Questions tagged [group-homomorphism]

For questions about a function from one group to another that respects the structures of the groups. In symbols, $\varphi$ is a group homomorphisms if for group elements $a$ and $b$, $\varphi(ab)=\varphi(a)\varphi(b)$. Consider also using the broader tags (group-theory) or (abstract-algebra).

Filter by
Sorted by
Tagged with
0
votes
1answer
17 views

Question regarding surjectivity of induced homormophism in an old version of Hatcher's proof of Prop. 4.13

So I am currently trying to understand the given proof of Hatcher's proof of proposition 4.13. It's this particular part (in the middle of the screenshot) I don't understand: The extended $f$ ...
0
votes
1answer
24 views

Induced homomorphism example

Give an example of a commutative ring $R$, $R$ -modules $M,N,$ and$W$, and an injective $R$ module homomorphism $g:M \rightarrow N$ such that the induced homomorphism $Hom_{R}(N,W) \rightarrow Hom_{R}(...
1
vote
1answer
28 views

Counterexample: Two groups $H$ and $G$, with surjective homomorphismus

So I need to find a counter-example, i.e. I need to find two groups $H$ and $G$, with $N$ being normal subgroup of $G$, with a NON-surjective Homomorphism $\phi: G \rightarrow H$, such that $\phi(N)$ ...
0
votes
1answer
47 views

prove that function mapping is injective iff ker (f) = {e} [duplicate]

Heyall, would appreciate some help with abstract algebra because my undergrad brain is fried from doing all the proofs my prof asked me to do. I've hit a bit of a wall with this one; it involves group ...
0
votes
1answer
53 views

Proving that the restriction of a homomorphic function is also a homomorphism

Suppose $f$ is a homomorphism from $A \to B$, and $C \subset A$. Further suppose that $r$ is a function that maps $C \to B$ $|$ $r(C) = f(C)$ and since I wish to prove that $r$ is also a homomorphism ...
0
votes
1answer
14 views

Number of homomorphisms from direct products of $\mathbb{Z}_n$ to $\mathbb{Z}_{18}$

How many homomorphisms are there from $\mathbb Z_3\times \mathbb Z_4\times\mathbb Z_9$ to $\mathbb Z_{18}$. I tried to find possible kernals. The answer is $54$ but I'm getting something else. Can ...
2
votes
1answer
79 views

Show $G\cong \ker(f) \times \mathbb{Z}$ for abelian $G$

First of all, I am aware of the First Isomorphism Theorem but I am not sure how to use it/if it is useful here $G$ is an abelian group and $f:G\rightarrow\mathbb{Z}$ is a surjective group ...
0
votes
1answer
51 views

Continuity of a Group Homomorphism, Representation of SU(2)

In page 82 of Brian C. Hall's Lie Groups, Lie Algebras and Representations, he establishes the following representation $\Pi_m$ of SU(2) on $V_m$, the space of homogeneous polynomials of degree $m$ in ...
1
vote
1answer
26 views

Identifying group homomorphisms

I am struggling to know exactly how to go about working out and proving that a mapping function is a group homomorphism for a particular example. I have found that in order for the mapping to be a ...
0
votes
2answers
41 views

What are some nice informations about the dihedral groups,alternating groups,symmetric groups.

I am an undergraduate student and I want to know some nice informations about some special groups like the dihedral group $D_{2n}$,of regular $n$-gon , alternating group $A_n$ and symmetric group $S_n$...
-1
votes
1answer
19 views

Symmetric groups and surjective homomorphisms [closed]

Let, $S(6)$ be a symmetric group(of order $6$). List all groups $H$ such that there exists a surjective homomorphism $f: S(6) \rightarrow H$ Any hints or suggestions on how to approach this problem?
0
votes
2answers
14 views

Epimorphism from free product to direct product of groups.

Let $G_1$ and $G_2$ be two non abelian groups and $G_1 * G_2$ be the free product of these two groups. Can we define an epimorphism from $G_1 *G_2$ to the direct product $G_1 \times G_2$ of these ...
3
votes
1answer
85 views

Show that there is an injective homomorphism from $G_2 = \langle a, b \mid aba^{-1}b^{-1}\rangle$ to $G_1 = \langle x, y \mid xyx^{-1}y\rangle$.

I'm trying to show the above as simply as possible; I suspect that the method I have is somewhat overcomplicated. From looking at the Cayley diagrams of the groups, it is clear that $\varphi: G_2 \to ...
6
votes
2answers
71 views

Determine the number of homomorphisms from $S_{3} \rightarrow \Bbb Z_{2} \times \Bbb Z_{4}$.

