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3
votes
1answer
38 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
40 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.
0
votes
0answers
21 views

Is there a term for an endomorphism defined up to conjugation by an automorphism?

Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that ...
2
votes
0answers
36 views

Question on morphism locally of finite type

The exercise 3.1 in GTM 52 by Hartshorne require to prove that $f:X \longrightarrow Y$ is locally of finite type iff for every open affine subset $V=\text{Spec}B$, $f^{-1}(V)$ can be covered by open ...
1
vote
1answer
49 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 ...
0
votes
2answers
34 views

Factor ring induced by the ideal generated by x(x-1)(x-2)

Consider the ring $R:=\mathbb C[x]/I$, where $I$ is the ideal in $\mathbb C[x]$ generated by $x(x-1)(x-2)$. Show that the evaluation map $\mathbb C[x]\to \mathbb C,\ p(x)\mapsto p(\alpha)$, for ...
2
votes
1answer
61 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
1answer
116 views

How to construct a nonzero homomorphism from a module to a proper submodule?

Let $M$ be a finitely generated module over a commutative ring and $N$ be a non zero proper submodule of $M$. Then is it always possible to have a non zero homomorphism $f$ from $M$ to $N$?
2
votes
2answers
105 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
116 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
81 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 ...
3
votes
1answer
69 views

Confusion about “horizontal composition” of natural transformations

I'm having trouble with an exercise from Rotman's Homological Algebra. It has to do with what Wikipedia calls "horizontal composition" of natural transformations. Namely, given $F, ...
2
votes
1answer
48 views

Smallest Graph that is Regular but not Vertex-Transitive?

I'm trying to find the smallest graph that is regular but not vertex-transitive, where by smallest I mean "least number of vertices", and if two graphs have the same number of vertices, then the ...
0
votes
3answers
51 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
102 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
1answer
48 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. ...
2
votes
0answers
66 views

Homeomorphism form $(-1,1)$ to $\mathbb{R}$

I want to show that every open intervall $(a,b)$ is homeomorph to $\mathbb{R}$. On $(a,b)$ I chose the relative topology $\mathcal{T}_{(a,b)}$ and on $\mathbb{R}$ the trivial topology $\mathcal{T}$ ...
0
votes
0answers
145 views

Collections of Homomorphic (defined) structures via $f$

Long ago I read a text about a collection of algebraic sturctures all homomorphic (or isomorphic) via a unique homomorphism An Example similar to the construction I found was this: Lets take define ...
-1
votes
1answer
79 views

What are morphisms in the category of sets $\mathbf{Set}$?

Do i understand correctly that morphisms in the category of sets $\mathbf{Set}$ are ordered triples $(f, A, B)$ where $f$ is a function $A\to B$? It seems that it is often claimed, even in the ...
4
votes
2answers
276 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
202 views

Evaluation morphisms of formal power series and nilpotent elements

Given a commutative ring $A$, and a finitely presented (associative) $A$-algebra $B$, show that a morphism of $A$-algebras $A[[x]] \longrightarrow B$ is given by evaluation at an nilpotent element $ ...
2
votes
2answers
108 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 ...
0
votes
0answers
26 views

Two Homomorphisms Question

As an exercise in automata and formal languages, we got two question which I would like to share with and ask you if I am on the right lane a) Give all homomorphism $\varphi:\mathbb{N} \rightarrow ...
0
votes
1answer
44 views

Is this morphism flat?

Suppose $X$ is a smooth projective curve over an algebraically closed field $k$. Is the morphism $ X \to \operatorname{Spec}(k) $ necessarily flat? What kind of conditions on the above morphism are ...
2
votes
1answer
79 views

Finite fiber- unramified morphisms

I'm in trouble understandig the proof of Proposition 3.2 Chapter 1 of Milne's Book "Étale Cohomology". Let $f:Y\rightarrow X$ be locally of finite-type. The following are equivalent. $(a)$ f is ...
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: ...
3
votes
1answer
49 views

How do morphisms in a comma category single out commuting squares?

I'm trying to teach myself the rudiments of Category Theory. I have a doubt about the definition of comma categories, more precisely about the morphisms. Suppose have two functors ...
7
votes
1answer
67 views

Space of morhisms of representations, its dimension in special case

The symmetric group $S_n$ acts linearly on $\mathbb{C}^n$, hence it brings up to the representation in $\Lambda^m\mathbb{C}^n$. The goal is to evaluate the dimension of morphisms ...
4
votes
2answers
116 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
1answer
97 views

Two isomorphism questions

Let G = (C - {0}, mult.), and let U be the subgroup U = {x+yi such that x^2 + y^2 = 1}. Use the Fundamental Theorem to show that: a) G/U is isomorphic to (R>0, mult.) b) G/R>0 is isomorphic to U. ...
2
votes
0answers
68 views

Epimorphism (abelian group)

Let $(G,\cdot), (H,*)$ Groups and $f: G\rightarrow H$ an Epimorphism. Show that: If G is an abelian group, then H is also an abelian group. Is the reversal of this proposition also true? My idea: ...
0
votes
0answers
32 views

Classification of endomorphisms in infinite dimension

I'm interested in the following question : let $E$ be an infinite dimensional vector space and $u$ some endomorphism of $E$. We consider endomorphisms equivalent to $u$, that is, of the form $\alpha ...
0
votes
2answers
128 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.
0
votes
2answers
68 views

How is this map a well-defined homomorphism?

