Abelian Groups - László Fuchs [Springer]

1 The Ulm-Zippin Theory 349
of C i . The group A will be defined as the group generated by fa i ; g i g i0) and the columns
by non-negative integers. The cardinal numbers in the th row are the invariants of
the th Ulm factor of A. The matrix U.A/ satisfies:
(i) it is an ˛ !-matrix with a countable ordinal ˛;
(ii) the entries are non-negative integers or @ 0 ;
(iii) every row (with the possible exception of the very last one) contains infinitely
many non-zero entries.
Ulm’s theorem can be interpreted as saying that two reduced countable p-groups
are isomorphic, A Š C, if and only if the corresponding matrices are equal: U.A/ D
U.C/, while Zippin’s theorem says that every matrix with properties (i)–(iii) is the
UK-matrix of some reduced countable p-group.
Corollaries We conclude this section with two corollaries.
Proposition 1.11. A countably infinite reduced p-group decomposes into the direct
sum of infinitely many non-trivial groups.
Proof. Let A be a group as stated, and let U.A/ be its UK-matrix. It is an easy
exercise in cardinal arithmetics to decompose U.A/ into the sum of countably many
matrices satisfying (i)–(iii). By Zippin’s theorem, each of these matrices defines a
group A n .n