All Questions
Tagged with quotient-group group-theory
692 questions
1
vote
1
answer
47
views
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} =...
- 39
0
votes
1
answer
129
views
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
-1
votes
3
answers
89
views
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
votes
2
answers
107
views
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
votes
0
answers
68
views
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
2
votes
1
answer
53
views
$G/P$ 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 ...
3
votes
2
answers
221
views
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
0
votes
2
answers
108
views
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{...
- 7,920
0
votes
1
answer
64
views
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
2
votes
1
answer
119
views
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 ...
- 101
4
votes
1
answer
46
views
(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
0
votes
1
answer
38
views
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)...
- 169
2
votes
1
answer
141
views
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
0
votes
0
answers
72
views
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 $\mathbb{Z}$ ...
- 531
4
votes
0
answers
81
views
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
2
votes
1
answer
129
views
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 ...
- 31
1
vote
0
answers
28
views
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
-2
votes
1
answer
73
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 $x\in\mathbb{R^*}$, ...
- 167
3
votes
1
answer
152
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/...
- 1,246
0
votes
0
answers
48
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 $|H \backslash G / K|$, $|H:H'|$...
- 611
0
votes
1
answer
75
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 ...
- 1,054
0
votes
0
answers
115
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.
$$\text{Prove that }U(40)/U_8(40) \text{ is cyclic but } U(40)/...
- 59
0
votes
1
answer
59
views
Every abelian quotient group is a quotient group of $G/G'$ [duplicate]
A group theory book (not in English) I'm reading states the following:
$G/G'$ is the largest abelian quotient group. In fact, every other abelian quotient group is also a quotient group of $G/G'$, ...
- 149
1
vote
0
answers
47
views
The free product has the direct product as a factor group. What's the corresponding normal subgroup?
Let $G$ and $H$ be groups. Consider the free product $G * H$ and the direct product $G \times H$. There is a particular way of identifying $G \times H$ as a factor group of $G * H$. Namely, the ...
- 2,049
0
votes
1
answer
67
views
Closed subgroups of the additive group of adeles and their "modulo 2" quotients
I was wondering if in the adeles ring over $\mathbb{Q}$, viewed as an additive
group, any closed subgroup H is such that $((1/2)H)/H$ is finite or compact, where $1/2H$ is the group of adeles $x$ such ...
- 41
1
vote
1
answer
53
views
Ordering of group elements in MultiplicationTable vs CosetTable in GAP
I construct a finitely presented discrete group $G$ in GAP, a normal subgroup $H\triangleleft G$ of finite index $N$, and the factor group $G/H$ of order $N$. Assume for simplicity that $G$ has two ...
- 109
1
vote
0
answers
44
views
Subgroups and Quotient Groups of Infinite bounded Abelian p-groups
I am looking to a reference to a solution of Exercise 13 (page 91) of L. Fuchs, Infinite Abelian Groups, Vol. 1, Academic Press, 1970. There is no solution to this exercise in a collection Exercises ...
- 11
0
votes
0
answers
38
views
Rules of Products of Quotient Groups [duplicate]
Suppose that you have 2 groups, $G$ and $H$, such that $H\trianglelefteq G$. I was wondering if we can say that
$$
(G/H) \times H \cong G
$$
I know some rules about product and quotient groups work &...
- 41
4
votes
2
answers
67
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 ...
- 4,559
1
vote
2
answers
59
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 ...
- 665
1
vote
1
answer
76
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 ...
- 968
2
votes
1
answer
84
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$,...
- 3,184
2
votes
1
answer
102
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 ...
user1127460
3
votes
1
answer
598
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 ...
user1266775
2
votes
1
answer
97
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 ...
- 21
3
votes
2
answers
238
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,...
- 53
1
vote
1
answer
209
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 ...
- 63
1
vote
1
answer
46
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 ...
- 341
0
votes
1
answer
121
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 ...
- 63
1
vote
0
answers
49
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 ...
- 11
2
votes
1
answer
80
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 ...
- 1,589
0
votes
1
answer
112
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 ...
- 341
4
votes
2
answers
517
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
$...
- 342
1
vote
0
answers
57
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 + ...
- 355
1
vote
1
answer
74
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 ...
- 1,364
0
votes
1
answer
96
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 $G$ ...
1
vote
0
answers
54
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 ...
- 305
3
votes
2
answers
206
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 ...
- 89
1
vote
1
answer
104
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 $N$."
- 37
0
votes
1
answer
110
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 ...
- 342