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.
3
votes
3answers
81 views
Does every category have a functor?
Is there any one (or more) categories that doesn't have a functor? Functors go between categories, so is there any category that only has an identity functor but no other functor that maps it to ...
1
vote
0answers
43 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 ...
1
vote
3answers
72 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 ...
7
votes
1answer
60 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 ...
2
votes
2answers
63 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 ...
2
votes
1answer
28 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
66 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.
5
votes
2answers
69 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
79 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 ...
2
votes
1answer
57 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 ...
7
votes
1answer
62 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 $Tx=x^{-1}~\forall~x\in G.$
0
votes
0answers
29 views
Example of Homeomorphism [duplicate]
Can anyone give me an example of a continuous bijective map between 2 path-connected topological spaces, which is not a homoemorphism? Thanks!
2
votes
2answers
94 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 ...
7
votes
0answers
112 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
86 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
75 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
72 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
72 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.
2
votes
1answer
62 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?
0
votes
1answer
28 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
31 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
121 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
91 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
0answers
94 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 ...
3
votes
1answer
61 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
105 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
43 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
58 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
42 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
57 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
84 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
112 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 ...
4
votes
5answers
245 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
94 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
55 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 ...
14
votes
2answers
281 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 ...
6
votes
3answers
198 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
111 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
53 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 ...
80
votes
2answers
2k 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
86 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
137 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 ...
12
votes
3answers
317 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
48 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
168 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
109 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 ...
3
votes
1answer
270 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 ...
2
votes
2answers
110 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
144 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
321 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 ...
