/N$? Motivation: For a finite extension $L/K$ of ...
1
vote
1answer
48 views

The order of an element in a quotient group

Suppose $G$ is a finite group, that $H$ is a subgroup of $G$, and that $N$ is a normal subgroup of $G$. Suppose that $|H| = n$ and $|G| = m|N|$, where $m$ and $n$ are coprime. Consider the quotient ...
0
votes
2answers
20 views

Factor square root out of quotient

How do I get from the first expression to the second? The only reason the limits are included is because on WolframAlpha it mentioned that this was the case for large negative numbers of x: $\lim_{x\...
0
votes
1answer
41 views

Order of taking quotients?

Let $G$ be a group and $M,N$ be its normal subgroups. I am thinking about the relationship between $(G/M)/N$, $(G/N)/M$, $G/(M/N)$ and $(G/M\cup N)/(M\cap N)$ (they are not proper notions but I am ...
1
vote
1answer
39 views

Quotient group of $C^*$ by Circle group is isomorphic to $R^+$

$C^*$ is the set of complex numbers except $0$ and $S'$ is the circle group. I have to show the quotient group $C^*/S'$ is isomorphic to $R^+$. Let's define $f:C^*→ R^+$, $z→|z|$. Why is $Kerf$ $S'$? ...
1
vote
1answer
34 views

Lift of a generator of a non-trivial quotient of a cyclic group with prime power order must be a generator of the whole group

Let $G$ be a cyclic group with prime power order and $H$ be a proper subgroup of $G$. Problem: Any element in $G$ whose image in the cyclic quotient $G/H$ is a generator, is already a generator of ...
0
votes
1answer
63 views

Multiplicative quotient - what's the correct notation? Does a quotient group contain singletons or cosets?

I want to learn the correct notation and language with which to communicate clearly about a quotient on the multiplicative group $G=(\Bbb Q^+,\times)$ In particular, I want to know the notation with ...
0
votes
2answers
40 views

Given that $G$ is cyclic then $G/N$ is cyclic.

$\newcommand\sg[1]{\langle#1\rangle}$I'm kind of lost in trying to write the proof. My book has an example involving a Cayley Table of how $\mathbb{Z} / 4\mathbb{Z} \approx \mathbb{Z}_4$. So I can see ...
3
votes
2answers
34 views

$GL(2,R) / SL(2,R)$ isomorphic to R*

I am needing to write a prove showing that $GL(2,\mathbb{R}) / SL(2,\mathbb{R}) $ is isomorphic to $\mathbb{R}^*$. I know that $SL(2,\mathbb{R})$ is a normal subgroup of $GL(2,\mathbb{R})$ but I'm ...
2
votes
0answers
38 views

Are two lifts of a generator of a cyclic quotient conjugated?

Let $G$ be a group and $I$ be a normal subgroup of $G$ such that $G/I$ is cyclic, and let $\bar{g}$ be a generator. Let $g, g' \in G$ be two lifts of $\bar{g}$, i.e. $g + I = g' + I = \bar{g}$. ...
0
votes
0answers
11 views

Intuitive explanation for: let $I_G$ be the group of > all inner automorphisms of $G$. Then $I_g$ is isomorphic to $G/C_G$

In the book of Fundamental Concepts of Abstract Algebra by G. Ehrlich, at page 106, it is given that Let $G$ be a group with centre $C_G$, and let $I_G$ be the group of all inner automorphisms of ...
0
votes
0answers
26 views

Let $K \subseteq G$. Nec. and suff. cond. for that $\exists$ a normal subgr $H$ of $G$ s.t $exists$ transversal $I$ s.t $I \subseteq K$.

I am trying to solve/understand the following question that I have come up with; Let $G$ be a group, and $K \subseteq G$ be given. What are the necessary and sufficient condition for that there ...
0
votes
0answers
12 views

Homogeneous space construction theorem and expected dimensions of quotient maps

I am reading the notes linked here. On page 3 we read at Theorem 2 that the left coset space $G/H$ is a topological manifold of dimension $dim(G)-dim(H)$. Here $G$ is a Lie group and $H$ a closed ...
0
votes
0answers
25 views

How do I write $\Bbb Z{\left[\frac16\right]}\cap\left[\frac23,\frac43\right),+$ modulo $\frac23$?

