Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [group-homomorphism]

For questions about the group homomorphism, a function between two groups statistifying $\alpha(ab)=\alpha(a)\alpha(b)$ and its applications in group theory. Should usually be used together with the tag (group-theory).

4
votes
1answer
35 views

Question about the proof of Second Isomorphism Theorem

The Second Isomorphism Theorem: Let $H$ be a subgroup of a group $G$ and $N$ a normal subgroup of $G$. Then $$H/(H\cap N)\cong(HN)/N$$ There is the proof of Abstract Algebra Thomas by W. Judson: ...
4
votes
1answer
45 views

Questions about the functions of the type $f:\Bbb Z_{40} \rightarrow \Bbb Z_{60}$

Given the function $f:\Bbb Z_{40} \rightarrow \Bbb Z_{60}$, how many of them are there such that $f([0]_{40})=[0]_{60}$ and $f([1]_{40})=[1]_{60}$? How many of them are homomorphism of additive groups?...
2
votes
1answer
48 views

Prove that $\frac{x+y}{1+xy}$ is an Abelian Group [duplicate]

Let $I=\left]-1,\ 1\right[$ be an interval, and $\left(I,\ \star\right)$ be a magma such that: $$\left(\forall\ \left(x,\ y \right) \in I^2\right)\ x \star y=\frac{x+y}{1+xy}$$ I need to prove that ...
3
votes
1answer
77 views

How many group homomorphisms are there from $(\mathbb{Q}^+, \cdot\ )$ to $(\mathbb{Z}_m,+)$

How many group homomorphisms are there from $(\mathbb{Q}^+, \cdot\ )$ to $(\mathbb{Z}_m,+)$? Of course, there is a trivial zero homomorphism. I think there are no more homomorphisms there but I am ...
2
votes
1answer
30 views

Exact sequence of groups: proof of injectivity

There must be a duplicate being the question very introductory, but I was not able to find it. We have the following diagram $$\begin{array}{ccccccccc} 1&\to& H &\to & G &\...
2
votes
2answers
32 views

Confusion about a proof about subgroups of dihedral groups

This article shows that every subgroup of $D_n = \langle r, s \rangle$ is cyclic or dihedral. Theorem 3.1. Every subgroup of $D_n = \langle r, s \rangle$ is cyclic or dihedral. A complete listing ...
0
votes
2answers
33 views

Homorphism between groups and generators

Let $f: G \rightarrow H$ be a homomorphism of finite groups. Since $G$ is finitely generated, $G = \langle x_1 , ... , x_n \rangle$. Is it then true that $H = \langle f(x_1), ... , f(x_n) \rangle$? ...
0
votes
1answer
25 views

Group representation preserving finitely many generators

Let $\rho : G \rightarrow GL_n(\mathbb{C})$ be a representation of $G$. If $G$ is finitely generated as a group, does that mean that $im(\rho) \leq GL_n(\mathbb{C}) $ is finitely generated? Because, ...
2
votes
1answer
41 views

Third isomorphism theorem on groups

From this wikipedia's page, I have found these important theorems. I would like to give it a try. Since I am self-learning Mathematical Analysis without teacher or tutor, it will be great if someone ...
2
votes
0answers
30 views

Second isomorphism theorem on groups

From this wikipedia's page, I have found these important theorems. I would like to give it a try. Since I am self-learning Mathematical Analysis without teacher or tutor, it will be great if someone ...
-1
votes
0answers
22 views

For what integers $a$ does the map $\psi_{a} \colon Z_{36} \to Z_{24}$ defined by $ \psi_{a} \colon x \mapsto y^{a}$ extend to a homomorphism? [on hold]

For what integers a does the map $$ \psi_{a} \colon Z_{36} \to Z_{24} $$ defined by $$ \psi_{a} \colon x \mapsto y^{a}$$ extend to a homomorphism?
-1
votes
0answers
14 views

