Abelian Groups - László Fuchs [Springer]

6 Theorems on p-Basic Subgroups 177
an index set J of cardinality . We quote Lemma 6.1 to argue that each C j contains
properly a basic subgroup B j < C j . The direct sum B 0 D˚j2J B j is evidently a
basic subgroup of B, and hence of A. ThisB 0 is lower, because by construction
rk.A=B 0 / rk.B=B 0 / D P j2J rk.C j=B j / .
ut
Another property of basic subgroups is featured in the next lemma.
Lemma 6.7. Let A be a reduced p-group, and B a basic subgroup of a subgroup G
of A. If B p A for some ordinal , thenalsoG p A.
Proof. A=B is viewed as an extension of p A=B by A=p A. The latter group
is reduced, so every divisible subgroup of A=B, in particular G=B, ought to be
contained in p A=B.
ut
Szele’s Theorem The next result provides a good illustration of how isomorphism
properties between a p-group and its basic subgroup are applied in practice.
Lemma 6.8 (Fuchs [3]). If the group G is an epimorphic image of the p-group A,
then every basic subgroup of G is an epimorphic image of every basic subgroup
of A.
Proof. The restriction of an epimorphism W A ! G to p n A is an epimorphism
n W p n A ! p n G which induces the epic map N n W p n A=p nC1 A ! p n G=p nC1 G.
Owing to (B), p n A=p nC1 A Š p n B=p nC1 B and p n G=p nC1 G Š p n C=p nC1 C are natural
isomorphisms (B; C denote the basic subgroups). Take into account that p n B=p nC1 B
is a direct sum of groups of order p, where the number of summands equals the
number of summands in B that are of order p nC1 , and the same holds for C.
The existence of an epimorphism p n B=p nC1 B ! p n C=p nC1 C implies an inequality
between the cardinalities of sets of components of order p nC1 in B and in C,
for every n. Considering that both B and C are †-cyclic groups, this suffices to
guarantee the existence of a desired epimorphism (argument with cardinalities is
needed).
ut
Incidentally, this theorem leads us to a very interesting corollary about †-cyclic
groups that are epic images of a p-group.
Corollary 6.9 (Szele [5]). If a †-cyclic group is an epimorphic image of a p-group
A, then it is an epic image of every basic subgroup of A.
Proof. Apply Lemma 6.8 with G D C.
ut
While we are still on this theme, let us reveal another piece of information on
basic subgroups of p-groups: a most surprising feature. (We reproduce Szele’s proof,
a shorter proof can be given by using large subgroups (Sect. 2 in Chapter 10.)
Theorem 6.10 (Szele [5]). In a p-group, basic subgroups are endomorphic images.