Determine the number of homomorphism from $S_{3} \rightarrow \Bbb Z_{2} \times \Bbb Z_{4}$. My attempt: A homomorphism from $S_{3} \rightarrow \Bbb Z_{2} \times \Bbb Z_{4}$ is a homomorphism into ...
1
vote
1answer
25 views

Suppose there is a group homomorphism φ : G → G0 . If |G| = 1013 and |G0 | = 55, what can you say about the image of φ?

i get that cardinality of image(phi) should divide 55, by lagrange thm. so it can be 1, 5,11 or 55. how to proceed from here
0
votes
3answers
32 views

If $m \mid n$, show that there is a one-to-one homomorphism $\mathbb{Z}_m \to \mathbb{Z}_n$.

If $m \mid n$, show that there is a one-to-one homomorphism $\mathbb{Z}_m \to \mathbb{Z}_n$. Give an explicit homomorphism $\varphi : \mathbb{Z}_6 \to \mathbb{Z}_{12}$ that is injective, i.e., one-to-...
2
votes
0answers
40 views

Dual to group algebra of a finite group as a Hopf Algebra

I have a group algebra of a finite group $G$ over $\mathbb{C}$ and $Fun(\mathbb{C}[G])$ represents the linear functions on $\mathbb{C}[G]$. $\mathbb{C}[G] \otimes \mathbb{C}[G]$ denotes the tensor ...
0
votes
0answers
28 views

The evaluation of a homomorphism. [duplicate]

If we define the evaluation homomorphism $\DeclareMathOperator{\ev}{ev}$ $$\ev: \mathbb{R} [X,Y] \longrightarrow\mathbb{R} [T],\quad f(X,Y) \longmapsto f(T^2,T^3)$$ I have to show that $\...
2
votes
3answers
71 views

Number of Ring homomorphisms $\phi:\mathbb{Z}_{12}\longrightarrow\mathbb{Z}_{12}$

How could I determine the number of homomorphisms of rings with identity $1_A$ from $\mathbb{Z}_{12}$ to $\mathbb{Z}_{12}$? And how this number change if I consider the homomorphisms of rings without ...
0
votes
1answer
38 views

Proving a continuous homomorphism from $(\mathbb{R}^2,+)$ to $(\mathbb{R}, +)$ is a linear transformation.

Let $\phi$ be a continuous homomorphism from from the group $(\mathbb{R}^2,+)$ to the group $(\mathbb{R}, +)$. The, how could one prove that $\phi$ is also a linear transformation? Given that $\phi$ ...
1
vote
1answer
63 views

A special type of Gauss sum

In the work of my thesis I came up with a problem that is elementary, but I can't figure out its proof. Let $p$ be an odd prime, let $(\mathbb{Z}/p^n\mathbb{Z})^\times$ denote the multiplicative ...
0
votes
2answers
54 views

The only surjective homomorphism from $S_n$ to $\{1,-1\}$ is $\text{sign}$

I want to show that if there is a surjective homomorphism $\phi: S_n \rightarrow \{1,-1\}$ for $n\geqslant 2$, then that homomorphism is $\text{sign}$, where $$\text{sign}\ \sigma = (-1)^{\text{no. of ...
0
votes
2answers
34 views

Finding Kernel for Group and checking homomorphism

Let function f: Z10 -> { 0, 2, 4, 6, 8} (additive group) be defined as f(a)= 2a. How can I check if f is homomorphism? And how to compute 5 + Ker(f) + 2 + Ker(f). After going through some books and ...
3
votes
3answers
43 views

Help on homomorphisms from $C_{12}$ to $C_2 \times D_5$.

I'm taking an abstract algebra course and ran into a problem classifying the possible homomorphisms $\phi: C_{12} \rightarrow C_2 \times D_5$. We were asked to find the possible homomorphisms ...
0
votes
0answers
18 views

Prove Aut$(V)\simeq S_3$, where $V$ is the Klein-4 Group. [duplicate]

Suppose $g,h,k\in V$ are distinct, and choose a bijection $f$ so that $1\mapsto1$. Then $gh=k$, and since $f$ is a bijection, $f(gh)=f(k)=f(g)f(h)$. If $g\in V$, then $f(g^2)=1$, and since $f$ is a ...
1
vote
1answer
22 views

Homomorphism of $G=GL(n,\mathbb R)$ and $H=GL(n,\mathbb R)$ where $f(A)=(A^{-1})^T$

Is this $f:G\rightarrow H$ a homomorphism. At first look it appears to be an automorphism given $G=GL(n,\mathbb R)$ and $H=GL(n,\mathbb R)$ where $f(A)=(A^{-1})^T$ So $f(AB)=((AB)^{-1})^T=(A^{-1})^...
0
votes
1answer
27 views

