2
votes
2answers
48 views

Homomorphism $f: \mathbb{C}^{*}\rightarrow \mathbb{R}^{+}$. Prove that kernel of f is infinite group.

First of all we need to prove that $\ker(f)$ is group by proving: That $\ker(f)$ contains $e\in\mathbb{C}^*$, That $\ker(f)$ is closed under multiplication for every $a,b \in \ker(f)$ That ...
0
votes
0answers
22 views

Endomorphisms of Groups - Book Recommendation

Which books dealing with group theory have considerable material on endomorphisms? The books I have seen usually have something on homomorphisms, isomorphisms, and automorphisms, but very little on ...
0
votes
2answers
29 views

Automorphism problem

For arbitrary group $(G,\cdot)$ let $Aut(G) = $ {$f: G \to G | f $ is a isomorphism} be set of all automorphisms of group $G$. We assume that $(Aut(G),\circ)$ where $\circ$ is addition of mappings is ...
2
votes
3answers
132 views

Group Homomorphism Questions (my attempts shown)

(a) Let $p$ be a prime. Determine the number of homomorphisms from $\Bbb Z_p \oplus \Bbb Z_p$ into $\Bbb Z_p$. Attempt: Suppose $\Psi:Z_p \oplus Z_p \rightarrow Z_p$ is an into homomorphism. ...
3
votes
1answer
34 views

a step in a proof in Samuel's Algebraic number theory

In the proof of Dirichlet's unit theorem, in Algebraic number theory by Samuel, there is a step in the proof that i don't understand. (p.73 in the french edition). He first introduces the logarithmic ...
1
vote
1answer
32 views

isomorphism between group and product of kernel by image [duplicate]

If $\phi$ is a morphism between groups $G$ and $H$, is $G$ isomorphic to $$ker(\phi)\times im(\phi)$$ ? Why ? Thanks.
1
vote
1answer
33 views

cokernel of a homomorphism of groups

Given injective homomorphisms of finitely generated abelian groups $\phi_i \colon G_i\rightarrow G$ and normal subgroups $N_i\subset G_i$ for $i\in\{1,\dots,r\}$ what is the cokernel of the ...
2
votes
1answer
54 views

Does an injective homomorphism always exists from $G$ into $GL_n(R)$ where order of $G$ is $n$?

We have a group $G$ of finite order $n$. Does a one to one homomorphism always exist from $G$ to general linear group?
2
votes
2answers
81 views

What is a homomorphism?

I am starting to see the term everywhere I look, but every time I do, I get confused and can't get past it. I've seen various definitions: ("linear" homomorphism, i think) $$f:S\rightarrow T $$ ...
1
vote
3answers
101 views

Automorphism groups of isomorphic groups are isomorphic

Say $G \cong H$ are isomorphic groups. Show $Aut(G) \cong Aut(H)$ I just made this up so I'm not sure if actually $Aut(G) \cong Aut(H)$ is true but I'm $99.9\%$ sure this should be true I'm having ...
5
votes
1answer
70 views

$Gal(\mathbb{Q}(\sqrt 2 + \sqrt 3)/\mathbb{Q})$

A basis for $\mathbb{Q}(\sqrt 2 + \sqrt 3)$ over $\mathbb{Q}$ is $\{1,\sqrt 2 , \sqrt 3 , \sqrt 6 \}$ The roots of $x^2 -2$ are $\pm \sqrt 2$ and the roots of $x^2 -3$ are $\pm \sqrt 3$ so to find ...
0
votes
3answers
44 views

Problem with understanding homomorphism

Let $G$ be the group of all matrices of the form $\begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{pmatrix}$, $a,b,c \in \mathbb R$, ...
1
vote
2answers
72 views

How do I show a mapping is a homomorphism?

I don't want to make this question too broad, or non-specific. I'll will discuss a simple situation so we can all share a common context, but my question is less about this particular group, and more ...
1
vote
0answers
33 views

Do normal group endomorphisms form a normal submonoid?

What it says on the tin. A group endomorphism $v\colon G\to G$ is called normal if $v(aba^{-1})=av(b)a^{-1}$ for all $a,b\in G$. Equivalently, the map $g\mapsto v(g^{-1})g$ is a group homomorphism. ...
4
votes
2answers
190 views

homomorphism $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ with multiplicative groups, prove that kernel of $f$ is infinite.

Let $f: \mathbb{C}^* \rightarrow \mathbb{R}^*$ be a homomorphism of the multiplicative group of complex numbers to the multiplicative group of real numbers. I need to show that the kernel of $f$ must ...
2
votes
2answers
103 views

How Many Homomorphisms $\Bbb{Z}_4 \to \Bbb{Z}_8 \times \Bbb{Z}_{12} \times \Bbb{Z}_{15}$?

