# Tagged Questions

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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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Problem with understanding homomorphism

Let $G$ be the group of all matrices of the form $\begin{pmatrix} 1 & a & b \\ 0 & 1 & c \\ 0 & 0 & 1 \\ \end{pmatrix}$, $a,b,c \in \mathbb R$, ...
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### 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 ...
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### 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. ...
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### 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 $\mathcal{T}$ ...
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### 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 ...
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### 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 ...
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### 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 ...