Abelian Groups - László Fuchs [Springer]

344 11 p-Groups with Elements of Infinite Height
In other words, a has the maximal height in its coset mod G. Thus an element a 2 A
of height is proper with respect to G if and only if a … p C1 A C G. In this case,
h A .a/ D h A=G .a C G/:
Of course such an element a need not exist in a coset, though it can be shown easily
that if the height of a coset a C G in A=G is a non-limit ordinal, then the coset does
contain an element of this height. Note that if a is proper with respect to G,thenwe
have
h.a C g/ D minfh.a/; h.g/g for all g 2 G:
Evidently, if G is finite, then every coset mod G contains elements proper with
respect to G. The same holds if G D p A for an ordinal .
Let now A be a reduced p-group, and G a subgroup of A. For an ordinal , we
introduce the notation
G./ D .p C1 A C G/ \ p AŒpI
this is a subgroup between p AŒp and p C1 AŒp. From our remark above it follows
that an element a 2 A of order p and of height belongs to G./ if and only if it is
not proper with respect to G. Consequently, representatives of the non-zero cosets
in the factor group
f .A; G/ D p AŒp=G./
are exactly the elements of A that are of order p, of height , and proper with respect
to G. ThisZ=pZ-vector space is called the th UK-invariant of A relative to G
or the th Hill invariant of A relative to G. Sometimes we understand by this
invariant the dimension of the vector space. We can allow this ambiguity, since from
the context it will always be clear what we mean: vector space or its dimension.
Evidently, f .A; G/ f .A/ and f .A;0/ D f .A/: Here f .A/ D p AŒp=p C1 AŒp
is the Ulm-Kaplansky- or UK-invariant of A.
Example 1.1. Let A D Z.p nC1 / and G D pA. ThenG.k/ D p k AŒp D AŒp for k D 0;:::;n, and
G.k/ D 0 for all k > n. Thus the Hill invariants f .A; G/ vanish for all .
Example 1.2. Let A be a reduced p-group, and B its basic subgroup. Then B.n/ D .p nC1 A C
B/ \ p n AŒp D A \ p n AŒp D p n AŒp, thus f n .A; B/ D 0 for all n