Tag Archives: books i found

Countably generated abelian groups

by

The following theorem is one that many of us use practically daily.

Theorem 1 If {A} is a finitely-generated abelian group, then {A} can be written as a direct sum

\displaystyle A \cong {\mathbb Z}^r \oplus \bigoplus_{p\text{ prime}} \bigoplus_{n=1}^\infty ({\mathbb Z}/p^n{\mathbb Z})^{e_{p,n}}

in a unique way — that is, the exponents {r} and {e_{p,n}} are uniquely determined by {A} (and all but finitely many of them are zero).

This theorem is really two important theorems in one. The first is a decomposition theorem, which lets us decompose our objects (finitely generated abelian groups) in a simple way (a direct sum of cyclic groups). The second is a uniqueness theorem, which tells us when two of our objects are the same (in this case, precisely when the exponents {r} and {e_{p,n}} are the same).

In particular, this theorem tells us all we’d ever need to know about finite abelian groups. When we go further, though, the situation becomes far more difficult. Here are a few radically different examples of infinite abelian groups (all of which are countable, even!):

  • {{\mathbb Z}}, which we’ve already taken care of.
  • {{\mathbb Q}}, the rational numbers.
  • {{\mathbb Q}/{\mathbb Z}}.
  • {{\mathbb Z}_{(p)}}, the set of rational numbers with denominators prime to {p}.
  • {{\mathbb Z}/p^\infty = {\mathbb Z}_{(p)}/{\mathbb Z} = \varinjlim {\mathbb Z}/p^n} along the maps that are multiplication by the obvious powers of {p}.

We’d like to generalize the above classification theorem to deal with infinite abelian groups. This is surprisingly difficult for high cardinalities, and as far as I can tell in my limited research, there’s no real hope for such a theorem for uncountable abelian groups. For countable abelian groups which are assumed to be torsion, however, there’s a rather nice theorem, due to Ulm in the 30’s [4], and in this talk we’ll attempt to prove it. More generally, we’ll exhibit a couple convenient ways of decomposing abelian groups.

Most of this is from Kaplansky’s little red book [2], which is a great and short read for people of all mathematical backgrounds. I’m going to omit the word `abelian’ a lot. Additionally, all of this applies mutatis mutandis for countably generated modules over a PID.

Continue reading →