Questions tagged [quotient-group]
This tag is for questions relating to "Quotient Group".
901 questions
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Group theory - Transfer function and commutator
I'm stuck with this one:
Let $J$ normal in $H$, $H$ normal in $G$, with $G/H$ finite and $H/J$ abelian. Let $\operatorname{tr}$ be the transfer $G \to H/J$ and let $\operatorname{Im}\operatorname{tr} =...
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I still don't know why does from $b\in \{a^n|a\in G\}$ follow that $b\in H$.
In my previous post: Question about normal subgroups I got an answer to help with canonical epimorphism $\pi:G\rightarrow G/H$. In this answer we didn't use anywhere that $H$ is a normal subgroup in $...
user1489207
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3
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Let $\mathbb{T}=\{z\in \mathbb{C}\mid|z|=1\}$ and $N=\{z\in \mathbb{C}\mid z^3=1\}$.
Let $\mathbb{T}=\{z\in \mathbb{C}\mid |z|=1\}$ and $N=\{z\in \mathbb{C}\mid z^3=1\}$. I was able to prove that $N$ is normal subgroup of group $\mathbb{T}$.
Furthermore, I show that $\mathbb{T}/N \...
user1390721
0
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2
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107
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Let $D_6=\langle x,y\mid x^6=y^2=e,xy=yx^5\rangle $ and $H=\langle x^3,y\rangle$, $K=\langle x^2\rangle$.
Let $D_6=\langle x,y\mid x^6=y^2=e,xy=yx^5\rangle$ and $H=\langle x^3,y\rangle$, $K=\langle x^2\rangle$.
a) Are $H, K$ normal subgroups of $D_6$?
b) Is the quotient group $D_6/Z(D_6)$ Abelian?
I first ...
user1468123
2
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0
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Trying to classify a quotient group
Consider a $3 \times 3$ matrix with $1$'s on the diagonal, an $a_{12}$ entry from integers, and $a_{13}$ and $a_{23}$ entries from real numbers. What's the quotient of this group with the Heisenberg ...
- 3,720
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$G/P$ is isomorphic to $C_4 \times C_4$
$G$ is the group of matrices
\begin{pmatrix}
a & b \\
0 & c
\end{pmatrix}
where $a, b, c \in \mathbb{Z}/5$ with $a, c \neq 0$.
This is a group of order $80$ with one (normal) $5$-Sylow ...
1
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1
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Example of abelian quotient $A/B\cong\mathbb{Q}$ that doesn't split?
Can someone show me an example of an abelian group $A$ and its subgroup $B$ such that $A/B\cong Q\leq\mathbb{Q}$ so that the short exact sequence
$$0\longrightarrow B\longrightarrow A\longrightarrow A/...
- 2,376
3
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2
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220
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Index of a subgroup is finite
Suppose I have a group $G$ and a commutative subgroup $H$ and another subgroup $K\subseteq H$. So I have $K\leq H\leq G$. Moreover, suppose I know that the index $\vert G:H\vert$ is finite. I have to ...
user1443877
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108
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Classify the group $(\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z})/\langle(1,1)\rangle$
Classify the group $(\mathbb{Z}\times \mathbb{Z}_3)/\langle(1,1)\rangle$ according to the fundamental theorem of finitely generated abelian groups.
My Attempt
\begin{align}
G=\mathbb{Z}\times \mathbb{...
- 7,910
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Notation question regarding quotient groups
I have stumbled upon the following question from the book 'Module theory - An approach to linear algebra' by T.S. Blyth. Question 10 in chapter 8 asks the following question:
My question is mainly ...
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Question about the quotient group $a\mathbb{Z}/b\mathbb{Z}$
I know that $a\mathbb{Z} \cap b\mathbb{Z}=lcm(a,b)\mathbb{Z}$ and $a\mathbb{Z}+b\mathbb{Z}=mcd(a,b)\mathbb{Z}$. Is there something similar for $a\mathbb{Z}/b\mathbb{Z}$?
