Page 8 - ALGEBRA STRUCTURE cyclic group BY MIFTAHUL JANNAH (4193311004) MESP2019
P. 8
Thus, it can be concluded that U(10) for the multiplication operation modulo 10 forms a group.
Furthermore, it will be proved that U(10) is a cyclic group by showing that U(10) has an element
as a generator.
Element 1 Element 3
0
0
1 = 1 3 = 1
1
1
1 = 1 3 = 3
2
1 = 1.1 = 1 3 = 3.3 = 9
2
3
1 = 1.1.1 = 1 3 = 3.3.3 = 27
3
1 = 1.1.1.1 = 1 3 = 3.3.3.3 = 1
4
4
{1 | ∈ } = {1} < 3 > = {3 | ∈ } = {1,3,7,9}
a = 1 not generator a = 3 is generator
Element 7 Element 9
0
0
7 = 1 9 = 1
1
1
7 = 7 9 = 9
2
2
7 = 7.7 = 9 9 = 9.9 = 1
7 = 7.7.7 = 3 9 = 9.9.9 = 9
3
3
4
7 = 7.7.7.7 = 1 9 = 9.9.9.9 = 1
4
< 7 > = {7 | ∈ } = {1,3,7,9} < 9 > = {3 | ∈ } = {1,3,7,9}
a = 7 is generator a = 9 is not generator
2
1
0
because, < 3 > = {3 , 3 , 3 , 3 }
3
< 3 > = {1,3,7,9}
< 7 > = {7 , 7 , 7 , 7 }
0
1
2
3
< 7 > = {1,3,7,9}
7
