All Questions
Tagged with group-isomorphism group-homomorphism
241 questions
0
votes
1
answer
113
views
Screwing up basic short exact sequence $0\to\mathbb Z\leftrightarrow\mathbb Z\to0$
Represent an isomorphism by $\leftrightarrow$.
HAVE exact sequence.
$$ 0 \rightarrow \mathbb Z \leftrightarrow \mathbb Z \rightarrow 0 $$
Then
$$ \text{img} \left( 0 \rightarrow \mathbb Z \right) = 0 =...
- 1,328
1
vote
0
answers
57
views
Is a Rational Map Possible Between Elliptic Curves of Orders 19 and 18?
I'm exploring the relationship between elliptic curves and their orders. I have two elliptic curves:
E1: An elliptic curve with order 19.
E2: An elliptic curve with order 18.
Since the order of E1 is ...
- 11
8
votes
1
answer
354
views
Existence of injective and surjective group homomorphisms in both directions implies existence of group isomorphism
Let $G$ and $H$ be groups such that there exist group homomorphisms $\iota_{HG}: G \hookrightarrow H$, $\iota_{GH}: H \hookrightarrow G$, $\pi_{HG}: G \twoheadrightarrow H$, $\pi_{GH}: H \...
- 3,275
0
votes
2
answers
85
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Prove that $ (\mathbb{Z}/n\mathbb{Z}) / G$ is isomorph to $\mathbb{Z}/ (n/d) \mathbb{Z}$ with $G$ a sub group of $\mathbb{Z}/n\mathbb{Z}$
Question:
Let $d$ a divisor of $n$ ($n,d \in \mathbb{N}$) and let $G$ the only sub group with cardinal $d$ of $\mathbb{Z}/n\mathbb{Z}$. Prove that $ (\mathbb{Z}/n\mathbb{Z}) / G$ is isomorphic to $\...
- 739
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
95
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Let $A,B, C,A',B'$ be an abelian groups. Let $f:A\to B$ and $g:B\to C$ be group homomorphisms.
Let $A,B, C,A',B'$ be an abelian groups.
Let $f:A\to B$ and $g:B\to C$ be group homomorphisms.
Suppose $A'\cong A$, $B'\cong B$ and $C\cong C'$ as abelian groups and $imf\subset kerg$.
Let $f':A'\to B'...
- 6,648
-2
votes
1
answer
78
views
Can anyone give me an example of a map between two groups that is homomorphic but not isomorphic?
I'm learning homomorphism and isomorphism in basic group theory. Can anyone give me an example of a map between two groups that is homomorphic but not isomorphic? I couldn't think of one.
- 11
31
votes
2
answers
2k
views
Orbit stabiliser theorem as an analogue to first isomorphism theorem
The notes I'm using to study group theory make a remark that another appropriate name for the "orbit stabiliser theorem" is the "first isomorphism theorem for group actions". For ...
- 413
1
vote
1
answer
119
views
Even and Odd Maps in Group Homomorphism
I'm curious about a particular property in a group homomorphism. Suppose that $\phi:S_{6}\rightarrow\mathbb{Z}_{2}$ is any map, not necessarily a group homomorphism, with $(S_{6},\circ)$ and $(\mathbb{...
- 47
0
votes
2
answers
265
views
Doubt in the proof of the First Group Isomorphism Theorem
Let $f:G\to H$ be a group homomorphism. Then $G/\text{ker}(f)\cong \text{im}(f)$.
To prove this, we define a map $\varphi:G/\text{ker}(f)\to \text{im}(f)$ by $\varphi(gK)=f(g)$, where $K=\text{ker}(f)$...
- 27
0
votes
0
answers
73
views
Prove that every group of order 6 is isomorphic to $C_6$ or $S_3$ [duplicate]
Prove that every group of order 6 is isomorphic to $C_6$ or $S_3$
So I've already been able to prove this (without what's written below) but I've been told that there's a way to prove this as well ...
user1148590
0
votes
0
answers
52
views
Notation $NM/M$ in the second isomorphism theorem
I've been solving the following problems from Herstein:
" Given $N$ and $M$ normal in a group $G$, show that $NM/M$ is isomorphic to $N/N∩M$."
The problem is not difficult, but my doubt ...
- 105
2
votes
2
answers
108
views
Show that the size of the pre-image $(\psi(x))$ is the same for all $x$. What is this size equal to?
Let $\psi: G \rightarrow H$ be a homomorphism, and let $K=\operatorname{ker}(\psi)$. Define the canonical homomorphism $\varphi: G \rightarrow G/K$ by $\varphi(x) = xK$.
Show that the size of the pre-...
- 785
1
vote
1
answer
95
views
Let $\psi:G \rightarrow H$ hom, $K = \ker(\psi)$, and $\varphi$ be canonical projection. Show that $\psi(x)=\psi(y) \iff \varphi(x)=\varphi(y)$.
