Abelian Groups - László Fuchs [Springer]

4 1 Fundamentals
Theorem 1.2. In a group A, the set T of elements of finite order is a subgroup. T is
a torsion group and A=T is torsion-free.
Proof. Since 0 2 T,thesetT is not empty. If a; b 2 T,i.e.ma D 0; nb D 0 for some
m; n 2 N, thenmn.a b/ D 0, andsoa b 2 T. Thus T is a torsion subgroup. To
show that A=T is torsion-free, let a C T be a coset of finite order, i.e. m.a C T/ T
for some m 2 N. This means that ma 2 T, son.ma/ D 0 for some n 2 N. Thus a is
of finite order, i.e. a 2 T, anda C T D T is the zero of A=T. Consequently, A=T is
torsion-free.
We shall call T the (maximal) torsion subgroup or the torsion part of A, and
weshalldenoteitbyt.A/ or tA. (If we refer to the torsion subgroup of A, thenwe
always mean the maximal torsion subgroup.)
The following notations are typical for abelian groups; they will be used all
the time without explanation. Given a group A and an integer n 2 N, definethe
subgroups:
nA Dfna j a 2 Ag and AŒn Dfa 2 A j na D 0g:
Thus b 2 nA if and only if the equation nx D b has a solution for x in A, and
c 2 AŒn exactly if o.c/jn. A fundamental concept is the pure subgroup. Purity will
be discussed in Chapter 5, here we just state the definition: a subgroup G of A is
a pure subgroup if nG D G \ nA for every n 2 N, i.e. if whenever the equation
nx D g 2 G admits a solution in A for x, then it is also solvable in G.
Ulm Subgroups The first Ulm subgroup of A is defined as
A 1 D\ n2N nA:
The second Ulm subgroup is A 2 D .A 1 / 1 , etc. We shall also need the th Ulm
subgroups for ordinals ; these are defined transfinitely by the rules: A C1 D .A / 1 ,
and A D\ < A if is a limit ordinal (see Sect. 4). (We view A D A 0 .) The
Ulm length of A is the smallest cardinal such that A C1 D A (which exists by
cardinality reason). The th Ulm factor of A is the factor group
A D A =A C1 :
Given a 2 A, the largest non-negative integer n for which the equation p n x D a
is solvable for x 2 A is said to be the p-height h p .a/ of a. If this equation is solvable
for every integer n >0,thena is of infinite p-height, h p .a/ D1. The element 0
is of infinite height at every prime. If it is completely clear from the context which
prime p is meant, then we may simply talk of the height of a and write h.a/. (In
Chapter 11, we shall discuss transfinite heights.)
The socle s.A/ of a group A is the set (actually the subgroup) of all the elements
a 2 A such that o.a/ is a square-free integer. If s.A/ D A, A is called an elementary
group. If A is a p-group, then s.A/ D AŒp.