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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

Determine whether there is an onto homomorphism from $(\mathbb{Z}_6,+)$ to $(\mathrm{Z}_3,+)$

Question: We have to determine if there exists a homomorphism from $(\mathbb{Z}_6,+)$ onto $(\mathrm{Z}_3,+)$. My efforts: Let $\phi$ be an onto homomorphism. Since $\phi$ is surjective, then by ...
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2answers
28 views

Internal direct sum of kernel of surjective homomorphism and cyclic subgroup

I'm studying for a qualifying exam in algebra, and my abstract algebra skills are quite rusty. I'm attempting to solve the following problem: Suppose that $\Phi:G\rightarrow\mathbb{Z}$ is a ...
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2answers
59 views

Does there exist an onto homomorphism from $(\mathbb{Z}_6,+)$ to $(\mathbb{Z}_4,+)$ and why?

We have to determine whether there exists an onto homomorphism from $(\mathbb{Z}_6,+)$ to $(\mathbb{Z}_4,+)$. To do so, let us consider a homomorphism $\phi:\mathbb{Z}_6\to \mathbb{Z}_4$. Then $\...
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1answer
42 views

Characterization of anti-homomorphisms

Let $G$ be a group and $G^{op}$ denotes its opposite group. It is well-known that the functor $F$ from $Grp$ to itself, defined by $$ \begin{aligned} G&\mapsto G^{op}\\ x&\mapsto x^{-1}\\ \...
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2answers
55 views

let $N_1$, $N_2$ be normal subgroups of group G. Define $f : G → (G/N_1) × (G/N_2)$ by $f(a) = (aN_1, aN_2)$. show f is a homomorfism

Let G be a group, and let N_1, N_2 be normal subgroups of G. Define $f : G → (G/N_1) × (G/N_2)$ by $f(a) = (aN_1, aN_2)$. (a) Prove that f is a group homomorphism with kernel N1 ∩ N2. (b) Prove that ...
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36 views

For a group morphism $G \rightarrow H$, give the image of the generators.

I have two group tables, both with order $24$. Now I have to write the images of the first generator as the second generator. The group elements are all letters, thus $G=\{A,B,C,...,V,W,X\}$ and the ...
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0answers
24 views

Prove that there is an isomorphism $\phi_n:H_n(C_*)\to\bigoplus_{\alpha\in\Lambda}H_n(C_*^{\alpha}) $

Let $\Lambda $ be a fixed set. For each $\alpha\in\Lambda $ is $\{C_n^{\alpha}\}_{n\in\mathbb{Z}}$ a complex of chains with homomorphism border $\partial^{\alpha}$. Prove that there is an isomorphism ...
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0answers
21 views

Prove that $\{C_n\}_{n\in\mathbb{Z}}$ is a chain complex with homomorphism border $\partial:=\bigoplus_{\alpha\in\Lambda}\partial^{\alpha} $.

Let $\Lambda $ be a fixed set. For each $\alpha\in\Lambda $ is $\{C_n^{\alpha}\}_{n\in\mathbb{Z}}$ a complex of chains with homomorphism border $\partial^{\alpha}$. For each $n\in\mathbb{Z}$ we ...
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17 views

homomorphism determined modulo some other morphism

What does it mean for homomorphism $f:A\to B$ of abelian groups to be determined modulo an arbitrary homomorphism $g:A''\to B''$ of abelian groups? Edit: We have homomorphisms $B''\to B$ and $A\to A''...
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1answer
43 views

Kernel, Sets and Logic

Given the groups G, H the kernel of a homomorphism $f : G \rightarrow H$ is defined as : $\{$ $g \in G$ : $f(g)=e_H$ $\}$. I was wondering, is there a way to express the kernel in terms of ...
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1answer
25 views

Equivariant homomorphism in a non-abelian group of order $pq$, where $p$ and $q$ are distinct primes.