I know that the number of homomorphisms between $Z_n$ and $Z_m$ is $\gcd(m,n)$. However, I don't know what to do with these two questions: How many different homomorphisms exist: $\Bbb{Z}_4 \to ...
2
votes
2answers
42 views

Order of $M(x)$, where $M\colon G\to H$ is an injective homomorphism

Let $M\colon G\to H$ be a homomorphism and let $x$ be in $G$. Suppose that $x$ has order $k$. Show that if $M$ is injective the order of $M(x)$ equals order of $x$. My approach: ...
4
votes
2answers
106 views

Generalizing central automorphism group condition to endomorphisms

Given a group $G$, we can define its central automorphism group by $$\operatorname{Aut}_c(G)= C_{\operatorname{Aut}(G)}(\operatorname{Inn}(G)) = \{ \phi\in\operatorname{Aut}(G) : \phi(g)g^{-1} \in ...
0
votes
2answers
114 views

How to show there is only one homomorphism between $\mathbb Z_{25}$ and $S_4$?

Show that there is only one homomorphism from $\mathbb Z_{25}$ to $S_4$. How do i approach this question? If anyone could give me a hint or some guidelines that would be great.
1
vote
3answers
107 views

Question about automorphisms

Let G be a finite abelian group and let n be a positive integer relatively prime to |G|. a. Show that the mapping ϕ(x)=x^n is an automorphism of G. b. Show that every x∈G has an nth root, i.e., for ...
0
votes
2answers
45 views

group,subgroup and isomorphism

I study group theory now but I could not understand isomorphisms very well. In the book that I study I have seen that; $\mathbb{Z}_6=\{0,1,2,3,4,5\}$ is given and $H=\{0,2\}$ is a subgroup of the ...
3
votes
1answer
148 views

Automorphism of $\mathbb{Z}\rtimes\mathbb{Z}$

I'm looking for a description of $\mathrm{Aut}(\mathbb{Z}\rtimes\mathbb{Z})$. I've tried an unsuccessfully combinatorical approac, does anymore have some hints? Thank you.
0
votes
1answer
64 views

number of automorphisms for group in order 169

Let $G$ be a group with order 169. Prove number of automorphisms is at least 143. I thought that 169 is 13 squared so maybe G isomorphic to $ Z_{169} $ but I dont have any idea. How can I solve ...
0
votes
1answer
90 views

Hypercube and dihedral group

Let $G_n$ denote the subgroup of the orthogonal group $O_n$ of elements that send the hypercube to itself, the group of symmetries $C_n$, including the orientation-reversing symmetries. It would ...
0
votes
0answers
36 views

Homomorphism on the group of isometries

Prove that the map $f: M \rightarrow \{1,r\}$ defined by $t_a \rho_{\theta} \mapsto 1$, $t_a \rho_{\theta}r \mapsto r$ is a homomorphism. M denotes the set of isometries of the plane; r the reflexion ...
12
votes
1answer
188 views

Problem on abelian group

Let $G$ be an abelian group, and $\Phi:G\to \mathbb{R}$ is a function with the following property: $$\forall a,b\in G,~~ |\Phi(a+b)-\Phi(a)-\Phi(b)|<c$$ The problem asks to prove the existence of ...
1
vote
0answers
88 views

Number of kernals of all $\mathbb{Z}_n$ to $\mathbb{Z}_m$ homomorphisms?

How many subgroups $K \le \mathbb{Z}_n$ are there with $K =\ker(\phi)$ for some homomorphism $\phi\colon\mathbb{Z}_n \rightarrow \mathbb{Z}_m$? Stuck and need a hint. Have so far that there are ...
3
votes
2answers
375 views

Prove the third isomorphism theorem

I'm trying to prove the third Isomorphism theorem as stated below Theorem. Let $G$ be a group, $K$ and $N$ are normal subgroups of $G$ with $K⊆N$. Then $$(G⁄K)⁄(N⁄K)≅G⁄N.$$ I look up for some ...
6
votes
3answers
141 views

Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$

I'm trying to prove that Aut $\mathbb Z_p\simeq \mathbb Z_{p-1}$ (p prime). I know that Aut $\mathbb Z_p$ has $p-1$ elements because $\mathbb Z_p$ has $p-1$ possiblities of generators, so intuitively ...
8
votes
1answer
84 views

Prove that $Tx=x^{-1}~\forall~x\in G.$ [duplicate]

Let $G$ be a finite group and suppose the automorphism $T$ sends more than $\dfrac{3}{4}th$ of the elements of $G$ onto their inverses. Prove that $Tx=x^{-1}~\forall~x\in G.$
3
votes
2answers
610 views

If $\phi:G\to\bar{G}$ is an isomorphism and if $H$ is a normal subgroup of $G$, then $\phi(H)$ is a normal subgroup of $\bar{G}$.

