问答题
设G={[1]，[2]，[3]，[4]，[5]，[6]}，G上的二元运算×
7
如表5-36所示．(G，×
7
)是循环群吗?若是，请找出它的生成元．
表5-36
×7
[1]
[2]
[3]
[4]
[5]
[6]
[1]
[1]
[2]
[3]
[4]
[5]
[6]
[2]
[2]
[4]
[6]
[1]
[3]
[5]
[3]
[3]
[6]
[2]
[5]
[1]
[4]
[4]
[4]
[1]
[5]
[2]
[6]
[3]
[5]
[5]
[3]
[1]
[6]
[4]
[2]
[6]
[6]
[5]
[4]
[3]
[2]
[1]
【正确答案】
显然，(G，×
7
)是封闭的，具有结合律．[1]为它的单位元，[2]与[4]、[3]与[5]互为逆元，[6]的逆元是[6]．故(G，×
7
)是群．
通过观察可知：
[3]
2
=[3]×
7
[3]=[2]，[3]
3
=[3]×
7
[3]
2
=[3]×
7
[2]=[6]，
[3]
4
=[3]×
7
[3]
3
=[3]×
7
[6]=[4]，[3]
5
=[3]×
7
[3]
4
=[3]×
7
[4]=[5]，
[3]
6
=[3]×
7
[3]
5
=[3]
5
×
7
[5]=[1]，
所以，[3]是群(G，×7)的生成元．
【答案解析】
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