Let $G$ be a group and $H$ be a normal subgroup of $G$. A homomorphism $\phi$ from $H$ to $G$ is said to be $G$-equivariant if $\phi (ghg^{-1})=g \phi (h)g^{-1}$, for all $g \in G$ and $h \in H$. I ...
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30 views

How to construct a homomorphism from an abelian group to $\mathbb{Q}$ [duplicate]

For any abelian group $A$. Is it possible to construct a nontrivial homomorphism from $A \to \mathbb{Q}$? Is it possible if $A$ is a finite generate abelian group?
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1answer
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Confusion on Artin's Theorem (Linear Independence of Group Homomorphism)

The Artin's Theorem states as follows: Let $G$ be a group. and let $f_1,\dots, f_n\colon G\to K^{\times}$ be distinct homomorphisms of $G$ into the multiplicative group of a field. Prove that these ...
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Why is there a one-to-one correspondence between homomorphic images of a group $G$ and normal subgroups of $G$?

I was reading about group theory in Herstein's book (mentioned below) and I came across a couple of propositions that were not clear to me, in the sense that I couldn't quite figure out why they were ...
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2answers
63 views

Why not $\mathbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}\cong \mathbb{Q}^2$ not isomorphic as additive groups?

We know that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as additive groups(as they are isomorphic as $\mathbb{Q}$ vector spaces) under the axiom of choice. But why not $(\mathbb{Q}, +)$ and $(\...
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0answers
34 views

Determinant is a morphism from $PGL_2(\Bbb F_p)$ to $(\Bbb F_p^*/\Bbb F_p^{*2})$

Prove that determinant is a morphism from $PGL_2(\Bbb F_p)$ to $(\Bbb F_p^*/\Bbb F_p^{*2})$ where $PGL_2(\Bbb F_p) = GL_2(\Bbb F_p)/\Bbb F_p^*.I_2$. What tools do we have from group theory to tackle ...
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22 views

Proving that the Galois group of the cyclotomic field $\mathbb{Q}(\zeta_n)$ is isomorphic to $U(\mathbb{Z}/n\mathbb{Z})$ [duplicate]

How to prove that the Galois group of the cyclotomic field $\mathbb{Q}(\zeta_n)$ is isomorphic to $U(\mathbb{Z}/n\mathbb{Z})$? The isomorphism is given by $U(\mathbb{Z}/n\mathbb{Z}){\rightarrow} Gal(\...
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1answer
29 views

Lie algebra - how to calculate dim of Hom(M,M)

I'm studying Lie algebra in English and it's not my language.. I'm trying to read about it more but there're lot of things I don't understand. I will be happy if someone know how to do this question ...
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1answer
45 views

Isomorphism between $\mathbb{Z}/8\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ and a subgroup [closed]

I have been trying to find an isomorphism between $\mathbb{Z}/8\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ and a subgroup of the symmetric group, may someone please help me? I don't know what that ...
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1answer
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The set of bilinear forms is a right $(R \otimes R)$-module

Let $V$ and $A$ be abelian groups. An $A$-valued bilinear form on $V$ is a $\mathbb{Z}$-module homomorphism $$\beta : V \otimes_{\mathbb{Z}} V \rightarrow A$$ Now, let $V$ be a left $R$-module, where $...
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1answer
30 views

Determine whether the statement is true: “If $g\in G$ has finite order $|g|=m$ then $f(g)$ has order $m$ in $H$.”

Determine whether the following statement is true or false: "Suppose $f : \{G, *\} \mapsto \{H, \circ \}$ is a homomorphism of groups, and let $f(G) = \{f(g)~|~g\in G\}$. If $g\in G$ has finite order $...
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2answers
59 views

Quotient group of dihedral group

Let $G=\{e,r^{2},...,r^{8},s,sr,...,sr^{8}\}$ and let $N=\langle r^{3} \rangle.$ Now let $\pi(g)=\bar{g}=gN$ be surjective with kernel $N$. I have to show that $G/N=\{\bar{e},\bar{r},\bar{r^{2}},\...
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0answers
74 views

The coset corresponding to permutation $(123)$ in $\Bbb Z /3\Bbb Z$.

