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Questions tagged [quotient-group]

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0
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1answer
26 views

Quotient group of dihedral group

Let $G=\{e,r^{2},...,r^{8},s,sr,...,sr^{8}\}$ and let $N=\langle r^{3} \rangle.$ Now let $\pi(g)=\bar{g}=gN$ be surjective with kernel $N$. I have to show that $G/N=\{\bar{e},\bar{r},\bar{r^{2}},\...
0
votes
1answer
20 views

Index of the projection of a subgroup on a quotient by finite normal subgroup.

Given a finitely generated group $G$ and a finite normal subgroup $N \leq G$. I am trying to compare finite index subgroups in $G$ and $G/N$. I know that $H$ is a subgroup of $G$ iff $H/N$ is a ...
2
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1answer
35 views

If we add more relations to a presentation will it always form a quotient group?

Specifically, if I have a presentation $\left<G|R\right>$, and I look at the presentation $\left<G|R,R_1\right>$ is always true that $$\left<G|R,R_1\right>\cong\left<G|R\right>...
0
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1answer
30 views

Group is soluble if and only if quotient is abelian

So a group G is soluble if and only if it has a subnormal series $$ \{ 1\} =G_0 \ \triangleleft \ G_1 \ \triangleleft \ ... \ \triangleleft \ G_n=G $$ where all quotient groups $G_{i+1}/G_i $ are ...
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0answers
40 views

If $(m, n) = 1$, then $\dfrac{\mathbb Z \times \mathbb Z}{\langle (m,n)\rangle}\cong\mathbb Z$

How to show that If $(m, n) = 1$, then $\dfrac{\mathbb Z \times \mathbb Z}{\langle (m,n)\rangle}\cong\mathbb Z$. (Note: We consider this in group theory.) I know that $(m, n) = 1$ means that ...
-1
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1answer
38 views

Is there a procedure to calculate the multiplicative inverse in a quotient by a maximal ideal?

An elementary result in ring theory is that if $R$ is a commutative ring with unity and $M$ is a maximal ideal of $R$, then $R/M$ is a field. There are many proofs of this, as you can see here. But ...
0
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2answers
24 views

Size of Quotient Group

For a quotient group G/H (where H is a normal subgroup), is |G/H|=|G|/|H|? i.e. is the size of the quotient group = (size of G)/(size of H) Thank you.
0
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1answer
41 views

What is the order of a coset?

I'm trying to read Cauchy's Theorem on wikipedia (The abelian part of Proof 1) and they say that $G/H$ contains an element of order $p$. What does the order of a right coset mean? Is it the number of ...
0
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1answer
53 views

How to calculate quotient $A/(0)$ and $A/A$? [closed]

What is the procedure or to calculate ( or simplify, I'm new in abstract algebra) a quotient group? I know that $A$ a group and $B$ a subgroup we can form the quotient $$A/B $$ for example $$ \mathbb{...
2
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1answer
32 views

Can you write out a multiplication table for $D_4/L$ if $L$ is not a normal subgroup of $D_4$?

So $D_4$ is the symmetries of the square, I am told $L=<P>$ which is really just $L=${$P,I$} where $I$ is the identity and $P$ is whatever flipping of the square you desire. Now, in my text it ...
-1
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2answers
49 views

What is the factor group $C_{12}/C_{6}$?

I know this factor group is isomorphic to $C_2$, but I have tried calculating it and I only get one coset.
1
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1answer
32 views

Homomorphism and image

I'll consider the function $\phi: \mathbb Z^3 \to \mathbb Z^3$, given by: $\phi(x,y,z) := (x+5y+3z,2y,7z)$ I have a couple of questions associated with this: 1) This might be a very dumb question, ...
0
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0answers
30 views

How single elements belong to partition though they can't satisfy equivalence relation?

Let $A = \{2, 3, 5, 15\}$ and $G$ is the equivalence relation of elements divisible by 3 and $H$ is the equivalence relation divisible by 5. Now the quotient set $A/G = \{\{3, 15\}, \{5\}, \{2\}\},$ ...
1
vote
1answer
72 views

Why $\mathbb Z/2\mathbb Z$ is not definable in $(\mathbb Z, +, 0)$?