How do I talk about the elements of $\Bbb Z{\left[\frac16\right]}$ in the interval $\left[\frac23,\frac43\right)$ as a set with addition reduced by $\frac23$? It would seem to be a torsion group with ...
1
vote
0answers
32 views

Construction of a Finite Abelian Group

I'm trying to work through the following problem. Let $M=(m_{ij})$ be a $3\times 3$ matrix with integer entries. Assume that det$(M)\neq 0$. Consider the group homomorphism $f:\mathbb{Z}^{3}\to\...
0
votes
1answer
58 views

Write down all the elements of the quotient group $Z_{18} / \langle 6\rangle$. Is any element of order $5?$ Give reasons for your answer.

Write down all the elements of the quotient group $Z_{18} / \langle 6\rangle.$ Is any element of order $5?$ Give reasons for your answer. I just know order of $Z_{18} / \langle 6\rangle$ will be ...
3
votes
2answers
58 views

If S is a normal subgroup, identify the quotient group G/S. What are the $\varphi(G)$'s?

The following is an exercise from Artin's Algebra: (Kiefer Sutherland's voice) Let G be the group of upper triangular real matrices $\begin{bmatrix} a & b\\ 0 & d \end{bmatrix}$ with a and ...
0
votes
1answer
27 views

Let $P$ be a partition of a group with $AB \subseteq C$. Why is $1 \in P_n$? $P_n$ is the equivalence class of $n \in N$ and $1 \in N=P_1$.

Let P be a partition of a group G with the property that for any pair of elements A, B of the partition, the product set AB is contained entirely within another element C of the partition. Let N be ...
5
votes
1answer
69 views

Find number of invertible elements in $\mathbb{Z}[i]/(220+55i)\mathbb{Z}[i]$

I was able to find the factorization $220+55i=11*(2+i)*(2-i)*(4+i)$. Also know this famous question Quotient ring of Gaussian integers But how to apply it in this case? I'm confused, please help. ...
0
votes
0answers
16 views

Arguing that $(A\cap B)/N = A/N \cap B/N $

Let $N\leq A, B\leq G$ where $N\trianglelefteq G$ so that $G/N$ is a group. With correspondence theorem, I am trying to show that the join and intersection of $A$ and $B$ has a unique correspondence ...
1
vote
1answer
32 views

If $H\subseteq G$ is a cocompact subgroup, then $G=KH$ for $K\subseteq G$ compact subgroup

The question is in the title. $H\subseteq G$ is called cocompact if $G/H$ is a compact space. This is a well-known fact but I couldn't prove it myself. I would be happy as well for a reference, ...
1
vote
1answer
50 views

Why is $\mathbb{Q}/ \mathbb{Z}$ called the groups of rationals modulo one?

In the book of Algebra by Hungerford, at page 27, it is given that The following relation on $(\mathbb{Q}, +)$ $$aR b \quad iff \quad a-b \in \mathbb{Z},$$ defined a congruence relation, and the ...
2
votes
1answer
60 views

Definition of a quotient group in Dummit-Foote

I'm reading the section on quotient groups in Dummit and Foote, and they give somewhat non-standard definition of a quotient group. I was wondering whether there is an easy way to see right away for ...
1
vote
1answer
60 views

Does the notation $(G/A)/B$ actually make sense?

I'm having a debate with a friend. Let $G$ denote an abelian group, and suppose $A$ and $B$ are subgroups of $G$. The question is whether $(G/A)/B$ makes sense with the standard definitions. My ...
0
votes
1answer
32 views

Every finite group has a a sequence of nested subgroups where adjacent elements are normal [duplicate]

A group $Q$ is called simple if $|Q|>1$ and the only normal subgroups of $Q$ are the trivial subgroups $\{e\}$ and $Q$. Prove that for any finite group $G$ there exists a sequence of nested ...
-3
votes
1answer
68 views

If $G= \mathbb{Z}/10\mathbb{Z}$ and $H= 2\mathbb{Z}/10\mathbb{Z}$, then the quotient group $G/H$ is isomorphic to … [closed]

Let $G= \mathbb{Z}/10\mathbb{Z}$. Let $H= 2\mathbb{Z}/10\mathbb{Z}$. Then, the quotient group $G/H$ is isomorphic to ... $\mathbb{Z}/10\mathbb{Z}$ $\mathbb{Z}/5\mathbb{Z}$ $\mathbb{Z}/2\...
2
votes
1answer
43 views

Show that $\phi : H\to HN/N$ defined by $\phi(h)=hN$ is injective.