Question :
Let $\psi : G \rightarrow H$ be a homomorphism, and set $K = \operatorname{ker}(\psi)$. Let $\varphi : G \rightarrow G/K$ be the canonical homomorphism defined by $\varphi(x) = xK$.
Show ...
- 785
0
votes
0
answers
28
views
Show that $SL(2,\mathbb{Z}) / \Gamma (n)$ is isomorphic to $SL(2,\mathbb{Z}_n)$. [duplicate]
$\Gamma (n)= \{ \begin{pmatrix} a&b\\c&d \end{pmatrix} \in SL(2,\mathbb{Z})| a,d \equiv 1 \, mod \, n, b,c \equiv 0 \, mod \, n \}$
$\Gamma (n)$ is a normal subgroup of $SL(2,\mathbb{Z})$.
...
- 15
3
votes
4
answers
315
views
Suppose we have a homomorphism $f:G\to\Bbb Z$ for a finite group $G$. Prove that $f(g)=0$ for all $g$ in $G$
Suppose we have a homomorphism from $G$, a finite group, to $\Bbb Z$, the set of integers, $f:G\to\Bbb Z$. To prove that $f(g)=0$ for all $g$ in $G$, what I did is defined an isomorphism - $h:G/\ker(f)...
1
vote
3
answers
315
views
For a finite abelian $G$, $f: G\to G$ defined by $f(g)=g^2$ is an isomorphism iff $|G|$ is odd [duplicate]
Let $G$ be an abelian group of finite order, and define $f: G\to G$ by $f(g)=g^2$. I would like to prove that $f$ is an isomporphism if and only if $G$ has an odd order.
I am able to prove that $f$ (...
- 1,037
0
votes
1
answer
90
views
Let $\phi:G\to H$ be a surjective hom. of groups. Let $\sigma:H\to G,\phi\sigma=id_H.$ Show $G\cong\ker(\phi)\rtimes H.$
Let $\phi: G \rightarrow H$ be a surjective homomorphism of groups. Let $\sigma: H \rightarrow G, \,\,\phi\, \sigma = id_H.$ I need to show that $G$ is isomorphic to the semidirect product $\ker(\phi)\...
- 928
-1
votes
2
answers
233
views
Show that $\Bbb Z_{16}$ is not a homomorphic image of $\Bbb Z_4×\Bbb Z_4$.
Show that $\Bbb Z_{16}$ is not a homomorphic image of $\Bbb Z_4×\Bbb Z_4$.
My solution goes like this:
We consider $f$ as an epimorphism. Now, $f:\Bbb Z_4×\Bbb Z_4\longrightarrow \Bbb Z_{16}$. Hence, ...
- 2,630
1
vote
2
answers
313
views
If $G$ is an infinite group such that $G$ is a homomorphic image of $\Bbb Z$ then prove that $G$ is isomorphic to $\Bbb Z$
If $G$ is an infinite group such that $G$ is a homomorphic image of $\Bbb Z$ then prove that $G$ is isomorphic to $\Bbb Z$.
In this question, I tried applying the First Isomorphism Theorem. But I don’...
- 2,630
-2
votes
1
answer
182
views
If there exists a homomorphism from group $G$ onto group $H$ and a second homomorphism from $H$ onto $G$. Does this imply $G$ and $H$ are isomorphic? [duplicate]
Suppose there exists an surjective homomorphism from group $G$ to group $H$ and a second surjectibehomomorphism from $H$ to $G$. Does this imply $G$ and $H$ are isomorphic?
I know there exists an ...
- 286
1
vote
2
answers
293
views
Show that $(\Bbb Q, +)$ and $(\Bbb R, +)$ are not isomorphic groups.
Show that $(\Bbb Q, +)$ and $(\Bbb R, +)$ are not isomorphic groups.
I dont how to proceed here. My general strategies include: trying to show that if one group has an element of a particular order ...
- 2,630
0
votes
0
answers
122
views
Show that $(\Bbb Z, +)$ and $(\Bbb R, +)$ are not isomorphic groups .
Show that $(\Bbb Z, +)$ and $(\Bbb R, +)$ are not isomorphic groups.
What I did so far:
To show that these groups are not isomorphic, we need first assume that, there exists an isomorphism $f$ from $(...
- 2,630
0
votes
0
answers
80
views
Find all homomorphisms from $(\mathbb{Z}_8,+)$ into $(\mathbb{Z}_6,+)$.
Find all homomorphisms from $(\mathbb{Z}_8,+)$ into $(\mathbb{Z}_6,+)$.
The solution given is as follows :
We have , $\mathbb{Z}_8=\langle [1]\rangle$ . Let $f:\mathbb{Z}_8\longrightarrow \mathbb{Z}...
