0
votes
1answer
36 views

Two free basis of a free abelian group

We have a free abelian group $A(X)$, where $X$ is its free basis, and let $Y$ another free basis for $A(X)$. We know that every $g\in A(X)$ can be expressed as $g=a_1x_1+...+a_nx_n$ where the $a_i$'s ...
1
vote
2answers
60 views

Let A be a finitely generated abelian group. Show that Hom(A,Z) is a free abelian group.

My question is Let $A$ be a finitely generated abelian group. The structure theorem says that $A$ is isomorphic to $F \times T$, where $F$ is isomorphic $\mathbb Z^m$, some $m \geq 0$, and $T $ is ...
1
vote
0answers
48 views

Specific basis of a subgroup of a free abelian group

I'm looking for clarification on Fraleigh's "A First Course in Abstract Algebra" Theorem 38.11. It states: "Let $G$ be a nonzero free abelian group of finite rank $n$, and let $K$ be a nonzero group ...
0
votes
2answers
72 views

All bases for a finitely-generated abelian group have the same cardinality.

I want to understand more about this proof from Lang's Algebra: Let $B$ be a subgroup of a free abelian group $A$ with basis $(x_i)_{i=1...n}$. It has already been shown that $B$ has a basis of ...
0
votes
1answer
112 views

Free abelian subgroup of index 2.

Let $G$ be a group with the following presentation $G=gp(x,y \mid x^2=y^2=1)$. I need to know, what further information about $G$ can be derived from knowing that $G$ has a free abelian subgroup of ...
0
votes
1answer
236 views

How can I show the free abelian group of rank $r$ is isomorphic to an $r$-copy of $\mathbb{Z}_\infty$?

Today, this problem was given to me. Let $F$ be a abelian free group of rank $r$. Show that it is isomorphic to an $r$-copy of $Z_{\infty}$. I could do some messy job about it but so far I ...
1
vote
2answers
76 views

Bases for $\Bbb Z^n$ containing a given vector

I am trying to prove the following theorem on finitely generated free abelian groups (which thus for simplicity may be assumed to be $\Bbb Z^n$): Let $\alpha \in \Bbb Z^n$ be such that for all $k ...
3
votes
3answers
409 views

Quotient of two free abelian groups of the same rank is finite?

Let $A,B$ be abelian groups such that $B\subseteq A$ and $A,B$ both are free of rank $n$. I want to show that $|A/B|$ is finite, or equivalently that $[A:B ]$ (the index of $B$ in $A$) is finite. For ...
3
votes
4answers
204 views

$F$ is a free abelian group on a set $X$ , $H \subseteq F$ is a free abelian group on $Y$, then $|Y| \leq |X|$

I am confused by the proof a proposition: $F$ is a free abelian group on a set $X$ and $H$ is a subgroup of $F$, then $H$ is free abelian on a set $Y$, where $|Y| \leq |X|.$ The proof is: ...