The Theory of Groups (Dover Books on Mathematics) by Marshall Hall

Author:Marshall Hall
Language: eng
Format: azw3
Publisher: Dover Publications
Published: 2018-01-10T00:00:00+00:00

13.2. Characters of Abelian Groups. Duality of Abelian Groups.

Given an arbitrary Abelian group A. A character χ of A is a homomorphism of A into the group R+ (mod 1). Thus our definition is

Here the addition a1 + a2 is the addition in A, the addition of the values of the characters is, of course, in R+ (mod 1). We shall also define an addition of characters. If χ1 and χ2 are two characters of A, we define

Then χ3 is also a character of A, since

We readily verify that if we use (13.2.2) to define an addition

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