Exercise about a family of homomorphisms between abelian groups

It is considered the category $\mathscr{Ab}$ of abelian groups; a family of homomorphisms: $\{ f_i : M_i \rightarrow N_i | i \in I, M_i,N_i \in \mathscr{Ab} \}$ and constructions $\prod_i f_i$, $\...
5
votes
1answer
42 views

A problem about number of functions and homomorphism

I'm trying to solve this problem. How many functions $f: \Bbb Z_{10}\rightarrow \Bbb Z_3$ are there such that $|f^{-1}([0]_3)| = 3$ or $|f^{-1}([1]_3)| = 4$? How many of them are such that the ...
1
vote
0answers
47 views

First isomorphism theorem on groups

From this wikipedia's page, I have found these important theorems. I would like to give it a try. Since I am self-learning Mathematical Analysis without teacher or tutor, it will be great if someone ...
4
votes
1answer
52 views

$x \mapsto x^n $ is a automorphism of group $G$, show that for all $x$ in $G$, $x^{n-1} \in Z(G)$.

If $x \mapsto x^n $ is a automorphism of group $G$, show that for all $x$ in $G$, $x^{n-1} \in Z(G)$. This mean $G=\{x^n:x \in G\}$ and $x^n=e$ if and only if $x= e$. Now let $y\in G$. I Want to ...
0
votes
1answer
16 views

Taking the Homomorphism of a Subgroup [duplicate]

Let $\phi : G \rightarrow G'$ be a homomorphism. We want to show that $H \leq G$ implies that $\phi(H) \leq G'$. This is one of those problems that seems obviously true to the intuition, yet I'm ...
0
votes
0answers
33 views

Group Homomorphisms into an Abelian Group

The following comes from Hungerford's Algebra. [Prove that if] $f: G \to H$ is a homomorphism, $H$ is abelian and $N$ is a subgroup of $G$ containing $\ker f$, then $N$ is normal in $G.$ A ...
4
votes
1answer
25 views

Is the free product of residually finite groups always residually finite?

Suppose groups $G$ and $H$ are residually finite. Does that imply, that $G \ast H$ is residually finite? What have I tried to prove this: Suppose, $a = g_1h_1g_2h_2…g_nh_n \in G \ast H$, $g_1, .. ...
2
votes
2answers
39 views

Number of homomorphisms between $S_3$ and $Z_4=C_4$

These questions are pretty standard, this is the first time I'm trying one on my own, I would like to check my progress: Using the First Isomorphism Theorem and Lagrange’s Theorem, find all the ...
2
votes
1answer
58 views

Homomorphism condition for a subset of a group generates the whole group

I wonder whether the following statement is true: Let F is a group and X is a subset of F. Then $\left< X \right> =F\quad \Longleftrightarrow \quad$ For any group G and any function $\phi :X\...
5
votes
1answer
63 views

Find the number of group homomorphism between $A_5$ and $S_5$.

The above question is based on this answer to a similar question. I just want to apply what has been pointed out in that answer to this question. So we are interested in number of homomorphisms $f:...
5
votes
1answer
237 views

Is every normal subgroup the kernel of some self-homomorphism? [duplicate]

Let $G$ be a group. If there is a homomorphism $f:G\to G$ (special case of the codomain being arbitrary group), then the kernel $f^{-1}(id)$ is a normal subgroup of $G$. But now the other way around: ...
3
votes
2answers
68 views

How to prove/show this actually defines a homomoprhism

We define the homomorphism $f: \text{SL}_2(\mathbb Z / 2 \mathbb Z) \to \text{SL}_2(\mathbb Z / 2 \mathbb Z)$ that maps the generators to: $ \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \to ...
0
votes
1answer
50 views

Show if a homomorphism has a non trivial kernel

For Q1(a) I have shown, by the property of distribution for matrices, that 1(a) is a homomorphism. I have tried to show that it only has a non-trivial kernel but cannot find an example. Would I be ...
2
votes
1answer
29 views

Constructing a homomorphism such that a given set is the kernel.

