✦ LIBER ✦

Affine complete Abelian groups

✍ Scribed by K. Kaarli

Publisher: John Wiley and Sons
Year: 1982
Tongue: English
Weight: 258 KB
Volume: 107
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.19821070118

No coin nor oath required. For personal study only.

✦ Synopsis

In this paper the classification of ABELian groups in terms of affine completeness initiated by LAUSH and NOBAUER in [2] and continued by N~BAUER in [3] and [4] is completed.

1. Preliminaries.

The word "group" will mean, throughout the paper, "ABELian group". If X is a subset of a group A. then (X) denotes the subgroup generated by X. Definition 1.1. Let A 1% a group and let X, F be subsets in A. Y SX. A mapping g~ : for any elements ui, wi< I' (i = I , . . . , n). Definition 1.2. Let A, X, Y be as in the definition 1.1. A mapping tp : X" -+A is called polynomial, if there exist integers kl, . . . , &* and an element caEA such that -A is cdled compatible on Y, if q(a,. . . . , W")--qJ(@I.. . . , w")E(u,-v~. -. .. a * -. , ! Q(Y~, . . a , ~~) = k l ~l + .

. . +knyn+a for any elements yi, . , . , yne 1'. \Ye denote the set of all mappings cp : X"-A which are compatible on X (polynomial on X ) by Pz(A) (PR,(A)). If X = A . then we use notations P ( A ) and P ( A ) instead of PA(A) and P",(A) correspondingly. Obviously, P&4) ZC'&(A) for any subset X and. in particular.

P ( A ) g C " ( A ) .

Now we give the main definition of this paper.

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