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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).

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1answer
36 views

Is D4 Isomorphic to Z2 [closed]

If We define a homomorphism from D4 to Z2 and apply first isomorphism theorem. Is D4/A4 Isomorphic to Z2 As we know that D4 is non cyclic and Z2 is cyclic. So how can we prove first isomorphism ...
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2answers
12 views

Group Homomorphism from D4 to Z2

Is there any Homomorphism exist from D4 to Modulo group Z2? If such a homomorphism exist how can we define this Homomorphism?
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1answer
49 views

Verifying a statement related to group homomorphism

I am studying from a book about group-theory. I got the chapter of normal groups and isomorphisms. There was a question: Let $X=\mathbb{Z}_{4}$,$X'=\{0,2\}$,$Y=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ ...
1
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1answer
19 views

Why is the kernel of $\phi : G\mapsto S_m$ included in $B\le G$ where $\phi(g)=g:xB\mapsto gxB$?

Let $G$ be a group with $B$ a subgroup of index $m$. There is a homomorphism that associates each element of $G$ to a permutation of the left cosets $xB\in G/B$, which is of cardinality $m$. So these ...
0
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2answers
54 views

Construct a non trivial homomorphism from $\mathbb{Z}_2\times\mathbb{Z}_4$ to $\mathbb{Z}_8$

I don't know how to solve this problem since the group $\mathbb{Z}_2\times\mathbb{Z}_4$ is not cyclic. I just know that $\mathbb{Z}_2 \times \mathbb{Z}_4= \{ (0,0), (0,1),(1,0),(1,1),(0,2),(1,2),(0,3),...
1
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1answer
38 views

When is it true that $\nexists g\ne e$ such that $\phi(g)=e$ for a group homomorphism $\phi: G\mapsto H$ and $|H|\not\mid|G|$?

I was thinking if it is possible to have $g\ne e$ such that $\phi(g)=e$ for a group homomorphism $\phi: G\mapsto H$ It's not always true because $\phi(2)=0$ when $G=\Bbb Z_4$ and $H=\Bbb Z_2$ So ...
2
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2answers
35 views

Showing that $o(\phi(g))|o(g)$ if $\phi:G_1\mapsto G_2$ is a homomorphism and $g\in G_1$

$o(g)$ denotes the order of $g$. This is how I think I proved it: Let $m:=o(g)$ Let $d=o(\phi(g))$. $~d\le m$ because $\langle\phi (g)\rangle=\{\phi(g),\phi(g)^2=\phi(g^2),...,\phi(g^m)=e_{G_2}\}$ ...
0
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0answers
32 views

Prove the following permutation mapping is a homomorphism.

Prove that $\varphi: S_n \to S_{n+1 }$, where for $\sigma\in S_{n}$: \begin{pmatrix} 1 & 2 & \dots & n \\ 1\sigma & 2\sigma & \dots & n\sigma\\ \end{pmatrix} $\varphi(\sigma)\...
2
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2answers
42 views

Understanding semidirect product by constructing a non-abelian group of order $21$

I just learnt semidirect product, but only know the basic definition, not gaining the true understanding of it. There is an example that asks the reader to construct a nonabelian group of order $21$. ...
1
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3answers
76 views

How is it possible that $\textrm{HK}=\textrm{G}$?

For the following problem: If $\textrm{H}$ and $\textrm{K}$ are distinct subgroups of $\textrm{G}$ of index $2$, then $\textrm{H}\cap\textrm{K}$ is a normal subgroup of $\textrm{G}$ of index $4$. ...
3
votes
3answers
61 views

There exists an element of order $5$ in $G$

Let $G$ be a finite group and assume $\varphi: \, G \longrightarrow \mathbb{Z}_{10}$ is a group epimorphism. I want to show that there exist an $a \in G$ s.t. $\text{ord}(a)=5$. At first, the cyclic ...
1
vote
1answer
28 views

Relation between set of Normal subgroups and set of Homomorphism images of a group $G$

let $G$ be any group.let $A$ is set of all normal subgroups of $G$, and $B$ is set of all homomorphic images of $G$. Is there any type of relation between these sets.Mean some type of bijection like ...
1
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1answer
29 views

Let $N$ be an Abelian normal subgroup of $G$, if $G/N$ is perfect, then also $G'$ is perfect.