- 2,692
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Visualizing a quotient monoid of $\mathbb N$
Let $m,n\in \mathbb N$ and $m\geq 1$. Let $R_{m,n}$ be the equivalence relation
$$
xR_{m,n}y \Leftrightarrow x=y \quad \text{or} \quad (x\geq n \text{ and } y\geq n \text{ and } m\mid x-y)
$$
in the ...
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119
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Difference between quotient group and quotient topology
Suppose $(X,\tau_X), (Y,\tau_Y),(Z,\tau_Z)$ are topology space, let $p: X\to Y$ is a surjective map, we have if $g: Y\to Z$ is continuous $\iff f:=g\circ p$ is continuous, then $\tau_Y$ is quotient ...
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(Weak) Commensurability is an equivalence relation on the class of groups
I got stuck trying to do this problem, it is Exercise 5.E.16 in Clara Löh's Geometric Group Theory: An Introduction. First, let me quote the definitions:
Two groups $G$ and $H$ are commensurable if ...
- 1,746
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Number of distinct decompositions of $Z/(p) \oplus Z/(p)$ in a direct sum of two proper subgroups
I have read some similar questions and answers but it was not easy for me to understand them. So I ask you to check my solution.
The question is to find the number of distinct decompositions of $Z/(p)...
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141
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Existence of an element of a group whose order is the same as that of its congruency class
Let $G$ a finite group , and $H$ a normal subgroup of G. Let $x$ be an element of $G$ and $\bar{x}$ its congruency class in $G/H$.
Let $c \in G/H$, such that : $ord(c) \wedge |H|=1$. Show that there ...
- 407
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Prove that $\frac{I\cap J}{IJ}\cong\frac{k[x,y]}{\langle xy\rangle}$ for $I=\langle x^2y \rangle$ and $J=\langle xy^2\rangle$
$R=k[x,y]$ is a $R$ module,where $k$ is a field. $I=\left<x^2y\right>$ and $J=\langle xy^2\rangle$. Prove that
$$\frac{I\cap J}{IJ}\cong\frac{k[x,y]}{\left<xy\right>}$$ as $R$ modules.
...
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72
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Quotient group simplification exercise
I am doing some abstract algebra and it's been a hot second since I've done group theory. For both exercises, like $a, b, c, d$ all be free variables that generate a group with $\mathbb{Z}$ ...
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If $G$ is finitely generated and $G/N$ is finitely presented, then $N$ is the normal closure of a finite set
Suppose $G$ is a finitely generated group and $N$ is a normal subgroup of $G$ such that $G/N$ is a finitely presented .Then show that $N$ is a normal closure of some finite set of $G$.
Self studying ...
- 1,746
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Quotient Groups and the Zero Element
I'm trying to understand tensor products from a rigorous mathematical perspective, and while there are definitely other ways to understand it, understanding it using quotient spaces will also help me ...
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How to find the generator matrix for the quotient group $C/C^{⟂}$ using a list of coset representatives of $C^{⟂}$ in $C$?
$C$ is the $[6,5,2]$ classical binary code.
I am trying to find the generator matrix for the quotient group $C/C^{⟂}$ from the list of coset representatives of $C^{⟂}$ in $C$.
My question refers to ...
- 341
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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^*}$, ...
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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/...
- 1,246
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57
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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
...
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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}...
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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'|$...
- 611
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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 ...
- 1,054
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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)/...
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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 $360$...
- 17.9k
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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 $G/G'$, ...
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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 ...
- 2,049
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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 ...
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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^*...
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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 ...
- 41
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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 ...
- 109
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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 ...
- 177
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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 ...
- 11
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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 ...
- 177
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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 ...
- 3,830
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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 &...
- 41
4
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2
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67
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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 ...
- 4,559
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2
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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 ...
- 665
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1
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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 ...
- 968
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1
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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 $X=U\cup V$,...
- 3,184
2
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1
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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 ...
user1127460
3
votes
1
answer
598
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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 ...
user1266775
2
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1
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97
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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
3
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2
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238
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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
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1
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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 ...
- 63
1
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1
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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 ...
- 341