For question about morphism between groups, ring, topological spaces, vector space, categories, etcs... Please also use the correspondent tags (e.g. (group-theory), (ring-theory)) in order to precise the involved structure.

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2
votes
0answers
38 views

On the back and forth conditions for a set of partial isomorphisms

I've recently begun reading Poizat's A Course in Model Theory and already in the first pages I had some doubts. One odd (not necessarily bad) thing is that he defines notions such as isomorphism only ...
3
votes
3answers
70 views

An epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$

I want an explicit example of an epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$. Thanks.
1
vote
2answers
31 views

Normal subgroups, direct product and monomorphism problem

Let $G$ be a group and let$H,K$ be normal subgroups of $G$. Let $\pi_H,\pi_K$ be the projections on $H$ and $K$ respectively. Show that the map $$f:G/(H \cap K) \to G/H \times G/K$$ defined as ...
0
votes
2answers
20 views

Isomorphism between quotient groups

Exercise Let $f:G \to G'$ be an isomorphism and let $H\unlhd G$. If $H'=f(H)$, prove that $G/H \cong G'/H'$. As I've shown that $H'\unlhd G'$, I thought of defining $$\phi(Ha)=H'f(a)$$I was trying ...
3
votes
2answers
41 views

Abelian group and morphism equivalent statement

Exercise Show that the following statements are equivalent: $(i) \space G \space \text{is abelian.}$ $(ii) \space \text{the map f: G} \to \text{G defined as} \space f(x)=x^{-1} \space \text{is a ...
0
votes
0answers
21 views

Graph of a regular function. When is the projection on the first component birational?

Let $X$ be an irreducible variety over a field $k$ and $f$ a regular function on some open subset $U\subseteq X$. Let $F\subseteq X\times \mathbb{A}^1_k$ be the graph of $f$ and suppose that the graph ...
1
vote
0answers
45 views

Epimorphisms in two directions

Let $\mathcal{C}$ be a category. Consider the following statement: (S1) Whenever $A,B$ are objects and $\iota_1:A\to B$ and $\iota_2:B\to A$ are monomorphisms, then there is an isomorphism $\phi:A\to ...
1
vote
1answer
30 views

Properties of varieties that can be lifted under etale morphisms

Let $X$,$Y$ be varieties over some field $k$ of char $0$ (not necessarily algebraically closed). Suppose we have an etale morphism $f : Y \to X$. My question: for what properties P can we say that ...
2
votes
1answer
48 views

Terminal objects of the category of morphisms

I'm reading Basic Category Theory for Computer Scientists by Benjamin C. Pierce and in exercice 1.4.6, he asks what the terminal objects are in $Set^\to$. Let $C$ be a category. The category ...
3
votes
0answers
23 views

Let $h: A \to B$ be a weak homomorphism. Is h$[A]$ a substructure of $B$?

A little bit more precise: let $\mathfrak{A}$ and $\mathfrak{B}$ be two structures. Define a weak homomorphism as a function $h: \mathfrak{A} \to \mathfrak{B}$ such that the folowing conditions are ...
0
votes
0answers
48 views

if a map is bijective, then it has inverse. is the inverse also bijective?

I asked this question because I want to know if given two groups that are isomorphic will always have the same order or not.
1
vote
0answers
40 views

Riesz homomorphism (Banach-Stone theorem)

Let $X,Y$ be compact Haussdorff spaces, and consider $$J:C(X)\to C(Y) $$ a bounded linear bijection such that $J(f\cdot g) = J(f)J(g)$. I know that if $T:C(X)\to C(Y)$ is a linear bijection, being ...
0
votes
0answers
21 views

Hypergraph notation and hypergraph morphisms

There are two parts to my question. The first part is about notation for hypergraphs. The sconed is about the notion of morphisms for hypergraphs. For the notation part, the context is that I make ...
1
vote
1answer
41 views

How do I show that an endomorphism is self-adjoint if and only if $\langle u, Tu \rangle \in \mathbb{R}$ for all $u \in \mathbb{V}$

Let $$(V,\langle \cdot , \cdot \rangle)$$ be a complex vector space. Let $T \in \mathcal{L}(V)$ be an endomorphism. Now I want to show, that $T \in \mathcal{L}(V)$ is self-adjoint if and only if ...
2
votes
2answers
51 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 ...
2
votes
1answer
40 views

Canonical isometric isomorphism of $l_{\alpha}^{2}$

Let $\alpha \in \mathbb{R}$ and $l_{\alpha}^{2}$ the vector space of bi-infinite sequences $(x_{n})_{n\in \mathbb{Z}}$ such that $||x||_{\alpha}:=\sum_{n\in\mathbb{Z}} ...
1
vote
1answer
38 views

Left shift operator $L: l^2 \rightarrow l^2$ on the sequence space $l^2$

$$L: l^2 \rightarrow l^2$$ is defined by $$b = (b_1,b_2,...) \mapsto Lb = (b_2,b_3,...)$$. $(Lb)_n = b_{n+1}$ respectively. How can I determine the adjoint endomorphism $L^*$? Kind regards George
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 ...
-4
votes
1answer
38 views

Question about nonisomorphic polynomial rings.

Let $n,k > 1$ be positive integers. Define the reduced polynomial rings : $g^k_n = \Bbb R[X_n]/(G^k_n(X_n))$ where $G^k_n$ is a real polynomial of degree $n$ (that keeps the ring reduced). (k is ...
0
votes
2answers
43 views

Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$

For an arbitrary group $(G,\cdot)$ let $\operatorname{Aut}(G) = \{f: G \to G \mid f \text{ is an isomorphism}\}$ be the set of all automorphisms of the group $G$. We assume that ...
2
votes
3answers
136 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. ...
2
votes
1answer
34 views

Isomophism between rings an two right ideals

Let I, J two right ideals of a ring R such that I+J =R. Show thath the direct sum of I and J is isomorphic to the direct sum of R and the intersection of I and J. Can anyone please give me at least ...
3
votes
1answer
36 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
33 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
29 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
34 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
27 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
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
1answer
109 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
85 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
74 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 ...
1
vote
1answer
49 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
41 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
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
76 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
35 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
60 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
142 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
77 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
201 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
187 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
105 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
20 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
39 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
75 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
43 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 ...