I'm reading Kurzweil & Stellmacher's "The Theory of Finite Groups", its 1.5.3 says: Let $N$ be an Abelian normal subgroup of $G$. If $G/N$ is perfect, then also $G'$ is perfect. Proof. ...
3
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0answers
74 views

On a characterization of abelian groups $G$ based on special commutator relations ($\exists n\in\Bbb N$ s.t. $[x^n,y]=[x,y^{n+1}],\forall x,y \in G$).

Let $G$ be a group. If $\exists n\in \mathbb N$ such that $x^n=yx^n(y^{n+1}x)^{-1}xy^n,\forall x,y\in G$, then how to prove that $G$ is abelian? Thoughts: the condition is same as saying $[x^n,y]=[x,...
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1answer
53 views

Is there exist a group homomorphism from the symmetric group $S_n$ to $S_{n-1}$ for $n \ge 5?$

Does there exist a group homomorphism from the symmetric group $S_n$ to $S_{n-1}$ for $n \ge 5?$ My attempt: I think not, because for $n \ge 5$ , $A_n$ is the only normal subgroup of $...
4
votes
3answers
86 views

If $\varphi$ is a homomorphism, $\text{ord}(\varphi(a))$ divides $\text{ord}(a)$

Assume $\varphi: \, G \longrightarrow G'$ is a homomorphism and $a \in G$ an element of finite order. If $m=\text{ord}(a)$ and $n=\text{ord}(\varphi(a))$, show that $n \big| m$. To begin with, it's ...
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3answers
32 views

(For any group $H$) Bijection between $H$ and the group consists of all the homomorphisms between $(\mathbb Z, +)$ and $H$.

My question is, for any group $H$, how to prove there exists a bijective function between $Z$ and the group consists of all the homomorphisms which I will call IS later on for convenience I just ...
4
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0answers
71 views

Existence of abelian group which has no “square-root” but whose “cube” has a “square-root”

Does there exist an abelian group $G$ such that $G \ncong H \times H$ for every abelian group $H$ but $G \times G \times G \cong K \times K$ for some abelian group $K$ ? Also see Existence of ...
9
votes
1answer
64 views

If $f$ has no non trivial fixed points and $f\circ f$ is the identity then $f(x)=x^{-1}$ and $G$ is abelian for $f$ an automorphism of $G$ [duplicate]

Let $f$ be an automorphism of the finite group $G$ such that $f\circ f=id$ and $f(x)=x\implies x=e$ Prove that $f(x)=x^{-1}~\forall x\in G$ If we can prove that $f(x)$ and $x$ commute for any ...
4
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0answers
50 views

Compute all homomorphisms of $\mathbb G_a$ to $SL_2$ over $\mathbb Z$

How to compute the set of homomorphisms $\mathrm{Hom}\left(\mathbb G_a , SL_2\right)$ between those two group schemes over $\mathbb Z$? Over $\mathbb C$, this can be done using classical algebraic ...
0
votes
3answers
29 views

Understanding proof about a group and homomorphism

I have a book about group theory and there was the following question: Let $G$ be a set of all the real matrices in the following form: $\begin{pmatrix}a & b\\ -b & a \end{pmatrix}$ when $a^...
0
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1answer
39 views

Construct a non-trivial homomorphism: $\mathbb{Z}_2 \times \mathbb{Z}_4 \longrightarrow \mathbb{Z}_8$

We shall construct a map $$ \varphi : \left\{0,1,2,3\right\} \times \left\{0,1\right\} \longrightarrow S \subseteq \mathbb{Z}_8 $$ which satisfies $$ \varphi (a+b \bmod4,c+d \bmod 2)=\varphi(a,c) +\...
2
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0answers
24 views

Normal groups and homomorphism are the same, and this gives an approach to isomorphism theorem?

I was reading a post here that give some interesting approach about isomorphism theorem (see quote). But there are some things I don't understand. What exactly does this mean? The Second ...
2
votes
2answers
29 views

Show that $R^n/Im(\rho)=R^{n-1}\bigoplus R/2R$, where $R$ is an abelian group and $\rho$ is the following function.