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
1answer
50 views

New outer automorphism for $G_1 \times G_2$

Suppose $G_1$ is a group, which has no outer automorphism. Suppose $G_2$ is a group, which has no outer automorphism. Main Question: What are the possible conditions to know can there be an outer ...
1
vote
2answers
51 views

Find all the group morphisms from $\Bbb Z$ to $D_4$

I haven't had any experience with finding group morphisms between different types of groups before and am wondering if anyone has any advice? Thanks in advance! Find all the group morphisms from $\...
-1
votes
1answer
48 views

Group homomorphism containing the trivial kernel only

Is it right to say that this is a group homomorphism and it only contains the trivial kernel? $$\Phi : (\mathbb{R}\setminus\{0\}, ×) \longrightarrow (\mathbb{R}\setminus\{0\}, ×) : x \mapsto |x|$$
0
votes
1answer
119 views

Non-trivial kernel

Am I correct in saying that this is a group homomorphism? If this is a group homomorphism does it have a non-trivial kernel? $$\Phi : (M(\mathbb{R},n), +) \longrightarrow (\mathbb{R}, +) : A \mapsto ...
0
votes
1answer
54 views

Group homomorphism with non-trivial kernel

Am I correct in saying that this group homomorphism below has a non-trivial kernel as it is not injective? $ φ : (\mathbb R, +) → (\mathbb Z, +) : x → b$, where $b$ is the largest integer which is ...
0
votes
2answers
60 views

Homomorphisms Group Theory [closed]

Does anybody know how I would go about proving this question ? Let G and H be groups and let φ : G −→ H be a group homomorphism. Suppose that G is abelian and φ is a surjection. Prove that H is ...
1
vote
1answer
38 views

Isomorphism between an elliptic curve and the additive structure of the field.

Let $C: y^2=x^3$ be a singular elliptic curve over a field K and define $C_{ns}(K)=C(K)/{0,0}$ the set of non singular points of C including $P_\infty$. I've proved that $C_{ns}$ is a group with the ...
0
votes
1answer
18 views

Injective and surjective homomorphisms between non cyclic group of order $4 $ to $Z_8$

Let $G$ be a non cyclic group of order $4$. Consider the following statements: $I:$ There is no one-one map homomorphism from $G$ to $Z_8$ $II:$ There is no onto homomorphism from $Z_8$ to ...
1
vote
0answers
27 views

Set of all R-homomorphism is finitely generated

I have the following proposition: Proposition: If $R$ is a principal ideal domain (PID) and $M$, $N$ two finitely generated (fg) $R-$modules, then $\text{Hom}_R(M,N)$ is finitely generated. My idea: ...
2
votes
1answer
72 views

An isomorphism between $\mathbb Z_n \times \mathbb Z_m$ and $ \mathbb Z_{mn}$

I am reading these lecture notes and they suggest the following generalisation of a specific example for $\mathbb Z_2 \times \mathbb Z_3 \cong Z_6 $: There exists an isomorphism between $\mathbb ...
2
votes
1answer
48 views

GAP semidirect product

I am newbie in the GAP and in the group theory. Now I am trying to make semidirect product if GL(3,2) and GL(3,2) inversed and transposed. I use code below ...
-1
votes
1answer
37 views

True or false: $\left(\mu\left[a,\bar{a}\right]=\mu\left[b,\bar{b}\right]\iff a+\bar{b}=b+\bar{a}\right)\iff\mu$ is a homomorphism?

Note: square brackets $\left[\dots\right]$ are use to indicate parameter lists in function signatures. Addition of ordered pairs in the domain is defined by $$\left<a,\bar{a}\right>+\left<b,...
1
vote
0answers
70 views

Hatcher Exercise 3.2.9

Show that if $H_n(X; \mathbb{Z})$ is free for each $n$, then $H^∗(X; \mathbb{Z}_p)$ and $H^∗(X; \mathbb{Z})⊗\mathbb{Z}_p$ are isomorphic as rings. I'm assuming the tensor product is taken over $\...
3
votes
2answers
48 views

Let $f : G → G_1$ be a surjective homomorphism (also called epimorphism) from $G$ to another group $G_1$. Prove that $f(Z(G)) \subseteq Z(G_1)$.

