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For endomorphisms  and  of a group G, define  +  by
( + )(x) =(x)(x).
Note:  +  need not be an endomorphism.
Lemma 6.33. If  and  are normal nilpotent endomorphisms of a finite indecomposable
group, and  +  is an endomorphism, then  +  is a normal nilpotent endomorphism.
Proof. It is obvious that  +  is normal. If it is an automorphism, then there exists a 
such that ( + ) %  = id. Set  =  and  = . Then  +  = id, i.e.,
 (x-1) (x-1) =x-1 =!  (x) (x) =x =  (x) (x) =!   =   .
Hence  +  =  +  . Therefore the subring of End(G) generated by  and  is commu-
tative. Because  and  are nilpotent, so also are  and  . Hence
m
( +  )m =  m +  m-1 + +  m
1
is zero for m sufficiently large.
GROUP THEORY 57
Proof. of Krull-Schmidt. Suppose G = G1 G2 Gs and G = H1 H2 Ht.
Write
i 
i
! !
Gi ! G1 G2 ׷ Gs, Hi ! H1 H2 ׷ Ht.
i i
Consider 1 11 + 1 21 + =idG . Not all terms in the sum are nilpotent, and so,
1 2 1
after possibly renumbering the groups, we may suppose that the first is an automorphism,
say  = 1 11 = -1. Thus (omitting subscripts)
1
    
(G1 ! G1 ! G ! H1 ! G ! G1) =idG .
1
Consider

   
(H1 ! G ! G1 ! G1 ! G ! H1) =.
Check % =  (use above factorization of idG ), and so  = id or 0. The second is impossible,
1
because  occurs in idG % idG . Therefore,  =idH . Hence 1 and 11 are isomorphisms.
1 1 1 1
On the other hand, 1(H2 ׷ ) = 1, but 11 =? is injective on G1. We conclude that
G1 )" (H2 ... Ht) =1. Henc e G1(H2 ׷ ) H" G1 (H2 ׷ ), and by counting, we see
that G = G1 H2 ׷ .
Repeat the argument.
Remark 6.34. (a) The Krull-Schmidt theorem holds also for an infinite group provided it
satisfies both chain conditions on subgroups, i.e., ascending and descending sequences of
subgroups of G become stationary. (See Rotman 6.33.)
(b) The Krull-Schmidt theorem also holds for groups with operators. For example, let
Aut(G) operate on G; then the subgroups in the statement of the theorem will all be char-
acteristic.
(c) When applied to a finite abelian group, the theorem shows that the groups Cm in
i
a decomposition G = Cm ... Cm are uniquely determined up to isomorphism (and
1 r
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