Download Infinite Abelian groups by Laszlo Fuchs PDF
By Laszlo Fuchs
Fuchs L. endless Abelian teams, vol.2 (AP, 1973)(ISBN 0122696026)
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3. (Cutler [ I ] ) If pA is a direct sum of torsion-complete p-groups, then so is A . 4. (Hill [ l o ] )There exist a countable direct sum A of torsion-complete p-groups and a separable p-group C such that A[p] and C[p] are isometric vector spaces, but A and C fail to be isomorphic. [Hint: Ex. 1 5. By a contraction of a metric space is meant a mapping of the space into itself that does not increase distances. Show that if n is a contraction of the vector space A[p] of a pure-complete p-group A which is at the same time a projection, then Im n supports a summand of A .
A ) EB\B. x, is of order p and of height in - 1. If y E B\B is of order p , then its height must be in - 1 for some n, thus y is of the form . , 0,pin- = (0, . I, with c, E BAl and infinitely many nonvanishing coordinates; let its nonzero coordinates be pin-lc,, pinz-~cnz, pin,-l c n 3 ,.... Notice that (c,,) is a direct summand of BA, ,and hence of Bij @ . * . @ Bij+,In the latter group, 4,- b,, + p i n j + l - i n i b , j ++l ... + pinj+l-inj bnj+l-l also generates a direct summand of the same order, thus B has an automorphism a mapping c;, upon c,, for everyj.
The following two theorems state that some classes of groups have the exchange property. 1 (Crawley and J6nsson [I]). If the reduced part of A is a torsion group with bounded p-components, then A has the exchange property. XI. 34 SEPARABLE P-GROUPS In view of (c) and (e), it suffices to show that divisible groups and groups which are direct sums of copies of the same Z(p")have the exchange property. Let A be divisible and assume (1). Select a subgroup E of M which is maximal with respect to the properties: E = O i E i with E i 5 C i , and E n A = 0.