Abelian Groups - László Fuchs [Springer]

664 17 Automorphism Groups
(2) If A is a p-group of order p m ,thenj Aut Aj is divisible by p m 1 .
(3) (Boyer, E. Walker, Khabbaz, Mader, Hill)
(a) If A is an infinite reduced p-group, then j Aut Aj D2 jBj ,whereB is a basic
subgroup of A.
(b) If A is a non-reduced p-group, then j Aut Aj D2 jAj .
(4) (Baer) The subgroup of elements in a p-group A that are left fixed under all
automorphisms of A is not f0g if and only if p D 2 and A has a unique element
g of order 2 of maximal height (in which case this subgroup is hgi).
(5) (Baer) Let A be a separable p-group and p 3. The set of elements of A left
fixed under all automorphisms that fix a subgroup C elementwise is C ,the
closure of C in the p-adic topology. (If p D 2, the exceptional subgroup hgi of
the preceding example, if exists, must be adjoined to C .)
(6) (Hill) Let A be a †-cyclic p-group. For n 1, every automorphism of AŒp n
that preserves heights (computed in A) extends to an automorphism of A.
[Hint: extend it to AŒp nC1 .]
(7) (Megibben) Every automorphism of a large subgroup of a p-group A that
preserves heights (computed in A) extends to an automorphism of A.
(8) If A is a p-group, and ˛ 2 Aut A induces the identity on p A=p C1 A for some
ordinal , then it also induces the identity on p Cn A=p CnC1 A for all integers
n 1.
(9) (Mader) Let A be a divisible p-group, and † n the normal subgroup of Aut A
that consists of those ˛ 2 Aut A which leave AŒp n element-wise fixed. Show
that .Aut A/=† n Š Aut AŒp n .
(10) (Hausen) The automorphism group of a finitely cogenerated group is residually
finite (i.e., every element ¤ 1 is contained in a normal subgroup of finite
index). [Hint: consider ˛ 2 Aut A fixing AŒp n element-wise.]
(11) (Tarwater) Let A be a homogeneous †-cyclic p-group. It has an automorphism
˛ carrying the subgroup G into the subgroup H if and only if G Š H and
AŒp=GŒp Š AŒp=HŒp.
(12) (Baer) For a p-group A with p 3, all the torsion subgroups of Aut A are finite
exactly if A is finitely cogenerated. [Hint: an infinity of independent summands
yields an infinite torsion group in the centralizer of the system of involutions;
conversely, reduce to divisible groups and consider matrix representation.]
(13) Suppose A D B˚C,andB is fully invariant in A. Then Aut A is the semidirect
product of the stabilizer of the chain 0