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Questions tagged [quotient-group]

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

0
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1answer
27 views

Is $HN/N$ equal to $H/N$?

The question might be stupid but I was wondering if $(HN)/N = H/N$. It seems true, but in the second isomorphism theorem, one can show that $H/(H\cap N)$ is isomorphic to $HN/N$. So it's weird if the ...
4
votes
2answers
96 views

Computing the index $\left(\mathbb Z\left[\frac{1+\sqrt{5}}{2}\right]:\mathbb Z \left[\sqrt{5}\right]\right)$?

If $\mathcal O=\mathbb Z\left[\frac{1+\sqrt{5}}{2}\right]$, I'd like to show that $\left(\mathcal O:\mathbb Z \left[\sqrt{5}\right]\right)=2$. It's to show that Dedekind's factorisation theorem doesn'...
2
votes
1answer
81 views

Is the Quotient Group Cyclic?

I'm just wondering how to show that a quotient group $H = (G/N)$ is cyclic? Let $G= \mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/6\mathbb{Z}$ Let $N = \left<(2,3)\right>$ , where N is a cyclic ...
0
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0answers
20 views

Find specific elements with a given order in quotient group

How can I deal with the quotient group generated by several element? And how can I find the element with a given order.
1
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1answer
33 views

Is there a name for the quotient of the symmetric group by the finitary symmetric group?

The finitary symmetric group on a set $S$ is the group of permutations that only move a finite set points. That is: $$FSym(S) = \{\phi:\{s : s \in S, \phi(s) \neq s\}\text{ is finite}\}$$ This is a ...
1
vote
1answer
30 views

Elementary question about well-definedness of induced homomorphism

I don't understand why this statement requires N to be subgroup of kernel $(N \leq \ker\Phi)$, not just kernel itself $(N = \ker\Phi)$ "φ is well defined on G/N if and only if N ≤ ker Φ" where N≤G, ...
0
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3answers
53 views

Quotient Group Is Finite Implies That Group Is Finite?

Let $A$ be an Abelian group and suppose $A$ has a subgroup isomorphic to $\mathbb{Z}/p^a\mathbb{Z}$, for some prime $p$ and positive integer $a$. Suppose that $A/(\mathbb{Z}/p^a\mathbb{Z}) \cong \...
2
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0answers
34 views

Quotient cancellation for invertible ideals of orders in quadratic fields

Let $K/\mathbb{Q}$ be an imaginary quadratic field, $m\ge 1$ be a positive integer and let $\mathcal{O}=\mathbb{Z}+m\mathcal{O}_K\subset \mathcal{O}_K$ be the unique order of index (equivalently, ...
3
votes
0answers
56 views

Find group isomorphic to the quotient group

I'm working on a few problems of a similar type. I'm pretty confident I got the right answer but I'm not sure how to provide satisfiable enough proof that the answer is correct. I'd be glad if you ...
0
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1answer
59 views

Which group is this quotient group isomorphic to?

Let $$G = \left\{\begin{pmatrix} q&0\\a+bi&q\end{pmatrix} \mid q \in \mathbb{Q}^\ast, a,b\in\mathbb{R}\right\}$$ and $$H = \left\{\begin{pmatrix} q&0\\a+ai&q\end{pmatrix} \mid q \in \...
1
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2answers
48 views

Proving 2 quotient rings are isomorphic

So I want to show $\mathbb{R}[x]/((x-r)^2)$ is isomorphic to $\mathbb{R}[x]/(x^2)$ where $r \in \mathbb{R} $. I thought of constructing a ring homomorphism $\phi : \mathbb{R}[x] \to \mathbb{R}[x]/(x^2)...
1
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2answers
48 views

Quotient group of the dihedral group by $\langle r^2 \rangle.$

Show that $G/H$ is abelian, where $G$ is the dihedral group $$ G={\langle r,\, f \mid r^n=f^2=1,\, rf=fr^{-1}\rangle}$$ and $H$ is the subgroup $\langle r^2 \rangle.$ I've tried showing that for $...
2
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0answers
28 views

Normal Subgroups and Quotient Groups help

Let $(G, ∗)$ be a group, let $H$ be a normal subgroup of $G$, and let $(G/H, ⋆)$ denote the quotient group of $G$ by $H$. (a) Prove that if $xH \in G/H$, then $(xH)^m = x^mH$ for all $m ∈ Z$. (b) ...
2
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2answers
88 views

Quotient ring local if Ring is local

I want to show: If $R$ is local and $I\neq R$ an ideal, then $R/I$ is also local. We already know: A Ring $R$ is local if and only if $R-R^{\times} = \{r\in R \, | r \notin R^\times \}$ is an ideal. ...
1
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0answers
32 views

Is G/H a group even if H is a subgroup of G but not a normal subgroup? [duplicate]

I know we can have left and right cosets without having the group be normal, but can we mod out by a subgroup without it being normal and still get a group? Thank you in advance!
2
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2answers
34 views

Given that $G/H=\{xH:x \in G\}$ is group under operation $(xH)(yH)=(xyH)$ . Then $H$ is normal subgroup of $G$

let $G$ is any group and $H$ is its subgroup , such that $G/H=\{xH:x \in G\}$ is group under operation $(xH)(yH)=(xyH)$ . Then show that $H$ is normal subgroup of $G$ if $H$ is not normal ,then there ...
1
vote
1answer
65 views

What does the multiplication in cosets mean?

I was getting ready to learn about order of an element, cosets, and langrange's theorem in group theory. Consequently this involves multiplying two elements of a group. After seeing several examples ...
4
votes
1answer
50 views

Let $G$ be a finite group and $M,N \lhd G$ such that $M \leq N\cap \Phi(G)$. Then $\frac{N}{M}$ is nilpotent iff $N$ is nilpotent.

Let $G$ be a finite group and $M,N \lhd G$ such that $M \leq N\cap \Phi(G)$. Then $\frac{N}{M}$ is nilpotent iff $N$ is nilpotent, where $\Phi(G)$ is the Frattini subgroup of $G$. The converse side ...
2
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2answers
36 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 ...
3
votes
1answer
56 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 ...
1
vote
1answer
58 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
32 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
20 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
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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
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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
75 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
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0answers
48 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
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1answer
18 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
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2answers
67 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
55 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
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0answers
48 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
57 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
57 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
23 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\...
1
vote
1answer
59 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
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1answer
36 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
65 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
43 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
149 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. $g + I = g' + I = \bar{g}$. ...
1
vote
1answer
16 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
29 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
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0answers
14 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
39 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
89 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
65 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
36 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
74 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 ...