Tagged Questions

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 ...
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 ...
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 $Tx=x^{-1}~\forall~x\in G.$
2
votes
2answers
96 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 ...
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.
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?
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 ...
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
5answers
246 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 ...
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
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 ...
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 $G'$.
3
votes
1answer
211 views

the image of $1$ by a homomorphism between unitary rings

let $R$ and $S$ be unitary rings and $\phi:R\rightarrow S$ a ring homomorphism. is the following correct: $\phi(1_R\cdot1_R)=\phi(1_R)\cdot\phi(1_R)$ so $\phi(1_R)(1_S-\phi(1_R))=0_S$ and so ...