1
vote
3answers
67 views

An abelian group is finite $\iff$ the kernel of a surjective homomorphism has rank $n$

I'm doing a course on lineare algebra and I have to show the following: let $H$ be a finitely generated abelian group and $g: \mathbb{Z}^n \to H$ a surjective homomorphism. I want to show that ...
0
votes
2answers
39 views

Show that given group is abelian

There's a set consisting of 2 elements: G = {a,b}. In this set we define an operation * in the following way: $$a*a=b*b=a$$ $$a*b=b*a=b$$ The question says: "Show that (G, *) is a commutative ...
2
votes
0answers
48 views

Epimorphism (abelian group)

Let $(G,\cdot), (H,*)$ Groups and $f: G\rightarrow H$ an Epimorphism. Show that: If G is an abelian group, then H is also an abelian group. Is the reversal of this proposition also true? My idea: ...
1
vote
2answers
46 views

Does a linear quotient map have sections

Suppose $V$ is a vector space with vector subspace $N$. Then there is a natural projection $$ \pi_N: V \to V/N $$ from the vector space $V$ to the quotient space $VN$ of $V$ modulo $N$. Does ...
7
votes
1answer
153 views

Finding the order of the automorphism group of the abelian group of order 8.

So I am given an abelian group of order $8$ such that for all non-identity elements $x^2 = e$ (all elements have order two). So I know the answer is gonna be $168$, but I gotta prove this. So far I ...
3
votes
1answer
153 views

Linear algebra of finite abelian groups

Let $\phi:G \to H$ be a surjective homomorphism of finite abelian groups, and let $g_1, \ldots, g_n$ be an irredundant set of generators (from now on, a basis) for $G$. be a basis for $G$, meaning a ...
1
vote
1answer
125 views

Dimension of subspace fixed by subgroup representation.

If $G$ is an abelian group with cyclic subgroup $H$ and $(\rho,V)$ is a (permutation) representation of $G$. Then I can form a representation of $H$ by considering the composition ...
2
votes
1answer
91 views

number of differents vector space structures over the same field $\mathbb{F}$ on an abelian group

My question here raised another one. How many differents vector space structures over a field $\mathbb{F}$ we may have on an abelian group? I know that there are abelian groups that we can not endow ...
6
votes
3answers
253 views

Is $\mathbb{Z}(p^{\infty})$ a vector space over some field $\mathbb{F}$?

I don't know how to write in good English, so I will follow Hungerford's word from his book Algebra. The following relation on the additive group $\mathbb{Q}$ of rational numbers is a congruence ...
5
votes
1answer
230 views

Non-degenerate alternating bilinear form on a finite abelian group

Ciao! Let $A$ be a finite abelian group, and let $ \psi : A \times A \to \mathbb{Q}/\mathbb{Z} $ be an alternating, non-degenerate bilinear form on $A$. Maybe I should say what I mean by these ...