All Questions
22 questions
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
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
1
vote
1
answer
84
views
A smart way of proving that $\langle A/\mathfrak{a}, +, . \rangle$ is a ring.
Here is the proposition I have:
Let $f: A \to B$ be a ring homomorphism, then $\operatorname{ker}f$ is an ideal of $A,$ and $\operatorname{Im}f$ is a subring of $B.$ And we have $A/\operatorname{ker}f ...
- 3,137
3
votes
0
answers
103
views
Measurable group's quotient is a measurable group?
A measurable group is a group equipped with a $\sigma$-algebra making the multiplication and the inversion measurable.
For any measurable space $X$ and an equivalence relation $\sim$ on $X$, there is ...
- 1,573
-1
votes
1
answer
43
views
Topological cone using cosets
The topological cone is defined as $(S^1 \times [0,1]) / (S^1 \times \{1\})$. This is a quotient group that which elements are the cosets of $S^1 \times \{1\}$. Now I'm trying to understand how do ...
-1
votes
1
answer
29
views
$\big(\frac{G}{N}\big)^{(n)}=\frac{G^{(n)}N}{N}$
$G$ is a group and $N\unlhd G$ then, $\forall n\in \mathbb N, \big(\frac{G}{N}\big)^{(n)}=\frac{G^{(n)}N}{N}$
$(G/N)^{(n)}=\left \{ Nx :x\in G \right \}^{(n)}=\left \{ Nx^{n} :x\in G \right \}$ but ...
- 435
3
votes
1
answer
760
views
Centralizer : center subgroup ~ normalizer : normal subgroup ~ ?: quotient group
Centralizer subgroup:
$$C_G(A)=\{g\in G\mid gag^{-1}=a,\forall a\in A\}$$
Center subgroup:
$$Z(G)=\{g\in G\mid ga=ag,\forall a\in G\}$$
Centralizer and center are subgroups of $G$. The centralizer ...
- 3,787
2
votes
1
answer
60
views
What is the cokernel of this map $\bar{1} \mapsto \bar{13}$?
I am calculating the cokernel of this map: $f: \mathbb Z_9 \rightarrow \mathbb Z_{39}$ defined by $\bar{1} \mapsto \bar{13},$ I know that its image is $\{\bar{0}, \bar{13}, \bar{26}\} = 13\mathbb Z_{...
user778657
0
votes
0
answers
18
views
Covering Space: definition of $\tilde{X}/Aut(\tilde{X},p)$
If $(\tilde{X},p)$ is a Covering Space of $X$ and $Aut(\tilde{X},p)$ is the group of the Automorphism in $\tilde{X}$, what is the correct definition of $\tilde{X}/Aut(\tilde{X},p)$?
I know that $p:\...
- 463
0
votes
1
answer
60
views
Proving that $\mathbb{Z}^{d}/n\mathbb{Z}^{d} = (\mathbb{Z}/n\mathbb{Z})^{d}$
This question is a continuation of my previous question on quotient groups. I know that:
\begin{eqnarray}
\mathbb{Z}/n\mathbb{Z} = \{[0],...,[n-1]\} \tag{1}\label{1}
\end{eqnarray}
and, from the ...
- 1,923
2
votes
1
answer
619
views
$f$ factors through $q$, where $q$ is the quotient map, and $f$ is a homomorphism.
Let $K$ be a normal subgroup of $G$ and $q:G \to G/K$ be the quotient map. Let $f:G \to H$ be a homomorphism with $K \subseteq \ker(f)$. Prove that $f$ factors through $q$, meaning that there exists ...
1
vote
1
answer
147
views
Is the following quotient group isomorphic to a subgroup?
Suppose $G$ is a compact connected Lie group and $H$ is a subgroup of $G$. Then is $G/H$ isomorphic to some subgroup of $G$?
Browsing around the internet, I came across the following link: https://...
- 779
1
vote
1
answer
102
views
cokernel of a map
I'm trying to compute the cokernel of $\mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \oplus \mathbb{Z}$ where $(a,b) \rightarrow (a+b, a-b)$. I realise that the image of this map is just $\...
- 553
3
votes
2
answers
55
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 $x,y \in H$ ...
- 4,336
0
votes
0
answers
65
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 =...
- 559
3
votes
1
answer
230
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 ...
- 12.1k
0
votes
0
answers
120
views
Isomorphism theorem like theorem for cosets
I would like to a get a result similar to the third isomorphism theorem, but I want to deal with sets of cosets, rather than a quotient subgroup.
Let me formalize it.
Say $G$ is a group, with ...
- 1,192
0
votes
2
answers
114
views
Why do we not have any subgroup of ${\mathbb Z}/{4 \mathbb Z}$ which maps isomorphically to the quotient group?
I have recently started watching the video lectures of the Harvard University Extension School regarding abstract algebra taught by Benedict Gross. I am now in lecture 9 where various results of ...
- 4,892
3
votes
1
answer
599
views
kernel of quotient is quotient of kernel?
Let $A,B,C$ groups and $f:A\to B$ a homomorphism with $C\subset \ker f \subset A$.
Then $f$ induces a map on the quotient $A/C\to B$
Is it then true that $\ker (A/C\to B)= (\ker A\to B)/C~$?
I tried ...
user506844
2
votes
1
answer
170
views
Quotient group of $D(2,3,7)$ under the Klein Quartic
The Klein Quartic is a quotient space of the hyperbolic plane. Let $k$ be its quotient map. Given two isometries $f,g$ of the hyperbolic plane, we say that $f \cong g$ iff $k \circ f = k \circ g$.
...
- 10.6k
0
votes
1
answer
342
views
Is $P/Z(P)$ cyclic when $Z(P)$ is not trivial?
I'm working my way through a proof given in Dummit and Foote.
Theorem: If $|P| = p^2$ for some prime $p$, then $P$ is abelian.
Proof: Since $Z(P)\not = 1$, it follows that $P/Z(P)$ is cyclic $\dots$
...
- 3,622
0
votes
1
answer
176
views
Is the quotient space O(3)/SO(2) a group?
I can't figure out how to demonstrate if it is a group or not. I should see if O(3) is a simple group or that SO(2) is a normal subgroup of O(3).
- 13