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
1answer
76 views

Homomorphical Equivalence is NP-complete

Two graphs $G,H$ are homomorphically equivalent if there are exists a homomorphism from $G$ to $H$ and a homomorphism from $H$ to $G$. The task is to prove that this decision problem ...
3
votes
3answers
94 views

Given a functor between categories, how to denote a morphism between particular objects of that category

I have a very common situation, for which I need both: (1) notation; and, if available, (2) a general relative term. Let's say that: there is a functor between categories, $f:C_1\to C_2$, $c_1$ is ...
8
votes
1answer
84 views

*-homomorphisms $M_n(\mathbb{C})\rightarrow M_m(\mathbb{C})$

I've heard that every *-homomorphism $\phi:M_m(\mathbb{C})\rightarrow M_n(\mathbb{C})$ is unitarily equivalent to some *-homomorphism of the form $A\in ...
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 ...
3
votes
1answer
71 views

Every injective function is an inclusion (up to a unique bijection)

Let $X$ be a set and let $A$ be a subet of $X$. Let $i:A\longrightarrow X$ be the usual inclusion of $A$ in $X$. Then $i$ is an example of an injective function. I want to show that every ...
0
votes
3answers
92 views

Let $p$ be a prime. Determine the number of homomorphisms from $\mathbb{Z}_p \oplus \mathbb{Z}_p$ into $\mathbb{Z}_p$.

Let $p$ be a prime. Determine the number of homomorphisms from $\mathbb{Z}_p \oplus \mathbb{Z}_p$ into $\mathbb{Z}_p$. How can I able to solve this problem?can anyone help me please.
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 ...
1
vote
3answers
147 views

Ring homomorphism question.

If $R$ is a ring, show that there is exactly one ring homomorphism $\phi: \mathbb{Z} \to R$. I can't grasp the idea that there can be only one ring homomorphism. Aren't there many (at least more than ...
3
votes
1answer
85 views

Algebra (Not *)-Isomorphisms of von Neumann algebras

Let $A$ and $B$ be any two infinite-dimensional von Neumann algebras, they are operator algebras with operator composition as the multiplication and as infinite dimensional vector spaces they're ...
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 ...
8
votes
1answer
157 views

Path Connectedness and continuous bijections

Mathoverflow. Are there any two topological spaces $X$ and $Y$ such that they are path connected and such that there exist continuous bijections $X\rightarrow Y$ and $Y\rightarrow X$, but and yet ...
5
votes
1answer
94 views

A question on morphisms of fields

Let $A,B$ be two fields. Let $\phi:A\rightarrow B$ and $\psi:B\rightarrow A$ be two morphisms of fields. Can i conclude that $A$ and $B$ are isomorphic fields? My guess is yes, because every morphism ...
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 ...
-1
votes
1answer
91 views

Problem of monomorphism of rings [duplicate]

Let $A$ a ring, for each monomorphism $f:A^m \rightarrow A^n$, I don't know how to prove that $m\leq n$. I can't start the problem, I have no idea, help me please.
4
votes
1answer
209 views

Image of a morphism of varieties

Suppose $A$ and $B$ are two algebraic varieties, and $f:A\to B$ is a morphism of algebraic varieties. I guess it is true that $\text{im}(f)$ is itself an algebraic variety. But how to prove it?
1
vote
1answer
197 views

If each component of a Cartesian product is homeomorphic to another space, are the Cartesian products homeomorphic

Assume we are given a space $A$ with a metric $d$. Assume $A = A_1 \times A_2 \times A_3 \cdots$, ie. $A$ is a Cartesian product of spaces $A_i$, where $i \in I$. $I$ is countable or countably ...
2
votes
1answer
43 views

What is the name for the intermediary object(s) of functional composition?

Consider two morphisms: $f : X \to Y$ and $g : Y \to Z$ , and their composition: $g \circ f : X \to Z$. What is the name given to the role of $Y$ with respect to $g \circ f$? Is there a naming ...
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?
3
votes
1answer
359 views

Isomorphic Hilbert spaces

As part of a broader proof , I need to show that every two separable Hilbert spaces (that contains a dense countable set) are isomorphic (the linear mapping from one space to the other is injective ...
5
votes
1answer
86 views

Morphism between matrices and linear equations

I'm currently a beginner at linear algebra. So, in some books I see authors start defining linear equations and then they define matrices and, supposedly, the definition of associative matrix is to ...
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 ...
0
votes
1answer
51 views

Extensions of $\sigma:\mathbb{Q}(\xi)\rightarrow\mathbb{Q}(\xi), \ \sigma(\xi)=\xi^{2}$ to $\mathbb{Q}(\xi)(\sqrt[3]{2})$

