Page 4 - ALGEBRA STRUCTURE cyclic group BY MIFTAHUL JANNAH (4193311004) MESP2019

Thus, it can be concluded that with respect to the addition operation of modulo 4 numbers forms
4
a group.

Let’s investigate the elements that are generator

Element 0

−1 1
1
0 = 0 0 −1 = (0 ) = (0) = 0
1
2
2
−1 2
0 = 0 + 0 = 0 0 −2 = (0 ) = (0) = 0 + 0 = 0
3
3
−1 3
0 = 0 + 0 = 0 0 −3 = (0 ) = (0) = 0 + 0 + 0 = 0
……………….. ………………..
……………….. ………………..

……………….. ………………..

{0 | ∈ } = {0}

Thus, 0 is not generator.

Element 1

1 = 1 1 −1 = (1 ) = (3) = 3
1
1
−1 1
1 = 1 + 1 = 2 1 −2 = (1 ) = (3) = 3 + 3 = 2
2
−1 2
2
−1 3
3
3
1 = 1 + 1 + 1 = 3 1 −3 = (1 ) = (3) = 3 + 3 + 3 = 1
1 = 1 + 1 + 1 + 1 = 0 1 −4 = (1 ) = (3) = 3 + 3 + 3 + 3 = 0
4
4
−1 4
−1 5
5
5
1 = 1 + 1 + 1 + 1 + 1 = 1 1 −5 = (1 ) = (3) = 3 + 3 + 3 + 3 + 3 = 3
……………….. ………………..
……………….. ………………..
……………….. ………………..

< 1 > = {1 | ∈ } =
4
Thus, 1 is generator

3