Homomorphism of groups $f: G \rightarrow H$ where $f(x)=e^{2\pi i x}$

Define a homomorphism as $f: G \rightarrow H$ where $f(x)=e^{2\pi i x}$ $G=\mathbb R$ and $H=\mathbb C\setminus\{0\}$ Checking that it is a homomorphism: $f(xy)=e^{2\pi i x y}=(e^{2πix})^y=f(x)^y$ ...
-2
votes
1answer
44 views

Is $f:\Bbb R^*\to\Bbb R^+: x\mapsto\lvert x\rvert$ homomorphic, and what is its kernel? [closed]

Let $\Bbb R^*$ be the group of real numbers under multiplication (number $0$ is not in $ \Bbb R^*$) and $\Bbb R^+$ be the group of positive real numbers under multiplication. Define $f:\Bbb R^*\to\Bbb ...
3
votes
0answers
45 views

determine whether the following are surjective homomorphisms and if so are they also isomorphisms by computing kernels

So I am looking to determine whether the following are surjective homomorphisms, and if so are they also isomorphisms by computing kernels: I am completely new to the topic of kernels in group theory ...
0
votes
1answer
52 views

Homomorphisms from $\mathbb Z \rightarrow \mathbb Z_{6}$

Homomorphisms from $\mathbb Z \rightarrow \mathbb Z_{6}$ I have proven that the number of Homomorphisms from $\mathbb Z_6 \rightarrow \mathbb Z=6$ Is the reverse just simply $6$ also? in $\mathbb ...
-1
votes
2answers
28 views

A question about the quotient group of images [duplicate]

$H\trianglelefteq G$. Let $\phi$ be a homomorphism from $G$ to another group, and ${\rm ker}~\phi\le H$. Assume that $G/H\cong \phi(G)/\phi(H)$(the third isomorphism theorem for example), and $\phi(G)\...
0
votes
2answers
39 views

Why does the surjectivity of the canonical projection $\pi:G\to G/N$ imply uniqueness of $\tilde \varphi: G/N \to H$

Let's look at the universal property of quotient groups: Let $\varphi:G \to H$ be a homomorphism, $N$ a normal subgroup of $G$ and $\pi:G \to G/N$ the canonical projection. If $N \le \ker \...
0
votes
2answers
29 views

Equivalence class as a homomorphism from a group to its quotient group

Given a normal subgroup $N$ of group $G$, the quotient group $G/N$is defined by the usual group multiplication $$[a]\star [b]=[a \cdot b]$$ where $\star$ is the group multiplication in $G/N$ and $\...
3
votes
2answers
86 views

Prove that $f∶N \times K \rightarrow G, f(a, b) = ab$ is an injective homomorphism where $N$ and $K$ are disjoint normal subgroups of $G$

Let $G$ be a group and let $N$ and $K$ be normal subgroups of $G$. Suppose $N \bigcap K = \{e_G\}$. Prove that $f∶N \times K \rightarrow G, f(a, b) = ab$ is an injective homomorphism. Is the ...
2
votes
1answer
49 views

Prove that there is an isomorphism between two groups.

I'm solving the following problem. Let H be a group and $\tau_1:H\rightarrow G_1, \tau_2:H\rightarrow G_2,\cdots\tau_n:H\rightarrow G_n$ homomorphims with this property: Whenever $G$ is a group ...
1
vote
2answers
55 views

How to prove $\langle x,y\rangle\cong\langle x\rangle+ \langle y\rangle$ in groups?

How to prove $\langle x,y\rangle\cong\langle x\rangle+ \langle y\rangle$ in groups? I am not sure if got the notations right, basically I was wondering given an additive group $G$, which is ...
0
votes
1answer
67 views

Induced Group Representation

Let $D_{\infty} = \langle a,t \mid a^2 = t^2 = 1 \rangle$ be the infinite dihedral group, and let $H = \langle at \rangle$. Given $\theta \in [0,2 \pi)$, let $f_{\theta} : H \to \Bbb{T}$ be defined ...
0
votes
1answer
28 views

Find homomorphism between groups [duplicate]

The task is to find all the Homomorphisms from group $\mathbb{Z}_{20}$ to $\mathbb{Z}_{16}$. My teacher told be that there is an sufficient way to do this, however I've just brute forced it. Do you ...
0
votes
0answers
19 views

Aschbacher finite group theory exercise 1.6 [duplicate]

I am reading Michael Aschbacher's Finite group theory textbook. Exercire 1.6 makes me confused. 1.6 Let H is normal subgroup in G. Then the map L->L/H is a bijection between the set of all subgroups ...
0
votes
2answers
21 views

Proof: If $ϕ: R \to R'$ is a ring isomorphism, then its inverse $ϕ^{-1} : R' \to R$ is a ring homomorphism.