If $\phi:G\to\bar{G}$ is an isomorphism and if $H$ is a normal subgroup of $G$, then $\phi(H)$ is a normal subgroup of $\bar{G}$. I am struggling with getting started with the problem. I know ...
1
vote
1answer
121 views

How I can find the inverse of an isomorphism?

The motivation of this question can be found in Can we extend the map $φ$ to $ℝ^{r}×C(ℚ)^{\text{tors}}→C(ℚ)$ as an isomorphism or not? My question is: How I can find the inverse of the ...
1
vote
0answers
125 views

Order of kernel of a homomorphism

Let $p,q$ be distinct primes. Prove that the kernel of the map $$f: (\mathbb{Z}/p^k\mathbb{Z})^* \rightarrow (\mathbb{Z}/p^k\mathbb{Z})^*$$ defined by $f(x)=x^q$ has order $\gcd(p-1,q).$ Thank ...
10
votes
1answer
281 views

Question on a homomorphism of a set G.

I'm having difficulty showing the given a map, say $\phi(z)=z^k$, is surjective. This question is from D & F section 1.6 - #19 Let $G$ =$\{z \in \mathbb C|z^n=1 \text{ for some } n \in \mathbb ...
1
vote
3answers
136 views

How can I conclude that an automorphism is the identity map?

Let $\phi$ be an automorphism on a group $G$. If $\phi$ maps any one non-identity element in $G$ to itself, is $\phi$ necessarily the identity map? What if $G$ is cyclic?
0
votes
5answers
181 views

Show that $S_2$ is isomorphic to $Z_2$

Show that $S_2$ is isomorphic to $Z_2$ I know that $S_2$ has two elements $\sigma_1$ and $\sigma_2$...and $Z_2$ has two elements 0 and 1. But isomorphism is a bijective function, right? So doesn't ...
4
votes
1answer
139 views

On automorphisms group of order $p^n$

Let $G$ be a finite group, such that $\mid Aut(G)\mid=p^n$. Then prove $G$ is p-group or $G\cong P\times C_{2}$, where $P$ is a p-group. Thank you
2
votes
1answer
113 views

On automorphisms group $C_{2}\times D_{8}$

Let $D_{8}$ be group dihedral of order 8 and $C_{2}$ be cyclic group of order $2$. Then determine the number all automorphisms of $C_{2}\times D_{8}$. Can you determine automorphisms group of ...
19
votes
2answers
516 views

Is every group the automorphism group of a group?

Suppose $G$ is a group. Does there always exist a group $H$, such that $\operatorname{Aut}(H)=G$, i. e. such that $G$ is the automorphism group of $H$? EDIT: It has been pointed out that the answer ...
0
votes
0answers
158 views

Homomorphisms and Automorphisms between cyclic groups of prime order

Let $A = B \times C$ where $B$ and $C$ are cyclic of order $p$ and $p^2$ respectively, where $p$ a prime. How many endomorphisms are there? How many of these endomorphisms are automorphisms?
0
votes
1answer
123 views

Homomorphisms between sets of homomorphisms

Let A,B be finite, commutative groups. Let $A^{*} = Hom(A, \mathbb{Q}/\mathbb{Z})$, the set of homomorphisms from $A$ to $\mathbb{Q}/\mathbb{Z}$. $A^{*}$ is abelian itself (take this for granted). Let ...
134
votes
2answers
4k views

Can we ascertain that there exists an epimorphism $G\rightarrow H$?

Let $G,H$ be finite groups. Suppose we have an epimorphism $$G\times G\rightarrow H\times H$$ Can we find an epimorphism $G\rightarrow H$?
5
votes
2answers
187 views

Where is the symmetric group hidden in the Yoneda lemma?

In extension to the question Yoneda-Lemma as generalization of Cayley`s theorem?, can someone point out to me where, in the categorical notation and analyzation of the Cayley's theorem, the symmetrc ...
4
votes
1answer
216 views

When is the image of a group morphism a normal subgroup?

Let $f : G \to G'$ be a group morphism. I need to find a necessary and sufficient condition such that $\operatorname{Im}(f)$ is a normal subgroup of $G'$.