We know that $V_4 \triangleleft A_4$ so $ A_4/V_4 \cong \Bbb Z /3\Bbb Z$. The coset corresponding to permutation $(123)$ is $(123)V_4$. Is it corresponding to $\overline{1}$ or $\overline{2}$ in $\...
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0answers
34 views

Homomorphism between Conjugacy classes and the Centralizer

Let $G$ be a group and $\sigma \in G$ some element. I want to prove that there is a injuctive homomorphism between the left cosets of the centralizer $C_G(\sigma)$ and between the Conjugacy classes of ...
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1answer
28 views

Prove that there is a unique homomorphism

$G, H, K$ are groups and let $\phi: G\to H, \psi: G\to K$ be homomorphisms such that $\phi$ is onto and $\ker\phi \subset \ker\psi$. Prove that there is a unique homomorphism $\alpha: H\to K$ such ...
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1answer
42 views

Find the Kernel of $\Psi : (\Bbb{Z}_{30},+_{30})\rightarrow (\Bbb{Z}_{20},+_{20})$

Find the Kernel of $\Psi : (\Bbb{Z}_{30},+_{30})\rightarrow (\Bbb{Z}_{20},+_{20})$ Provided that $\Psi([b])=[4b]$ My understanding of the kernel is that it should be: $\ker(\Psi)=\{[b]\in\Bbb{Z}_{...
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1answer
43 views

Is there any injective homomorphism (i.e. monomorphism) from a non-cyclic group of order $4$ to $\mathbb{Z}_8$?

The only such possible group is $V$ (up to isomorphism). If $\phi$ be such an into homomorphism, then $o(\phi(V))=4$ and $\phi(V)$ being a subgroup of $\mathbb{Z}_8$, it must be cyclic with a ...
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1answer
50 views

Are projection maps homomorphisms?

I have the following problem: Let $(G, +)$ and $(H, *)$ be groups. Proof that the projection maps $\pi_{1}: G×H\to G$ and $\pi_{2}: G×H\to H$ are homomorphisms. My attempt: We know by definition, ...
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0answers
41 views

If $(m, n) = 1$, then $\dfrac{\mathbb Z \times \mathbb Z}{\langle (m,n)\rangle}\cong\mathbb Z$

How to show that If $(m, n) = 1$, then $\dfrac{\mathbb Z \times \mathbb Z}{\langle (m,n)\rangle}\cong\mathbb Z$. (Note: We consider this in group theory.) I know that $(m, n) = 1$ means that ...
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1answer
115 views

Confusion about a ring homomorphism theorem in textbook Analysis I by Amann/Escher

I am reading Section 7: Groups and Homomorphisms, Chapter 1: Foundation, textbook Analysis I by Herbert Amann and Joachim Escher. First of all, I am so sorry for posting many screenshots. Since the ...
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2answers
37 views

Help with what a homomorphism means in $\Phi: (\Bbb{Z}_{18},+_{18})\rightarrow(\Bbb{Z}_{12},+_{12})$

Let $\Phi: (\Bbb{Z}_{18},+_{18})\rightarrow(\Bbb{Z}_{12},+_{12})$ be a homomorphism such that $\Phi([1])=[a]$ Determine values of $a$ for which such a homomorphism exists. I understand what a ...
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1answer
18 views

Help finding the Kernel of $\Psi (x_1,x_2,x_3)=(x_1+x_3,0,x_2+x_3)$

Consider the real vector space $\mathbb R^3$ and define $\Psi:\mathbb R^3\rightarrow\mathbb R^3$ by $$\Psi(x_1,x_2,x_3)=(x_1+x_3,0,x_2+x_3)$$ How do I got about calculating what the kernel of $\Psi$ ...
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2answers
61 views

If $M$ is a maximal subgroup and $f$ a surjective homomorphism then $f(M) = H$

Let $G$ be a group and let $M \le G$ be a maximal subgroup of $G$. Let $f:G \to H$ be a surjective homomorphism such that $M$ doesn't contain $kerf$. The task is to prove that $f(M) = H$. My attempt:...
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1answer
97 views

$\ker(\phi)$ is a normal subgroup.

