This is for questions about abelian groups, each with a basis.
Questions tagged [free-abelian-group]
190 questions
2
votes
1 answer
Basis of subgroup of a free abelian group
We have this theorem:
** Let $F$ be a free abelian group of rank $n$ and let $H$ be a subgroup
of $F.$ There exists a basis $\{x_1,...,x_n\}$ of $F$ and integers $d_1,...,d_r > 0 $ such
that
• $di\vert d_{i+1}$ for $i = 1,...,r$
•…
saba wazir
- 23
1
vote
0 answers
About order of Base of free abelian group
I am not sure about the connection between order of a base of free abelian group, to the order of the minimal generator set which generates the group.
I would like to inherit the conclusion from linear algabra, says that a base holds the minimal…
Ron Abramovich
- 618
0
votes
2 answers
Free abelian group - exercise
Can someone please help me understand how to do this exercise?
Show that multiplicative group of positive rationals is a free abelian group of countably infinite rank.
So, I know i can begin with:
Let Q be multiplicative group of positive…
user797618
0
votes
0 answers
What is the free abelian group of an infinite set?
Given a set $S$, the free abelian group is defined as the set of formal sums of elements of $S$. Is this restricted to formal sums of a finite subset of elements of $S$? For example, is $S = \mathbb{R}$, then is $\sum_{z\in \mathbb{Z}}a_z z$ a…
Anon
- 1,039
0
votes
0 answers
Conditions under which a union of free abelian groups is free
Let $(A_\nu)_{\nu<\mu}$ be an increasing sequence of free abelian groups such that for any limit ordinal $\lambda<\mu$, $A_\lambda=\bigcup_{\nu<\lambda}A_\nu$. Let $A=\bigcup_{\nu<\mu}A_\nu$. Let $\kappa$ be the cofinality of $\mu$.
Assume that…
Tri
- 387