- 2,630
2
votes
4
answers
140
views
Is the number of homomorphisms between two isomorphic groups equals to the number of Automorphisms for each group?
Is the number of homomorphisms between two isomorphic groups equals to the number of Automorphisms for each group?
Let us first divide to cases:
finite groups
infinite groups
Let $G,H$ be some ...
- 2,817
3
votes
0
answers
68
views
Let $G=\mathbb{Z}\big /4\mathbb{Z}$ and $H=2\mathbb{Z}\big /4\mathbb{Z}$. Is $H$ isomorphic to $G\big / H$?
In Harvard Abstract Algebra lectures (video here, minute 50:06), professor Gross shows the groups $G=\mathbb{Z}\big /4\mathbb{Z}$ and $H=2\mathbb{Z}\big /4\mathbb{Z}$ as an example in which a subgroup ...
- 289
0
votes
2
answers
37
views
Showing that $(\mathbb{Z}_6,+)$ and $(U(\mathbb{Z}_7), \cdot)$ are isomorphic groups
We have to prove that $(\mathbb{Z}_6,+)$ and $(U(\mathbb{Z}_7), \cdot)$ are isomorphic.
I know that they are and the isomorphism is as follows: $f(0)=1, f(1)=3, f(2)=2, f(3)=6, f(4)=4, f(5)=5$.
It is ...
- 51
5
votes
1
answer
143
views
Prove that $\{ (x, x): x\in G\}$ is a subgroup of $G\times G$, and is isomorphic to $G$.
Given the diagonal $H = \{(x, x): x\in G\}$, show that
(i) $H$ is a subgroup of $G\times G = G^2$ and; (ii) $H \cong G$
My attempt:
Proof.
(i) $G$ is a group, so an identity element $e\in G$ exists. ...
user1092432
4
votes
2
answers
1k
views
Question about the definition of a homomorphism
In Fraleigh's abstract algebra book, he gives definitions on how structure carries over between two isomorphic binary structures. The first definition is given relatively early in the book (Section 3) ...
- 1,328
1
vote
0
answers
53
views
Need help with a linear-algebra based proof for maximum transpositions in $S_n$
I have "roughed out" an idea for a proof for Fraleigh's Abstract Algebra book, but I am having trouble fully articulating and fleshing out what I want to write.
I need to show that every ...
- 1,328
-1
votes
2
answers
81
views
First Isomorphism Theorem: Does each homomorphism has to be surjective? Is it possible to define an homomorphism $\phi:G\to H$ such that $|G|<|H|$? [closed]
I have a little bit of misunderstanding about homomorphism and the first isomorphism.
Does each homomorphism has to be surjective? Is it possible to define an homomorphism $\phi:G\to H$ such that $|G|&...
- 2,322
1
vote
0
answers
62
views
How to see isomorphisms fast [duplicate]
As one goes deeper in abstract algebra, for example in topics like algebraic topology, there are often arguments when the authors just write things like
So we have Z + Z / (2,-2)Z is isomorphic to Z + ...
- 107
3
votes
0
answers
136
views
Isomorphism to the group of all infinite binary sequences.
The following question is Question 12.4 from Abstract Algebra by Dan Saracino.
Suppose $G=Z_2 \times Z_2 \times Z_2 \times Z_2 \times \cdots $, where $Z_2$ is the additive group of integers modulo 2.
...
- 2,596
-1
votes
1
answer
61
views
Homomorphism between two groups
I was solving some questions about homomorphism and isomorphism but then I made up a question. Here is the question: "If $\dfrac{G}{Ker(\phi)} \cong H$ then can we say there must be a ...
1
vote
1
answer
234
views
Prove that the conjugate of $h$ by $g$ is an isomorphism
This is part $2$ of a $3$ part problem, I'll link the other parts once I've created questions for them and they've been resolved. I believe I have a proof for the problem but I'm a little unsure on ...
- 2,250
-1
votes
1
answer
276
views
Homomorphism and Isomorphism importance
From group theory, two groups $(G,\cdot)$ and $(S,*)$ are homomorphic if there is a map $f$ such that $f(a\cdot b)=f(a)*f(b)$. While these groups are isomorphic if the map $f$ is homomorphism and ...
- 49
-1
votes
1
answer
48
views
Why does injectivity imply to $|G/(H \cap K)|\leq |G/H|\cdot|G/K|?$
I'm looking through some proof about the inequality in the title, the one defines:
$$\phi: G/(H \cap K)\rightarrow G/H\times G/K$$
$$\phi(g(H \cap K))=(gH,gK)$$
Note that $\phi$ is injective, I'd like ...
