交換群者,乘法合交換律之群也,亦曰阿貝爾群。可以加法謂之,則其單位元曰「零」,逆改曰負。
初,阿貝爾以置換之不可交換,證五次多項式無簡約解(阿貝爾–魯芬尼定理)。然天妒英才,終年不過三十。泰西疇人念其功績,名交換群為阿貝爾群也。
整數合加法,分數合加法,正數合乘法,非零複數合乘法,同餘集合加法,俱交換群也。
取一集合,含此集之群最小者,曰集之生成群。有限集之生成群,曰有限生成群。
「有限生成交換群基本定理」云:有限生成交換群,乃整數群與同餘群之直積也。(「 」)
因而得證,有限交換群乃同餘群之直積也。
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