Consider the following group homomorphism $\rho$, where $R$ is an abelian group, \begin{align*} \rho:&R\rightarrow R^n\\ \rho(r)=&(2r,2r,\cdots,2r). \end{align*} Show that $R^n/Im(\rho)=R^{n-...
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2answers
31 views

If $g:\mathbb{Z_{10}}\rightarrow U_{20}$ is a group homomorphism, then the order of $g(1)$ is either $1$ or $2$. [closed]

Why is that if $g:\mathbb{Z_{10}}$$\rightarrow$$U_{20}$ is a group homomorphism, then the order of $g(1)$ is either $1$ or $2$? Also, $g$ is a function, $\mathbb{Z_{10}}$ is the group of integers ...
0
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1answer
43 views

The matrix corresponds to a homomorphism $\mathbb{Z^2} → \mathbb{Z^3}$

In Aluffi's Algebra in 4.3. Reading a presentation it says: For example, take $M=\begin{pmatrix}1&3\\2&3\\5&9\end{pmatrix}$; this matrix corresponds to a homomorphism $\mathbb{Z^2} → \...
2
votes
2answers
60 views

Showing that there is a surjective map from $\Bbb Z \ast \Bbb Z$ to $C_2 \ast C_3$ just using universal property of coproduct

I am solving Allufi chapter $0$ exercise $3.7$. There is a easy way to solve this if we know how the coproduct of $\Bbb Z \ast \Bbb Z$ and $C_2 \ast C_3$. I was wondering if there is an abstract ...
0
votes
1answer
20 views

If $G$ is a cyclic group of order $d$ with generator $a$, show that $\dfrac{\mathbb{Z}}{d\mathbb{Z}}$ is isomorphic $G$. [duplicate]

We have to $|G|=d$ and $G=\left<a\right>=\{a^n : n \in \mathbb{Z}\}$ by definition of cyclic group. To show what you are asking, I want to define a function, not necessarily this $$\phi : \...
2
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1answer
46 views

Showing $\phi: \mathbb Z \to \mathbb Z_{12}$ is a homomorphism and determining it's $Ker(\phi)$ and $Im(\phi)$

Question: Group Theory: Let $\phi: \mathbb Z \to \mathbb Z_{12}$ be defined by $$\phi(x) = x\;(\text{mod}\; 12)$$ a). Show that $\phi$ is a homomorphism. My work: Let $\phi: \mathbb Z \to \mathbb ...
3
votes
2answers
28 views

What exactly is $\phi(G)$ in a homomorphism?

I'm learning about group homomorphisms and I'm confused about what the $\phi$ transformation is exactly. If we have some group homomorphism $\phi : G\rightarrow H$ what exactly does $\phi(G)$ mean? ...
1
vote
1answer
36 views

Finding all group homomorphisms from $\mathbb{Z}_7$ to $\mathbb{Z}_{12}$

Suppose that I'm interested in finding all group homomorphisms from $\mathbb{Z}_7$ to $\mathbb{Z}_{12}$. The textbook has provided a brief explanation: Let $\phi$ be such a homomorphism. Since the ...
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2answers
38 views

Finding Homomorphisms from dihedral groups to cyclical groups

Ok so there was another question very similar to this on here however it leaves me a little confused. $\bf{Question}$ Let G = $D_{14}$ the Dihedral group order 14 and A = $c_7$ be the cyclical ...
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votes
1answer
31 views

matrix homomorphism

Prove that $ϕ$ is a homomorphism and describe its kernel. $$ ϕ:ℝ→GL(2,ℝ),\qquad ϕ(x)= \begin{bmatrix}\cos(2x)&\sin(2x)\\-\sin(2x)&\cos(2x)\end{bmatrix} $$ where $x∈ℝ$ I have begun by saying $...
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votes
1answer
28 views

Prove the existence of homomorphism.

I am trying to answer the following question. Is there any group homomorphsim $\phi: D_4 \rightarrow S_5$?
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0answers
21 views

Homomorphism and relative order questions

1) I have to prove that if $\exists$ a nontrivial homomorphism $\phi:A\rightarrow B$, where A and B are finite and Abelian, then $|A|$ and $|B|$ are not relatively prime. I know that $\phi(A)$ is a ...
1
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1answer
40 views

If all homomorphisms $f:G→H$ are trivial or injective, then G is simple.

Let $G$ be a nontrivial group. Show that $G$ is simple if and only if, for every group $H$ and homomorphism $f:G→H$, either $f$ is trivial or $f$ is injective. So I have already proved that if $f$ ...
0
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3answers
33 views

Prove any ring homomorphism between $M_2(\mathbb{R})$ and $\mathbb{R}$ is trivial

Prove any ring homomorphism between $M_2(\mathbb{R})$ ($2\times 2$ matrices) and $\mathbb{R}$ is trivial. I am not looking for answers, I just want to know how to approach these types of problems (...
3
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1answer
57 views

What is p-adic logarithmic map of an elliptic curve? How to compute it?