I am working on a school assignment and have been stuck on this question for some time. Let $f : G \rightarrow G_1$ be a surjective homomorphism (also called epimorphism) from $G$ to another group $...
0
votes
3answers
43 views

Show that a function from $\Bbb Z / 3600\mathbb{Z}$ to $\mathbb{Z} / 1200\mathbb{Z}$ is a well defined group homomorphism of additive groups

So, my question is: "Show that \begin{align} f : \mathbb{Z}/3600\mathbb{Z} &\longrightarrow \mathbb{Z}/1200\mathbb{Z}\\ [x]_{3600} &\longmapsto [88x]_{1200} \end{align} is a well defined ...
3
votes
0answers
45 views

Are $\mathbb{Z}$ and $\mathbb{Z}_n$ the only rings (with identity) whose modules are equivalent to abelian groups?

Let $R$ be a ring with identity. Let $M$ and $N$ be $R$-modules. Let $f$ be an (arbitrary) group homomorphism from $M$ to $N$. Under what conditions on $R$,$M$, and $N$ is $f$ also a $R$-module ...
0
votes
1answer
69 views

Show that $Ax=0$.

I need a hint to help me get started with this problem: Given the sequence of homomorphisms, $\mathbb{Z}^{m}\to \mathbb{Z}^{n} \to M\to 0$, where $M=\mathbb{Z}^{n}/K$ and $K=im(\phi_{A})\subseteq \...
1
vote
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, ...
2
votes
2answers
40 views

Excercise with the normal subgroup $K$ of $G$ with $K:=\bigcap\{H \text{ subgroup of } G: \forall_{x,y\in G}:xyx^{-1}y^{-1}\in H\}$

So far I have showed that $K$ is a normal subgroup and that the Operation defined on $G/K$ is abelian. Now I have to show that if $\phi:G\rightarrow A$ is a group homomorphism and $A$ is abelian....
1
vote
1answer
42 views

How do the elements of the set $(\mathbb{Q}\backslash\{0\},\cdot)/(\mathbb{Q}^{+},\cdot)$ look like?

I have already asked that Kind of Question previously. I have the Problem that if I have to answer a Question like this than I can Always find a homomorphism for which the desired quotientgroup is the ...
3
votes
1answer
59 views

Show that $f^{-1}\{f(a)\}=a\text{Ker}f$

Let $G,G'$ be groups and assume $f: \space G \longrightarrow G'$ is a homomorphism. Show that $\forall f(a) \in \ \operatorname{Im}f$: $$ f^{-1}\{f(a)\}=a \operatorname{Ker}f $$ To begin with, the ...
-1
votes
4answers
217 views

Do isomorphic groups share a binary operation? [closed]

Suppose you have two isomorphic groups. Does the binary operation defined on each group need to be the same operation?
2
votes
2answers
74 views

Homomorphism $f: \mathbb{Z}_{12} \longrightarrow \mathbb{Z}_{30}$

Suppose that we want to construct a non-surjective homomorphism $$ f: \mathbb{Z}_{12} \longrightarrow \mathbb{Z}_{30} $$ Since $\mathbb{Z}_{12}$ is cyclic, $f$ is completely determined from the ...
0
votes
0answers
26 views

Let G be a group. Prove that the function: $G→G, x→x^2$ is a group homomorphism if and only if G is an abelian group. [duplicate]

Let G be a group. Prove that the function: $G→G, x→x^2$ is a group homomorphism if and only if G is an abelian group. Is this the same as if $f:G→G$ is defined by $f(x) = x^2$ ?
5
votes
1answer
118 views

Kernel of $G\ast H\to G\times H$ is free

Let $G$ and $H$ be groups. The identity homomorphism $G\to G$ and the trivial homomorphism $H\to G$ give a homomorphism $G\ast H\to G$ by the universal property of the coproduct. Similarly, we obtain ...