利用交换群研究群在集合上的作用的像集

所有讨论都是在φ是群G在集合Ω上的作用这一前提下进行的,得出了群G是交换群与群Gφ是交换群之间的充分非必要条件,对其充分性加以了证明,并通过反例来说明其非必要性.考虑到群G在集合Ω上的作用与其逆作用在定义上的区别,得出群Gφ是交换群与群在集合上的作用φ是其逆作用,两者之间是充要条件,并加以了证明.

Studying Image Set of Action of Group on Set with Abelian Group

All questions are discussed under this premise which let φ be an action of group G on set Ω.A sufficient and unnecessary condition between that G is an Abelian group and that G^φ is an Abelian group has obtained,the proof of its sufficiency has given and some counterexamples are given to explain its non-necessity.Considering distinction on their definition between an action of group G on set Ω and its inverse action,a sufficient and necessary condition between that G^φ is an Abelian group and φ which is an action of group on set is its inverse action has also obtained and proved.