- 221
2
votes
1
answer
198
views
Quotient group by normal closure of union
Let $G$ be a group and $H, N \subseteq G$ be subsets of the group. Let $\overline{A}$ denote the normal closure of any subset $A\subseteq G'$ in some group $G'$. Let $\pi: G \to G/\overline{N}$ denote ...
- 884
2
votes
2
answers
372
views
Prove that ${\rm Inn}(S_n)$ isomorphic to $S_n.$
Show that ${\rm Inn}(S_n)$ isomorphic to $S_n$ for $n\ge3$.
To do this, if I define some isomorphic function say $\phi$, where $\phi: S_n \to{\rm Inn}(S_n)$, then show that $\phi$ is bijective (by ...
- 503
0
votes
1
answer
65
views
Showing Groups of Homomorphisms are Isomorphic
While looking through prior exam problems for a group theory course, I encountered this question and am having some difficulty getting started.
Let $A,B,C$ be abelian groups. Let ${\rm Hom}(A \times B,...
0
votes
2
answers
226
views
Can we make sure two groups are isomorphic if there exists a function which is a bijective homomorphism?
I'm confused that is a function which is a bijective homomorphism enough to prove the two groups are isomorphic, or we need all possible maps to be bijective homomorphism?
I saw a statement that even ...
- 48
0
votes
1
answer
53
views
What Should the Formula for this Map Be?
We have $\phi:$${R/H\to G}$, where $(R,+)$ is a group, $H = ${$2n$$\pi$$:n$$\epsilon$$Z$} is a subgroup and $G$ is a group of matrices given as $\begin{bmatrix}cos{\theta}&-sin{\theta}\\sin{\theta}...
- 389
1
vote
0
answers
51
views
Consequence of the first isomorphism theorem
Here is the set up for this problem.
Let $A$ and $B$ be groups, $\phi: a \to B$ a group homomorphism, $A' \subset A$ a normal subgroup where $\phi$ sends $A'$ to $1_B$. I proved there is a unique ...
- 2,268
2
votes
0
answers
64
views
Is every surjective group homomorphism $f:\mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \oplus \mathbb{Z}$ also injective? [duplicate]
I would like to know if every surjective group homomorphism $f:\mathbb{Z} \oplus \mathbb{Z} \rightarrow \mathbb{Z} \oplus \mathbb{Z}$ is also injective. I suspect it is true, but I'm not sure how to ...
- 1,523
-1
votes
2
answers
494
views
Find the number of homomorphisms from $A_4\to\mathbb Z_3$ and $A_4\to S_3$?
$\alpha :G\to G'$ is a homomorphism. Then
$G/\ker(\alpha)\cong\alpha(G)$ according to the first isomorphism theorem. This implies
$\vert G\vert = \vert \ker(\alpha )\vert \vert \alpha ( G)\vert$.
...
- 178
0
votes
0
answers
47
views
Strategies to find easy isomorphism between basics groups
I am currently taking introduction to algebra and I am struggling to find rapidly isomorphism between groups. I know my question is a bit broad in a sense that there is no singular answers but I find ...
4
votes
1
answer
88
views
Prove $\phi:(\mathbb{Z}/n\mathbb{Z})^{\times}\to\text{Aut}(\mathbb{Z}_n)$ such that $[x]\mapsto \phi_{[x]}([t])=x[t]$ is an isomorphism
I want to prove $(\mathbb{Z}/n\mathbb{Z})^{\times}\cong \text{Aut}(\mathbb{Z}_n)$ so
(1) I proved $f_s:G\to G $ such that $g\mapsto sg=g+\overset{s}{\dots}+g\,$ is an automorphism on $G $. ($\gcd(s,|...
- 639
2
votes
2
answers
67
views
Let $t\in\Bbb N,t\ge 1,(G,+,0)$ abelian group $|G|=n$ and $\gcd(t,n)=1$ Prove $f(x)=tx$ is an automorphism
Let $t\in\mathbb{N}, t\geq 1,\, (G,+,0)$ abelian group $|G|=n$ and $\gcd(t,n)=1$
Let $f(x)=tx=\underbrace{x+\cdots+x}_{t\text{ summands}}.$
Prove $f\in \text{Aut}(G)$
I proved $f$ is an homomorphism ...
- 639
-1
votes
1
answer
155
views
How many homomorphisms are there from one group to another group?
In our lecture we were asked to answer the following questions:
How many are there homomorphisms from $\mathbb{Z}_{2} \times \mathbb{Z}_{2}$ to $S_{3}$? And vice-versa?
How many are there ...
user856333
3
votes
1
answer
460
views
Preimage of a homomorphism preserves the index?
Let $G, H$ be groups and $K \leq H$. Also, $\phi : G \rightarrow H$ is a homomorphism.
Assume the following
$|H : K| = 2$
$\phi(G)$ is not contained in $K$ (i.e. $\phi(G) \nsubseteq K$)
From these ...
- 785