I am trying to show the following: Show that $\phi : H\to HN/N$ defined by $\phi(h)=hN$ is injective. Note that $H\leq G$ and $N$ is normal in $G$. My attempt so far: Let $\phi(h_{1})=\phi(h_{2})$...
2
votes
1answer
29 views

A group $G$ is torsion-free, if a normal subgroup $H$ and $G/H$ are both torsion-free

I'm stuck with the following statement. Let $G$ be a group, $H$ a normal subgroup of $G$ and $G/H$ the quotient group. If $H$ and $G/H$ are both torsion-free, then $G$ is also torsion-free. I think ...
2
votes
2answers
46 views

Quotient of Quotient Group

Say I have $N_1$ a normal subgroup of $G$, $N_2$ a normal subgroup of $G/N_1$, $N_3$ a normal subgroup of $((G/N_1)/N_2)$, ... $N_r$ normal subgroup of $((((G/N_1)/N_2)/N_3)/ ... )/N_{r-1}$. What ...
0
votes
0answers
43 views

Group Z/3Z Proof of Subgroup.

I am looking at the quotient group G = Z/3Z. The equivalence classes are: [0] = {...,0,3,6,...} [1] = {...,1,4,7,...} [2] = {...,2,5,8,...} I want to prove [0] is a normal subgroup, N, by showing ...
1
vote
1answer
46 views

Polynomial in quotient ring

Given the ring $Z_5[x]$, the ideal $I=(x^2+3)$ and the polynomial $f(x)=14x^2+k$, for which $k \in \{0,1,2,3,4 \} $ it holds true that f(t)=1 in $R/I$? (where $t$ is the class of $x$ in $R/I$) I am ...
2
votes
2answers
101 views

The minimal size of generating set of quotient group

True or false: Let $G$ be a finite group, and we'll denote $d(G)$ as the minimal size of the generating set of $G$. Let $Q$ be a quotient map of $G$, so it than it must be that $d(Q) \le d(G)$. I ...
1
vote
0answers
40 views

Notation for quotient space obtained by collapsing a subset to a point?

If $X$ is a topological space and $A$ is a (closed, usually) subset of $X$, then the quotient space obtained by "collapsing $A$ to a point" is often denoted by $X / A$. Unfortunately, that notation ...
1
vote
1answer
56 views

Quotient Group $U(1)/\mathbb{Z} $

Define a homomorphism $f_t$ from $\mathbb{Z}$ to $U(1)$ by $$f_t: n \rightarrow \exp {(i2\pi n t)},\,\,\,\,\,\, n\in \mathbb{Z}$$ where $t\in[0,1)$. Obviously if $t=1/m$ with $m\in\mathbb{Z}^+$, the ...
2
votes
1answer
69 views

Confusion about the quotient $\mathbb Z^4/H$

Let $f:\mathbb Z^3\to\mathbb Z^4$ be the group homomorphism given by $$f(a,b,c)=(a+b+c,a+3b+c,a+b+5c,4a+8b).$$ Let $H$ be the image of $f$. Find an element of infinite order in $\mathbb Z^4 /H$ and ...
8
votes
7answers
214 views

Conditions for cyclic quotient group

Let $G$ be an arbitrary finite group and $H$ a normal subgroup. What are some good conditions on $H$ that make the quotient $G/H$ cyclic? I want to avoid any further restriction on $G$.
-3
votes
4answers
45 views

Discrete quotient is it finite? [closed]

Let $G$ be a countable set, with a group law $(G,*)$. Let $H \leq G$ be a subgroup of $G$. Is it right that $G/H$ is finite? EDIT. I am in fact interested in the following particular situation. ...