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).

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Let $\phi : G\to G'$ be a homomorphism and let $S'\subseteq G'$. Is $\phi^{-1}(\langle S'\rangle ) = \langle \phi^{-1}(S')\rangle$?

I've shown that $\phi^{-1}(\langle S'\rangle ) \supseteq \langle \phi^{-1}(S')\rangle$ and was wondering whether the other inclusion holds, although I've not been able to prove it nor find a ...
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1 vote
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A link between multiplicative function on endomorphisms and their determinants

Take $E$ a $K$-vector space with $K$ a field with characteristic $0$. Then take $$ F:\operatorname{End}_K(E) \to K $$ a map such that $$F(1)=1$$ and $$ F(\phi\circ\psi) = F(\phi)F(\psi) $$ Then there ...
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7 votes
1 answer
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Is $\text{GL}_2(\mathbb{R})/\mathbb{R}^{\times}$ isomorphic to $\text{SL}_2(\mathbb{R})$?

Let $\text{GL}_2(\mathbb{R})$ be the set of real $2\times 2$ invertible matrices (where the operation is matrix multiplication). It has $\text{SL}_2(\mathbb{R})=\{ A \in \text{GL}_2(\mathbb{R}| \det A ...
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-1 votes
1 answer
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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 ...
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-4 votes
0 answers
33 views

If $f$ is an injective group homomorphism from $G\to G$ where $G$ is a finite abelian group, then $G$ has odd order. [closed]

If f is an injective group homomorphism defined as f(x)=x² from G to G where G is a finite abelian group then G has odd order.How do we prove it?
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1 vote
0 answers
20 views

When two semilinear morphisms are said to be equal? [closed]

Suppose $(S,A)$ and $(T,B)$ are two left semigroup acts. A pair of mappings $(\mu,f):(S,A) \to (T,B)$ is called a semilinear morphism if $\mu$ is a semigroup homomorphism and $f$ is a function ...
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0 votes
0 answers
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Image of finitely generated group in an injective group homomorphism

Suppose I have an injective homomorphism $\varphi:F_n\to F_m$ between free groups, and suppose $G = F_n/\langle r_1, \dots, r_s\rangle$ is some finitely generated group with relations $r_i$. Is it ...
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3 votes
1 answer
77 views

Does there exist a surjective group homomorphism $\varphi:A_4\rightarrow\mathbb{Z}/4\mathbb{Z}$?

Does there exist a surjective group homomorphism $\varphi:A_4\rightarrow\mathbb{Z}/4\mathbb{Z}$? Edit: I have narrowed it down to the problem of whether $A_4$ has a normal subgroup of order 3 with 3-...
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1 vote
0 answers
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Homomorphism between $\mathbb{Z}/m$ and $\mathbb{Z}/n$

The lecture notes I am working through assert, but leave as an exercise, that if $n\mid m$, then the map $f:\mathbb{Z}/m\to \mathbb{Z}/n$ sending$$x\mapsto x\pmod n$$is a surjective homomorphism. My ...
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1 vote
0 answers
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A surjective morphism of groups takes finitely generated subgroups into finitely generated subgroups

Let $f:G\to H$ be a surjective morphism of groups and $f^{*}:[\textrm{ker}(f), G]\to [\{e_{H}\},H]$ defined by $f^{*}(L)=f(L)$. Show that $f^{*}$ takes finitely generated subgroups of $G$ into ...
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1 vote
0 answers
61 views

Prove that there is no homomorphism from $S_5$ onto a group of order $24$.

Prove that there is no homomorphism from $S_5$ onto a group of order $24$. My solution: Let $G$ be a group such that $|G|=24.$ Denote $\phi: S_5\to G$. The normal subgroups of $S_5$ are $S_5,A_5,\{e\}$...
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0 votes
0 answers
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Prove that $D$ is a field. [duplicate]

Here is the question I want to answer: Let $F$ be a field and $D$ an integral domain which is a finite dimensional vector space over $F.$ Prove that $D$ is also a field. Here are my thoughts: Since $F$...
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0 answers
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For the group $G$ below, prove that ${\rm sgn}:G\to H=\Bbb Z/2\Bbb Z$, mapping $e,d,f$ to $[0]$ and $a,b,c$ to [1], is a group homomorphism

Using the notation of the $G$ group below, prove that the map $\operatorname{sgn}: G \rightarrow H=\mathbb{Z} / 2 \mathbb{Z}$, mapping $e, d, f$ to $[0]$ and $a, b, c$ to [1], is a group homomorphism $...
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3 votes
1 answer
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Prove that $G\cong H\times K$ if and only if there are homomorphisms...

