Page 5 - ALGEBRA STRUCTURE cyclic group BY MIFTAHUL JANNAH (4193311004) MESP2019

Element 2

1
−1 1
1
2 = 2 2 −1 = (2 ) = (2) = 2
2 = 2 + 2 = 4 = 0 2 −2 = (2 ) = (2) = 2 + 2 = 0
2
−1 2
2
3
−1 3
3
2 = 2 + 2 + 2 = 6 = 2 2 −3 = (2 ) = (2) = 2 + 2 + 2 = 2
−1 4
4
4
2 = 2 + 2 + 2 + 2 = 8 = 0 2 −4 = (2 ) = (2) = 2 + 2 + 2 + 2 = 0
5
5
−1 5
2 = 2 + 2 + 2 + 2 + 2 = 10 = 2 2 −5 = (2 ) = (2) = 2 + 2 + 2 + 2 + 2 = 2
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{2 | ∈ } = {0,2}
Thus, 2 is not generator

Element 3

−1 −1
1
3 = 3 3 −1 = (3 ) = 1
2
3 = 3 + 3 = 2 3 −2 = (3 ) = (1) = 1 + 1 = 2
−1 2
2
−1 3
3 = 3 + 3 + 3 = 1 3 −3 = (3 ) = (1) = 1 + 1 + 1 = 3
3
3
−1 4
4
4
3 = 3 + 3 + 3 + 3 = 0 3 −4 = (3 ) = (1) = 1 + 1 + 1 + 1 = 0
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< 3 > = {3 | ∈ } =

4
So that is cyclic group
4

Example 2

Z = Set of integers

With ordinary addition operation < , +> is group

4