Let $G_1$ and $G_2$ be groups and suppose $\phi: G_1\mapsto G_2$ is a homomorphism. Then $\ker (\phi)\unlhd G_1$. Need some feedback and help proving this. I am still new to proofs, but here's my ...
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3answers
63 views

An application of the first (group) isomorphism theorem

Suppose that $G$ is a group and $N$ is a normal subgroup of $G$. How do we find out what $G/N$ is isomorphic to? For example, $C_m$ is a normal subgroup of $D_{2n}$ generated by a rotation of angle $...
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1answer
34 views

If $H$ is not normal subgroup of $G$ then there are two left cosets of $H$ which multiplication is not a left coset of $H$

Let $G$ be a group and $H$ be a not normal subgroup of $G$. I need to prove that there are two left cosets of $H$: $g_1H,g_2H$ such that $g_1Hg_2H$ is not a left coset of $H$. I tried to assume the ...
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1answer
33 views

Homomorphism and image

I'll consider the function $\phi: \mathbb Z^3 \to \mathbb Z^3$, given by: $\phi(x,y,z) := (x+5y+3z,2y,7z)$ I have a couple of questions associated with this: 1) This might be a very dumb question, ...
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1answer
44 views

Homomorphism $\phi: G \rightarrow H$, $\phi$ surjective, $\exists a \in H : |a| = 5$, show $\exists x \in G : |x| = 5$

Homomorphism $\phi: G \rightarrow H$, $\phi$ surjective, $\exists a \in H : |a| = 5$, show $\exists x \in G : |x| = 5$, where $|G|$ is finite. I'm not sure if this proof is correct, but here's what I ...
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2answers
31 views

Homomorphism $\phi : (\mathbb{Z} \oplus \mathbb{Z}) \rightarrow (G, +)$ via $(3,2) \mapsto x$ and $(2,1) \mapsto y$. Find $\phi((4,4))$

$\phi : (\mathbb{Z} \oplus \mathbb{Z}) \rightarrow (G, +)$ via $(3,2) \mapsto x$ and $(2,1) \mapsto y$. Find $\phi((4,4))$ where $\phi$ is a homomorphism Since we know $\phi$ is a homomorphism: $\...
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1answer
37 views

Does the function $ \{ x \in G_1 \ : \ x^k = a\} \rightarrow \{ y \in G_2 \ : \ y^k = \phi(a) \}$ have a name?

So I'm stuck on a problem and I'd like to do more research. But I'm afraid I don't know what to even type into Google to start. The problem is this: Let $\phi: G_1 \rightarrow G_2$ be a ...
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1answer
28 views

what is $\sum_{n \in \mathbb{Z}/p \mathbb{Z}^* } \chi(n)$?

Given a no trivial homomorphism $\chi$ from $\mathbb{Z}/p \mathbb{Z}^*$ to $\mathbb{C}^*$, what is $\sum_{n \in \mathbb{Z}/p \mathbb{Z}^* } \chi(n)$? is it $0$? why?
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0answers
20 views

Image of a subgroup is subgroup under homomorphism.

Image of a subgroup under group homomorphism is a subgroup of "codomain." Image of a normal subgroup under group homomorphism is a normal subgroup of "range." Image of a subring under ring ...
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0answers
45 views

If $H$ is a subgroup of $G$ and $a \in G$. Show that $H$ and $a^{-1}Ha$ are isomorphic.

I feel like I did something wrong, because I never actually used $a \in G$. Proof: Define $\alpha: H \rightarrow a^{-1}Ha$ by $\alpha(h)=a^{-1}ha$ Let $x,y \in H$ then, $\alpha(xy) = a^{-1}xya\\ =...
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2answers
40 views

Similarities between the idea of Group Homomorphisms and Linear Transformations

I am not sure if this question has been asked before, but my search did not return me any answers. While reading several online notes that attempt to give an intuitive understanding of group ...
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2answers
25 views

Generators of the image of a homomorphism

Let $G = \langle x_1, ... , x_n \rangle$, a finitely generated group. Given $\rho : G \rightarrow GL_{n}(\mathbb{C})$, a homomorphism, we know that $H = im(\rho)$ is a subgroup of $GL_{n}(\mathbb{C})$...
4
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1answer
61 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
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1answer
51 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
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1answer
53 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
88 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
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1answer
34 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 &\...