Let $L$ be the splitting field of $T^{3}-2$ and let $\sigma:\mathbb{Q}\left(\xi\right)\rightarrow\mathbb{Q}\left(\xi\right)$ be a morphism defined by $\sigma\left(\xi\right)=\xi^{2}$. Find all ...
0
votes
1answer
62 views

Extensions of $\mathbb{Q}\left(\sqrt{2}\right)\rightarrow\mathbb{Q}\left(\sqrt{2}\right),\ \sqrt{2}\mapsto -\sqrt{2}.$

I have to solve an exercise that requires to find all extensions $\hat{\sigma}:\mathbb{Q}\left(\sqrt{2},\xi\right)\rightarrow\mathbb{Q}\left(\sqrt{2},\xi\right)$ (with $\xi$ being a third root of the ...
4
votes
0answers
51 views

Extending a homeomorphism between two curves [duplicate]

Possible Duplicate: Is there “essentially only 1” Jordan arc in the plane? Let $\gamma : [0,1] \to \mathbb{R}^2$ be a simple curve with $\gamma(0)=(0,0)$ and $\gamma(1) =(1,0)$. Clearly, ...
1
vote
2answers
69 views

Morphisms generated by functions

Given a function $f: A \to B$, I can construct a morphism $g : A^* \to B^*$ where $X^*$ denotes some free structure generated by $X$ (Could be monoid, group, module, etc.). I'd like to study ...
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
157 views

proof that sum of ramification degrees is degree of morphism between curves?

If $X$, $Y$ are irreducible smooth projective curves over an algebraically closed field and $\alpha:X\rightarrow Y$ is a morphism, how do we prove that ...
5
votes
5answers
1k views

In a ring homomorphism we always have $f(1)=1$? [duplicate]

Possible Duplicate: the image of $1$ by a homomorphism between unitary rings I'm studying the Atiyah's commutative algebra book and I realized that in the beginning of the book, the author ...
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 ...
0
votes
0answers
82 views

Verifying that a function is a morphism by checking a generating set

In some categories, to verify that a map $f : X \to Y$ is a morphism, it suffices to check only a generating set for $Y$ (or, rather, a generating set for some structure on $Y$ such as a topology or a ...
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 ...
7
votes
3answers
245 views

Category theory without codomains?

A surjection is a function whose range equals its codomain. Thus, the distinction between functions and surjections requires the notion of a codomain. Similarly, a bijection is an injection whose ...
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 $G\rightarrow H$?
1
vote
3answers
180 views

Are monomorphisms of rings injective?

Let $R$ and $S$ be rings and $f:R\to S$ a monomorphism. Is $f$ injective?
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 ...
20
votes
3answers
899 views

Intuition for étale morphisms

Currently working on algebraic surfaces over the complex numbers. I did a course on schemes but at the moment just work in the language of varieties. Now i encounter the term "étale morphism" every ...
0
votes
0answers
72 views

Applied Math question: Is there a way to use the set of GPS satellite positions as a lattice?

Okay, I am wondering if it is possible to take a satellite constellation, say the gps satellites, and use the satellites' positions (at time $t$) as points (shown below) and use them as a lattice. ...
-2
votes
2answers
213 views

Morphims with unique domain

A morphism $m$ of a category has the following property: No morphism (except of the identity morphism) of the category has codomain equal to the domain of $m$. In other words, $m$ cannot be composed ...
1
vote
1answer
146 views

Unique decomposition of a mapping by an equivalence relation

I have a math question from computer science. The following should be a fundamental fact from mathematics. Can you the mathematicins tell me how you would say it in a more elegant way? Given a ...
7
votes
1answer
504 views

History behind Exact Sequences.

I am very much interested in listening to the history behind the exact sequence. We know that the exact sequence is sequence of objects with morphisms such that image of one morphism equals to the ...
3
votes
2answers
183 views

Basic questions about definitions in category theory

I'm just getting started in category theory, and I'm not understanding the basic definitions. For example, a common example of a category is a poset. So suppose I have a trivial poset $P$ of 10 ...
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 $G'$.
12
votes
1answer
349 views

Working with Morphisms in Local Coordinates

In light of the holiday, I would like to air a grievance. I have no good way to recoordinatize a morphism of varieties as I move between coordinate neighborhoods. Let me explain what I mean with ...
1
vote
1answer
62 views

Do nonsingular points get mapped to nonsingular points in a branched cover

Let $\pi:C\to D$ be a finite surjective morphism of noetherian integral schemes. Let $x\in C$ be a nonsingular point. Does it follow that $\pi(x)$ is nonsingular? What if we impose some conditions ...
3
votes
3answers
520 views

Examples of categories where epimorphism does not have a right inverse, not surjective

Epimorphism is defined as following: $f \in \operatorname{Hom}_C(A,B)$ is epimorphism if $\forall Z. \forall h', h'' \in \operatorname{Hom}_C(B, Z)$ the following holds: $h' f = ...