I have some questions for the following following exercise which came from Hungerford's undergraduate Abstract algebra An introduction 3rd edition text in chapter 9, section 1. Let $G$ be an additive ...
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3 votes
4 answers
137 views

Prove $f$ injective implies $o(f(g)) = o(g)$.

Suppose $G_1,G_2$ are groups and $f:G_1 \to G_2$ is a homomorphism. Prove $f$ injective implies $o(f(g)) = o(g) \space\forall g\in G$. Suppose $g\in G , g'\in G$ such that $o(g)=n, f(g)=g'$. Then $$\...
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  • 1,561
0 votes
1 answer
108 views

$H \lhd G$, $\pi:G \to H$ is a group homomorphism with $\pi(h)=h$ show that $G \cong H × G/H$

Let $H$ be a normal subgroup of a group $G$ such that there is a group homomorphism $\pi:G \to H$ with $\pi(h)=h$ for all $h \in H$. Prove that $G$ is isomorphic to $H × G/H$ Don't know how to solve ...
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0 votes
1 answer
53 views

left coset of Kernel and Image relationship.

if we have Groups G, and H, and homomorphism F between them and left cosets g*Ker(F) Why is it that "there is one left coset [g*Ker(F)] for each element of Im(F)" (This is part of the answer ...
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0 votes
1 answer
98 views

Existence of an element in a group $G$ [closed]

Let $G$ be a finite group and $\phi$ is an automorphism of $G$ such that $\phi(x)=x$ if and only if $x=e,$ where $e$ is the identity element of $G.$ Show that for every $a\in G$, there exists $x\in G$ ...
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0 votes
0 answers
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Subspace and Dimension of a Homomorphism/Linear Mapping

I am very confused about the following exercise: Let $V, W$ be vector spaces over a field $F$. Show that $Hom_F(V, W)$ is a vector subspace of the set of all mappings $Maps(V, W)$ from $V$ to $W$. It ...
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0 votes
1 answer
21 views

Abelian group with surjective pushfoward maps and decompsotion of free abelian group

Let $A,B,C$ be abelian groups. $A$ has the property that for any homomorphism $\alpha:A\to B$ and surjective homomorphism $\phi:C\to B$ there exists a homomorphism $\beta:A\to C$ such that $\alpha = \...
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1 vote
1 answer
38 views

Name for equivalent mapping from a function in $\mathbb{R}$ to $\mathbb{R}^2$

Suppose there could be a function $f: \mathbb{R} \rightarrow \mathbb{R}$, for instance $f(x) = x^2$, and an equivalent set of points in $\mathbb{R}^2$ that represent the same $(x,y)$ points as this ...
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1 vote
1 answer
38 views

Exactness of sequence induced by $\operatorname{Hom}(G,\cdot)$.

$\newcommand{\Hom}{\operatorname{Hom}}$ For this problem, we let $0\rightarrow A\xrightarrow{i} B \xrightarrow{j}C\rightarrow 0$ be a short exact sequence of groups, and $G$ an abelian group. It's not ...
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1 vote
1 answer
85 views

How do we find the homorphism from $\mathbb{Z_2} \to{\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$ [closed]

How do we find the homorphism from $\mathbb{Z_2} \to {\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})?$ I know that ${\rm Aut}(\mathbb{Z_3} \times \mathbb{Z_3})$ is isomorphic to $GL_2(\mathbb{Z_3})$. We ...
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1 vote
1 answer
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Is there a more general way to prove homomorphism between two algebraic objects?

Seems all the proofs I saw are by construction, what if the construction is so hard that one can not possibly construct it by hand. Is it possible to prove the homomorphism without having to construct ...
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0 votes
1 answer
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$p,q$ primes, $p\mid q-1$. Weaker assumption in the proof of the existence of non-trivial $C_p\ltimes C_q$?

Motivated by the fact that the non-existence of non-trivial $C_p\ltimes C_q$, for $p\nmid q-1$, can be proven without any piece of information on the structure of $\operatorname{Aut}(C_q)$, not even ...
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  • 1,011
0 votes
1 answer
38 views

Why is $f(1)=0$ on $f:(\mathbb{Q},+) \rightarrow (\mathbb{Z},+)$, where $f$ is a homomorphism?

