Abelian Groups - László Fuchs [Springer]

2 Finite and Finitely Generated Groups 81
thus 0 ¤ ra 0 Cb 0 D sg for some r; s 2 Z; b 0 2 B. This can happen only if .r; p/ D 1,
since pa 0 2 B.Butthenpa 0 ; ra 0 2 A implies a 0 2 A , a contradiction. ut
Fundamental Theorem on Finite Abelian Groups The first structure theorem
in the history of group theory was the famous Basis Theorem on finite abelian
groups.
Theorem 2.2 (Frobenius–Stickelberger [1]). A finite group is the direct sum of a
finite number of cyclic groups of prime power orders.
Proof. Thanks to Theorem 1.2 in Chapter 2, the proof reduces at once to p-groups.
In a finite p-group A ¤ 0, we select an element g of maximal order. By the preceding
lemma, A Dhgi ˚B for some subgroup B. SinceB has a smaller order than A, a
trivial induction completes the proof.
ut
There is a uniqueness theorem attached to the preceding result. Again, it suffices
to state it for p-groups.
Theorem 2.3. Two direct decompositions of a finite p-group A into cyclic groups
are isomorphic.
Proof. In a direct decomposition of A collect the cyclic summands of equal orders
into a single summand to obtain a courser decomposition A D B 1 ˚˚B k where
each B i is 0 or a direct sum of cyclic groups of fixed order p i . Evidently, p k 1 A D
p k 1 B k is the socle of B k ,itisanelementaryp-group, its dimension (as a Z=pZvector
space) tells us the number of cyclic components in B k . As this socle depends
only on A, the number of cyclic summands of order p k is independent of the choice
of the direct sum representation of A. In general, p i 1 AŒp D p i 1 B i Œp ˚ ˚
p i 1 B k Œp modulo p i AŒp D p i B iC1 Œp ˚ ˚ p i B k Œp is a Z=pZ-vector space Š
p i 1 B i Œp whose dimension is equal to the number of cyclic summands (of order p i )
in B i . The same argument shows that this dimension is independent of the choice of
the selected direct decomposition of A.
ut
Finitely Generated Groups We proceed to the discussion of finitely generated
groups. We start with a preliminary lemma.
Lemma 2.4 (Rado [1]). Assume A Dha 1 ;:::;a k i, and n 1 ;:::;n k are integers such
that gcdfn 1 ;:::;n k gD1. Then there exist elements b 1 ;:::;b k 2 A such that
A Dhb 1 ;:::;b k i with b 1 D n 1 a 1 CCn k a k :
Proof. We induct on n Djn 1 jCCjn k j: If n D 1,thenletb 1 D˙a i for any i,and
the claim is evident. Next let n >1.Thenatleasttwoofthen i are different from
0, say, jn 1 jjn 2 j >0. Since either jn 1 C n 2 j < jn 1 j or jn 1 n 2 j < jn 1 j,wehave
jn 1˙n 2 jCjn 2 jCCjn k j < n for one of the two signs. gcdfn 1˙n 2 ; n 2 ;:::;n k gD1
and the induction hypothesis imply that A Dha 1 ;:::;a k iDha 1 ; a 2 a 1 ;:::;a k iD
hb 1 ;:::;b k i with b 1 D .n 1 ˙ n 2 /a 1 C n 2 .a 2 a 1 / C n 3 a 3 CCn k a k D n 1 a 1 C
Cn k a k :
ut