If $f: R \rightarrow S$ is a homomorphism of rings with kernel $K$, and $I$ is an ideal in $R$ such that $I \subset K$. The hypothesis is that the map $\overline{f}: R/I \rightarrow S$ given by ...
1
vote
3answers
103 views

No isomorphism between $(\mathbb R,+)$ and $(\mathbb R^*, \times)$

My goal is to disprove the existence of an isomorphism between $(\mathbb R,+)$ and $(\mathbb R^*, \times)$. I proceeded by contradiction. Suppose $f$ is such a map. Then $$f(0-0)=f(0)f(-0)=-f(0)^2$$ ...
1
vote
3answers
176 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 ...
2
votes
3answers
52 views

Boolean endomorphisms vs endofunctions on finite sets

I stumbled upon a funny fact: Let $\mathbf{Bool} = \{0, 1 \}$. For all functions $f: \mathbf{Bool} \to \mathbf{Bool}$ it is the case that $f^3 = f$. This got me excited and I was wondering whether ...
6
votes
1answer
96 views

How to use flatness here?

Let $X\to S$ be a scheme. Definition: A relative effective Cartier divisor on $X/S$ is a closed subscheme $D\subset X$ such that the ideal sheaf $I$ of $D$ is invertible and $D\to S$ is flat. Let ...
5
votes
2answers
87 views

Differences between nilpotent and pointwise nilpotent endomorphisms.

Consider an endomorphism of a module $f:M\rightarrow M$. We have that $f$ is pointwise nilpotent if $\forall x\in M,\ \exists n,\ n\in \mathbb N$ such that $f^{n}(x)=0$. I already know that the ...
5
votes
1answer
74 views

Relative effective Cartier divisors

I have two different definitions of a relative effective Cartier divisor. The first one is a bit outdated and defines the notion over analytic spaces, in the following way: Definition 1: Let $X$ be ...
3
votes
1answer
79 views

Morphisms, Splitting Fields, and Primitive Cube Roots

Let $\alpha=\sqrt[3]2$ and $\omega=e^{2\pi i/3}$, and let $K=\mathbb{Q}[\alpha,\omega]$ be the splitting field of $f(x)=x^3-2$ over $\mathbb{Q}.$ Determine all morphisms $K\to K$. Ok, so I have ...
0
votes
2answers
92 views

$k$-algebra homomorphisms

I would like to ask if the following is true: Let $A$ be a $k$-algebra where $k$ is any field. If we have a $k$-algebra homomorphism $f:A\rightarrow k$, does it follow that $\ker(f)$ is a maximal ...
2
votes
2answers
73 views

Proving the Frobenius map is an endomorphism

I have prime $p$, and $K$ a field such that $p \cdot 1 = 1+1+\cdots+1 = 0$. I am asked to prove that $F: K \rightarrow K$, $a \mapsto a^p$ is a ring homomorphism. I can prove this for ...
0
votes
1answer
44 views

Say $K=\mathbb Q(2^{1/3})$. Determine all endomorphisms of $K$.

Say $K=\mathbb Q(2^{1/3})$. Determine all endomorphisms of $K$, and justify your answer. Hint: Say $f(x)= x^3-2$. How many roots of $f$ are in $K$? For this I know $x^3-2$ has 1 real root, ...
0
votes
1answer
166 views

Isomorphism of R-modules

Does somebody has an example where the left $R$-modules $R^m$ and $R^n$ are isomorphic for all positive integers $m$, and $n$?
3
votes
3answers
205 views

Isomorphisms and the Fundamental Homomorphism Theorem

Let $$ R=\left\{ \begin{bmatrix} a & b \\ 0 & a \end{bmatrix} : a,b∈ℝ\right\}⊂M_2(ℝ) $$ and $$ I=\left\{ \begin{bmatrix} 0 & b \\ 0 & 0 \end{bmatrix}: b∈ℝ\right\}. $$ Identify the ...
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
156 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
91 views

Finding homomorphisms and kernels from a given ring R

Give the following rings $R$ and ideals $I$ find a ring $S$ and a homomorphism $f:R \rightarrow S$ with kernel $I$ i) $R=\mathbb{Q} [x], I=(x^{2}-2)R$ ii)$R=\mathbb{Z}[i], I=2R$ (Gaussian Integers) ...
0
votes
1answer
65 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 ...