Why is $f(1)=0$ on $f:(\mathbb{Q},+) \rightarrow (\mathbb{Z},+)$, where $f$ is a homomorphism? My professor stated that $f(1)=0$ in the such case because: $$f(1)=f\left(\frac{1}{n}+...+\frac{1}{n}\...
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-1 votes
1 answer
66 views

Show that the function is homomorphism with kernel φ

If $H_1,\dots, H_s$ are normal subgroups of $G$, show that the function: $$\varphi: G\to G/H_1 \times\dots\times G/H_s$$ given by: $$\varphi(g) = (gH_1,\dots, gH_s)$$ is a homomorphism with kernel: $$\...
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-6 votes
1 answer
55 views

Let $G$ be a group and $H\le G$. If there exists a homomorphism $f:G\to H$ such that $f(h)=h$ for all $h\in H$, then is $H$ normal in $G$? [closed]

Let $G$ be a group and $H$ be a subgroup of $G$. If there exists a homomorphism $f:G\to H$ such that $f(h)=h$ for all $h\in H$, then is $H$ normal?
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5 votes
1 answer
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Homomorphisms from $S_3$ to $\mathbb{Z}/10\mathbb{Z}$

I want to check if my line of thought is correct. We need to find all homomorphisms $\phi: G=S_3\rightarrow H=\mathbb{Z}/10\mathbb{Z}$. We already know that $\phi(g)=\bar{0}$ for all $g\in G$ is a ...
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3 votes
0 answers
52 views

Are there two non-isomorphic finitely presented groups $G_1$ and $G_2$ so that $G_1$ is an r-image of $G_2$ and $G_2$ is an r-image of $G_1$?

By Are there two non-isomorphic groups $G_1$ and $G_2$ so that $G_1$ is an r-image of $G_2$ and $G_2$ is an r-image of $G_1$? there are two non-isomorphic groups which are r-images of each other. I ...
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-1 votes
2 answers
66 views

Constructing a Surjective Homomorphism to Satisfy The First Isomorphism Theorem [closed]

Let $H=\langle (1,2)\rangle$ be a subgroup of $\mathbb{Z}_4\times\mathbb{Z}_8$ and $\left|(\mathbb{Z}_4\times\mathbb{Z}_8)/H\right|=32/4=8$. By the First Isomorphism Theorem, $(\mathbb{Z}_4\times\...
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-5 votes
2 answers
84 views

Yes/No: There always exists an injective homomorphism from $G$ into $S_n$. [closed]

Let $G$ be a finite group of order $n\ge2$. Is the following statements true/false? There always exists an injective homomorphism from $G$ into $S_n$. My attempt: I found the answer here. I think ...
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  • 183
4 votes
0 answers
63 views

Proving isomorphism between two vector spaces of homomorphisms

Let $(\rho,V)$ be a representation of $G$, and we consider the following action of $G\times G$ on $\hom(V,V)$:$$\forall(g,h)\in G\times G:(g,h).\phi=\rho_g\circ\phi\circ\rho_{h^-1}$$ Find an ...
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3 votes
1 answer
156 views

Every action gives rise to a homomorphism

I'm reading Chapter 3 from the book "The theory of finite groups" by Kurzweil and Stellmacher where they say the following: Let $G$ act on a set $\Omega$. That is, for each $x\in G$ and $\...
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  • 549
3 votes
1 answer
41 views

For any finite group $H$ and homomorphism $\alpha:BS(2,3)\to H$, prove $\alpha([bab^{-1},a])=1$

$BS(2,3)=\langle a,b \mid ba^2b^{-1}=a^3 \rangle$. Let $H$ be a finite group and $\alpha:BS(2,3)\to H$ a homomorphism. Let $g=[bab^{-1},a]$. Prove $\alpha(g)=1$ My Attempt $$\begin{align} \alpha(g)&...
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-2 votes
1 answer
41 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 ...
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0 votes
1 answer
56 views

Is $\ker\varphi^2=(\ker\varphi)(\mathop{\rm im}\varphi\cap\ker\varphi^2)$ always true for group endomorphism $\varphi$?

