Page 6 - ALGEBRA STRUCTURE cyclic group BY MIFTAHUL JANNAH (4193311004) MESP2019
P. 6
Solution
1. The first axiom (closed property) is fulfilled because the all result of the operation are on
set Z.
2. The second axiom (associative property) in ordinary addition is fulfilled in integers.
3. The third axiom (identity element) is fulfilled ∃ 0 ∈ as the identity element because
4
∀ a ∈ fulfilled a * 0 = 0 * a = a
4
4. The fourth axiom (inverse element) is fulfilled, that is :
−2 the inverse is 2; −1 the inverse is 1; 0 the inverse is 0; 1 the inverse is −1, 2 the inverse
is −2
Therefore, it can be concluded that Z to ordinary addition operation formed a group.
Element 1
1 = 1 1 −1 = (1 ) = (−1) = −1
1
−1 1
1
2
−1 2
1 = 1 + 1 = 2 1 −2 = (1 ) = −1 + −1 = −2
−1 3
3
1 = 1 + 1 + 1 = 3 1 −3 = (1 ) = −1 + −1 + −1 = −3
1 = 1 + 1 + 1 + 1 = 4 1 −4 = (1 ) = −1 + −1 + −1 + −1 = −4
4
−1 4
−1 4
5
1 = 1 + 1 + 1 + 1 + 1 = 5 1 −5 = (1 ) = −1 + −1 + −1 + −1 + −1 = −5
……………….. ………………..
……………….. ………………..
……………….. ………………..
< −1 > = {−1 | ∈ }
= {… , −5, −4, −3, −2, −1,0,1,2,3,4,5, … }
=
Therefore, −1 is generator
So that it can be concluded that Z is a cyclic group, with < 1 > and < −1 >
Example 3
5