Marker states in his textbook: It is possible that $G/H$ does not correspond to a definable group in our structure. Our instructor gave $\mathbb Z/2\mathbb Z$ as an example. But definability, ...
0
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1answer
66 views

Derivative of the quotient map $\mathbb R \to \mathbb R / T\mathbb{Z}$

We consider the quotient space $\mathbb R / T\mathbb{Z}$ and the quotient map $\pi:\mathbb{R} \to \mathbb R / T\mathbb{Z}\ $ defined by $\pi(t):= t\bmod T:=t+T\mathbb{Z}$. In a journal i read that $\...
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0answers
44 views

How to think intuitively quotient of quotient set in equivalence relations

If $G$ and $H$ are arbitrary equivalence relations in $A$, prove that $$A/(G\circ H) ≈ (A/G)/(G\circ H/G).$$ How to think $(G\circ H)/G$ intuitively, especially if $(G\circ H)$ is empty set, and ...
6
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2answers
76 views

G is homeomorphic to $G/H\times H$

Let G be a topological group and $H\le G$. Let $\pi: G\to G/H$ be the canonical projection and a continuous $\sigma: G/H\to G$ such that $\pi \circ \sigma = Id$. Prove that G is homeomorphic to $G/...
1
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4answers
89 views

$\mathbb{Q}\!\left(\sqrt{2}+\sqrt{3}\right)$ definition question

I know the definition of $\mathbb{Q}\!\left(\sqrt{2}\right)=\{a+b\sqrt{2}\mid a,b\in\mathbb{Q}\}$. Why can't you similarly say $\mathbb{Q}\!\left(\sqrt{2}+\sqrt{3}\right)=\{a+b\!\left(\sqrt{2}+\sqrt{3}...
2
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1answer
42 views

Find the quotient group of an infinite subgroup.

From what I understand, to find the quotient group requires knowing all cosets of a subgroup. I know what a coset is defined, but when it comes to infinite groups, I'm a bit confused. For example, ...
0
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0answers
37 views

Quotient space not the same as the original space

I am trying to understand why quotient space is not the same as the original space. Let $V$ be a vector space and $W$ be its vector subspace. If I define $a+W:=\{x=a+w; w\in W\}$ then the quotient ...
5
votes
1answer
67 views

Proof of $G$ is solvable implies $G/N$ is solvable.

I want to show that if $N$ is normal in $G$ then $G$ is solvable implies $G/N$ is solvable. Now, $G$ is solvable implies there exists a chain $\{e\}=G_0 \trianglelefteq G_1 \trianglelefteq G_2 \...
3
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2answers
50 views

$H/K$ when $K$ is not a subgroup of $H$.

Let $G$ be a group and $H$, $K$ ($K$ is normal) two subgroups of $G$ but neither $H$ is subgroup of $K$ nor $K$ is subgroup of $H$. What would be the problem with $H/K$? If I define for $x,y \in H$ ...
2
votes
1answer
35 views

$G/H$ cyclic and $H$ subgroup of $Z(G)$, then $G$ cyclic.

Gallian, in his Contemporary Abstract Algebra, first proves this theorem: Theorem: Let $G$ be a group and let $Z(G)$ be the center of $G$. If $G/Z(G)$ is cyclic, then $G$ is Abelian. Proof: ...
1
vote
1answer
27 views

quotient module $\mathbb{Z}^3/(2,0,3)\mathbb{Z}$

Is $\mathbb{Z}^n/(m,a_2,...,a_2)\mathbb{Z}\cong\mathbb{Z}/m\mathbb{Z}\oplus\mathbb{Z}^{n-1}$ for $0<m<a_2,...,a_n\in\mathbb{Z}$ (as $\mathbb{Z}$-modules)? If not, how can you compute something ...
0
votes
1answer
44 views

Coset representatives of a group

Given the matrices with coefficients in $\mathbb Z_5$, I am asked to consider: $G= \left\{\begin{pmatrix} a & 0 \\ 0 &d \end{pmatrix} \mid ad\neq 0 \bmod 5 \right\}$ and: $ H= \left\langle \...
2
votes
2answers
34 views

Prove that $S_3/C_3$ is a (quotient) group

Consider $S_3$ to be the symmetries of a triangle, and let $C_3$ be a subgroup that cycles the three corners, so generated by: $(1 \ 2 \ 3 )$. Using Lagrange's theorem, compute $k = |S_3/C_3|$. ...
0
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1answer
36 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 ...
6
votes
2answers
135 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
95 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 ...
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0answers
21 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.
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1answer
35 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
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1answer
31 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
66 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
37 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
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0answers
57 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
votes
1answer
60 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
vote
2answers
53 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)...
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2answers
60 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 $...
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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
181 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. ...
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0answers
33 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
votes
2answers
48 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 ...
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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 ...
5
votes
1answer
69 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
votes
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
65 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
76 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
34 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
33 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 ...