Page 12 - ALGEBRA STRUCTURE cyclic group BY MIFTAHUL JANNAH (4193311004) MESP2019

2. The second axiom (associative property) in the addition of modulo 6 is satisfied in integers,

because of that is satisfied.
6
3. The third axiom (identity element) is satisfied ∃0 ∈ as identity element because ∀ ∈
6
is satisfied a*0=0*a=a
6
4. The fourth axiom (inverse) is satisfied, namely :
0 the inverse is 0

1 the inverse is 5
2 the inverse is 4

3 the inverse is 3

Because all of axioms are satisfied, so it is proven that is group.
6

1 is the generator of group
6

1
1 = 1
1 = 1 + 1 = 2
2
1 = 1 + 1 + 1 = 3
3
1 = 1 + 1 + 1 + 1 = 4
4

5
1 = 1 + 1 + 1 + 1 + 1 = 5
1 = 1 + 1 + 1 + 1 + 1 + 1 = 6
6
7
1 = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 7

So it is obtained {(1) | ∈ } =

6
5 is the generator of group
6
1
5 = 5
5 = 5 + 5 = 10 = 4
2

3
5 = 5 + 5 + 5 = 15 = 3
4
5 = 5 + 5 + 5 + 5 = 20 = 2
5
5 = 5 + 5 + 5 + 5 + 5 = 25 = 1
So it is obtained {(5) | ∈ } =

6
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