Absolutely and Relatively free abelian groups.

Question

I see that there are the notions of absolutely free abelian group and relativley free abelian group. Could you please explain the difference between the two notions. Thanks!!

Answer 1

According to the online Encyclopedia of Mathematics:

A free group in a variety $\mathcal D$ of groups is defined analogously to a free group, but within $\mathcal D$. It is also called a $\mathcal D$-free group, or a relatively-free group (and also a reduced free group).

Presumably, if the variety $\mathcal D$ only consists of abelian groups, then it would be a relatively-free abelian group.

For example, the variety $\mathcal D$ defined by the single relationship $xyx^{-1}y^{-1}=1$ describes the variety of abelian groups, so a $\mathcal D$-free group would be a standard free abelian group.

If $\mathcal D$ contains more relations, you'd get different classes of relatiely free groups. For example, with the relations:

$$xyx^{-1}y^{-1}=1, x^3=1$$

then you'd get the variety of vector spaces over $\mathbb Z/3\mathbb Z$, and the relatively free ones would be the ones with bases (which is all of them, if you use use the axiom of choice.)

The varieties of groups $\mathcal D$ that only include abelian groups are all of the form:

$$xyx^{-1}y^{-1} =1, x^d=1$$

for some integer $d\geq 0$. In this case, the variety is the variety of $\mathbb Z/d\mathbb Z$-modules. and the "relatively free groups are the free modules.