I was reading about elliptic curves in this pdf. Page 55 of the pdf states that if number of points on elliptic curve #$E(F_p) = p$, then there exists a p-adic logarithmic map that homomorphically ...
2
votes
1answer
34 views

Describing $Hom(\mathbb{Z}^2, G)$ as a subset of $G \times G$

I am asked to describe $Hom(\mathbb{Z}^2, G)$ as a subset of $G \times G$. I interpret this as describing a relationship between the two (i.e. showing they are isomorphic) G is finite, not ...
0
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2answers
45 views

Proving a ring-homomorphism using a group-homomorphism

Let f : R → R' be a group homomorphism. Show that the induced map φ : R[x] → R'[x], where φ(anxn + . . . + a0) = f(an)xn + . . . + f(a0), is a ring homomorphism. I know that &#...
0
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1answer
38 views

$Z(G)=\left \langle e \right \rangle$ if and only if $N_\Gamma (\operatorname{Im}(\varphi))=\operatorname{Im}(\varphi)$ [closed]

Let $G$ a group and let a group $\Gamma=G\times G$ with $\forall (a,b),(c,d)\in\Gamma, (a,b)(c,d):=(ac,bd)$. Let a map $\varphi:G\to\Gamma$ by $\varphi(g)=(g,g)$. Prove that $Z(G)=\left \langle e \...
1
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1answer
25 views

The group homomorphism $z \mapsto (z/|z|)^2$ with an application of the homomorphism theorem.

Question Let $G = (\mathbb{C}\setminus\{0\}, \cdot)$. 1) Show that $N = (\mathbb{R}\setminus \{0\},\cdot)$ is a normal subgroup. 2) Show that $f:G \rightarrow G$ with $z \mapsto (z/|z|)^2$ is a ...
0
votes
1answer
30 views

Let H be a subgroup of finite group G. G acts on G/H by left multiplication. This induces a homomorphism. Show that its kernel is in H

Let G be a finite group and H is a subgroup of G. We have G acts on the set of left co-sets of H (G/H) by left multiplication x(gH)=xgH. This action induces a homomorphism from G to perm(G/H). Show ...
1
vote
2answers
54 views

Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H \times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$.

Suppose $H$ and $K$ are normal subgroups of $G$. Prove that $G/H \times G/K$ has a subgroup that is isomorphic to $G / (H∩K)$. Also prove that if $G = HK$, then $G/(H∩K)$ is isomorphic to $G/H \...
1
vote
0answers
46 views

Isomorphism $\{1\}\times \mathbb C^*\to \{1\}\times \mathbb C^*$

I am reading a text where I have trouble understanding an argument: Let $f: \mathbb C^*\times \mathbb C^*\to \mathbb C^*\times \mathbb C^*$ an isomorphism, such that $f(\{1\}\times \mathbb C^*)= \{1\}...
1
vote
1answer
38 views

If there exists an epimorphism from a group $G_1$ to $G_2$ that is not one-to-one, then can $G_1$ and $G_2$ be isomorphic? [duplicate]

I am trying to show that two groups are isomorphic only if a certain condition holds. I can show that a specific epimorphism between the two groups is one-to-one only if this condition holds, but it ...
-2
votes
1answer
82 views

Isomorphism on a torsion group - automorphism or endomorphism?

Let $f:G\to G$ be a surjection from a torsion group $G$ onto itself. Let the kernel have infinite cardinality: $\lvert\ker(f)\rvert=\aleph_0$ What category of function on groups is this? To my ...
0
votes
1answer
26 views

Find two distinct group homomorphisms between $(U_{11}, *)$ and $(\mathbb{Z}_{10}, +)$

Find two distinct group homomorphisms between $(U_{11}, *)$ and $(\mathbb{Z}_{10}, +)$, where $U_{11} =$ set of units in $\mathbb{Z}_{11}$. My observations: We want to create a function $f: (U_{11}, *...
0
votes
1answer
24 views

the group homomorphism $f:G\rightarrow H$ is exactly then an injection when $ker(f)=eG$ [duplicate]

I have to show that a group homomorphism $f:G\rightarrow H$ is a injective function when $ker(f):=[u\in G:f(u)=eH]=eG$,$\:e$ is the neutral element. Attempt: If we take $ker(f)=\{a\}$ $f(a\#b)=f(a)*...
1
vote
0answers
41 views

How to find a homomorphic map in following question?

Let $S1$ and $S2$ be two sets. Suppose that there exists a one-to-one mapping $J$ of $S1$ into $S2$ . Show that there exists an isomorphism of $A(S1)$ into $A(S2)$, where $A(S)$ means the set of ...