Proposition. If $ϕ: R \to R'$ is a ring isomorphism, then its inverse $ϕ^{-1} : R' \to R$ is a ring homomorphism. How would you start a proof for this proposition?
4
votes
1answer
55 views

Homomorphic image of an alternating group

I'm solving the following problem: If $f:S_n\rightarrow S_n$ is a group homomorphism, prove that $f(A_n)\subseteq A_n.$ (Here, $S_n$ is a symmetric group of degree $n$, and $A_n$ is an ...
0
votes
1answer
52 views

How would I prove the following theorem on quaternions?

The theorem states The map that takes $q$ to the map $[q] : x \to q^{-1}xq$ is a 2-to-1 homomorphism from the group of unit quaternions to $SO(3)$. How would I prove this?
1
vote
1answer
32 views

Condition for triviality of group morphism $\tau : \mathbb{Z}_p \to\mathrm{Aut}(\mathbb{Z}_q)$ : $\tau_\overline{k}(\overline{n}) = \bar{r}^k \bar{n}$

I'm trying to show the following: Let $p,q$ be prime numbers, $\tau : \mathbb{Z}_p \to \operatorname{Aut}(\mathbb{Z}_q)$ be a group morphism such that $$\overline{k} \mapsto \tau_\overline{k}, \...
-3
votes
1answer
34 views

If $f:\mathbb{Z}\times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}$ is homomorphism, show that $f$ is monomorphism. [closed]

Here's the complete problem. If $f:\mathbb{Z}\times \mathbb{Z} \rightarrow \mathbb{Z} \times \mathbb{Z}$ is homomorphism such that $f(0,1)=(-1,5)$ and $f(1,0)=(2,-3)$, show that $f$ is monomorphism. ...
5
votes
1answer
90 views

Compute the kernel of the group hom $\Omega : \Bbb{Q}^{\times} \to \Bbb{Z}^+$.

The $\Omega$ function is the counting function that returns precisely the number of primes $\Omega(n)$ (including multiplicity) that divide a natural number $n \in \Bbb{N}$. For example $\Omega(6) = ...
2
votes
1answer
34 views

General form of a group morphism $\tau : \mathbb{Z}_p \to \operatorname{Aut}(\mathbb{Z}_q)$ such that $\bar{k} \mapsto \tau_{\bar{k}}$

I'm trying to prove the following proposition: Let $p,q$ be two prime numbers. Then every group morphism $\tau : \mathbb{Z}_p \to Aut(\mathbb{Z}_q)$ such that $\overline{k} \mapsto \tau_{\overline{...
1
vote
1answer
31 views

Let $f:\mathbb{Z} \rightarrow \mathbb{Z}_{10}$ is homomorphism. Determine $f(18)$ such that $f(1)=6$.

Let $f:\mathbb{Z} \rightarrow \mathbb{Z}_{10}$ is homomorphism. Find $\ker f$ and $f(18)$ such that $f(1)=6$. Here I tried. $$\ker f = \lbrace 10q \;|\; q \in \mathbb{Z} \rbrace.$$ Next, may I ...
0
votes
2answers
34 views

If $T=\lbrace -1,1 \rbrace$, show that $\mathbb{R}^{*}/\mathbb{R}^{+} \cong T$ is group under multiplication.

If $T=\lbrace -1,1 \rbrace$, with the Fundamental Theorem of Homomorphism group, show that $\mathbb{R}^{*}/\mathbb{R}^{+} \cong T$ is group under multiplication, where $\mathbb{R}^*$ is set of all ...
0
votes
0answers
56 views

Number of homomorphism from $\mathbb Z_5 \to S_6$

I am finding the number of homomorphisms from $\mathbb Z_5 \to S_6$. We see the only subgroups of $\mathbb Z_5$ are $\mathbb Z_5$ and ${0}$. $\frac{\mathbb Z_5}{\mathbb Z_5}$ is isomorphic to onle ...
0
votes
4answers
73 views

Prove any surjective homomorphism $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is injective [closed]

I'm not sure how to even start this one if I'm being completely honest. All I have is $f$ is a homomorphism so $f(ab)=f(a)f(b)$ and $f$ is surjective so $\forall y \in \mathbb{Z} : \exists x \in \...

1
2 3 4 5
28