Abelian Groups - László Fuchs [Springer]

656 17 Automorphism Groups
Example 1.3. As End Z.p 1 / Š J p , it follows that Aut Z.p 1 / is isomorphic to the multiplicative
group of p-adic units. (This group will be explicitly described in Sect. 8 in Chapter 18.) The same
holds for Aut J p .
Example 1.4. Let R be a rational group of type t D .k 1 ;:::;k n ;:::/.Thena 7! p n a .a 2 R/ is
an automorphism of R if and only if k n D1. Since each endomorphism is a multiplication by a
non-zero rational number, it follows that Aut R must be isomorphic to the multiplicative group of
all rational numbers whose nominators and denominators are divisible by primes p n only for which
k n D1. Thus Aut R is isomorphic to the additive group which is the direct sum of Z.2/ and as
many copies of Z as the number of k n D1in t.
Example 1.5. Let A be an elementary p-group of order p m .ThenA is an m-dimensional vector
space over the prime field of characteristic p. Since automorphisms of A are linear transformations
of the vector space, Aut A is isomorphic to the general linear group GL.m; p/ (non-commutative).
Elementary Facts We list below some facts that we have found useful to keep in
mind.
(a) If C is a characteristic subgroup of A and ˛ 2 Aut A, then ˛ Cisan
automorphism of C, and a C C 7! ˛a C C .a 2 A/ is an automorphism of
A=C.
(b) An isomorphism W A ! C between groups induces an isomorphism W
Aut A ! Aut C via .˛/ D ˛ 1 .
(c) If A is a direct sum, and if we represent the endomorphisms by matrices, then
the automorphisms correspond to the invertible matrices.
(d) If A D B ˚ C, then Aut B may be regarded as a subgroup of Aut A, identified
with the set of all ˛ 2 Aut A satisfying ˛ C D 1 C . (This identification depends
on the choice of the complement C.)
(e) If A D˚i2I A i , then under the identification indicated in (d),eachofAut A i may
be considered as a subgroup of Aut A. The cartesian product Q i2I Aut A i is a
subgroup of Aut A, it consists of those ˛ 2 Aut A that carry each A i into itself.
If all the A i are characteristic subgroups in A, then Aut A Š Q i2I Aut A i. Hence:
(f) If A is a torsion group, then Aut A Š Q p Aut A p.
(g) Every automorphism of p n A extends to an automorphism of A. It suffices to
prove this for n D 1, andwhenA has no summand of order p. Let be an
automorphism of pA. Iffb i g i2I is a p-basis of A, then write a 2 A in the form
a D P m
iD1 k ib i C pc with k i 2 Z n pZ and c 2 A. Note that the terms k i b i
and pc are uniquely determined by a. For each i 2 I, choose a i 2 A such that
pa i D .pb i / 2 pA, and define W A ! A by
.a/ D
mX
k i a i C .pc/:
iD1
It is clear that is an endomorphism of A such that pA D . To see that
is monic, assume .a/ D 0 for some a 2 A. Then.pa/ D 0, sopa D 0.
Hypothesis on A implies a 2 pA, therefore, .a/ D .a/ D 0 is impossible.
Finally, to show that is epic, for a given x 2 A pick a y 2 A such that .py/ D
px. Thenx .y/ is of order p, thus x .y/ D pc for some c 2 A. Ifz 2 A
satisfies pz D .pc/,thenx D .y C z/. Consequently, 2 Aut A.