Is $\ker\varphi^2=(\ker\varphi)(\mathop{\rm im}\varphi\cap\ker\varphi^2)$ always true for group endomorphism $\varphi$? It is trivial that $\ker \varphi^2 \supseteq (\ker \varphi) ( \mathop{\rm im}\...
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  • 3
2 votes
1 answer
36 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 ...
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  • 171
-2 votes
1 answer
65 views

Understanding homomorphism and non-homomorphism from $A_4$ to $S_6$

Could anyone help me how to find an example of a function $f: A_4\to S_6$ that is not a homomorphism and one $f$ that is a homomorphism? And is $A_4\times S_6$ isomorphic to $S_6\times A_4$? Thanks so ...
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1 vote
2 answers
69 views

How to construct an injective homomorphism from $GL(2,\mathbb{Z}_2)$ into $S_4$?

I am struggling to devise an injective homomorphism from $GL(2, \mathbb{Z}_2)$ to $S_4$, in particular construction which actually allows me to verify whether its a homomorphism or not. Eg, we could ...
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1 vote
1 answer
48 views

What’s going on when I restrict a homomorphism to a smaller quotient group?

I have $N\lhd G$, $K\lhd G$ and $N\le K$. Let $f:G\rightarrow H$ be a homomorphism with kernel $K$. I want to write $f$ as a composition $h\circ\pi$ where $\pi:G\rightarrow G/N$ is the projection and $...
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2 votes
2 answers
63 views

Existence of a canonical isomorphism for ($\operatorname{Hom}$-$\otimes$ adjunction).

Here is the question I want to answer: Let $U,V, W$ be finite dimensional vector spaces over the field $F.$ Prove that there is a canonical isomorphism $$\operatorname{Hom}(U \otimes V, W) = \...
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  • 1,163
0 votes
0 answers
46 views

What is the kernel of this homomorphism?

Let $$q : \mathbb{C}[a_1,a_2,a_3,a_4] \rightarrow \mathbb{C}[u,v]$$ $$a_1\rightarrow u+3v,a_2\rightarrow 3uv+v^2, a_3\rightarrow v^2(3u+v), a_4\rightarrow uv^3$$ be a ring map. My goal is to ...
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  • 168
1 vote
1 answer
30 views

Draw a covering of $S^1 \vee S^1$ whose fundamental group is isomorphic to $\ker \Phi : F_2 \to \mathbb Z$ with $a\mapsto 2, b\mapsto 3$

$\newcommand{\Z}{\mathbb Z}$ Let $K\leq F_2 = \langle a,b\rangle$ be the kernel of the map $\Phi: F_2 \to \Z$ sending $a$ to $2$ and $b$ to $3$. Draw a cover of $S^1 \vee S^1$ whose fundamental group ...
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1 vote
2 answers
105 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 ...
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  • 409
2 votes
1 answer
68 views

Help understanding the use of the First Isomorphism Theorem with regards to the special linear group

I'm reading a textbook (Contemporary Abstract Algebra, 9th Edition, Joseph A. Gallian), and it gave the following example: Recall that $SL(2,\mathbb{R})=\{A\in GL(2,\mathbb{R}\ |\det A=1)\}$ and let $...
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  • 1,137
3 votes
2 answers
174 views

A conditioned morphism in a group

Let $(G,+)$ be an abelian group of at least $3$ elements and $f:G \to G$ a homomorphism such that $f(x) \in \{0, x, -x\}$ for all $x \in G$. Show that $f \in \{0_G, 1_G, -1_G\}$. I tried proving that $...
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0 votes
1 answer
54 views

Is there an element $f$ of $\operatorname{End}(V)$ which is not an element of $\operatorname{End}_K(V)$?

I am reading "An Introduction to Algebraic Systems (in Japanese) by Kazuo Matsuzaka. Let $V$ be a vector space over a field $K$. Let $\operatorname{End}_K(V)$ be the set of all linear mappings ...
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  • 6,149
3 votes
1 answer
42 views

Multiset image of a set under a homomorphism

I know the basic group theory definition of the image of a subset of group elements under a homomorphism is the following: The image of any subset $X \subseteq G$ is given by $\phi(X) = \{\phi(x) : x ...
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4 votes
1 answer
50 views

Finding the number of the group homomorphisms $G\to S_4$ ($|G|=6$) by group actions.

A homomorphism from a group $G$ of order $6$ to $S_4$ is equivalent to an action of $G$ on the set $X=\{1,2,3,4\}$. By the orbit-stabilizer theorem, every orbit must have size either $1$, or $2$, or $...
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