Page 2 - ALGEBRA STRUCTURE cyclic group BY MIFTAHUL JANNAH (4193311004) MESP2019

Cyclic Group

Definition B-1

Let G group with operation ∗, ∀a, b ∈ G and n, m ∈ Z then :

= a a a……………. A (m factor)

=
∗ +
−1
)
− = ( ) = ( −1
0
= c (Identity element)

Theorem B-1

Let < ,∗> group and ∈ then H = { |n ∈ Z} is the smallest subgroup of G which contains

a

Proof

We use theorem A-3 about subgroup

0
1) H ≠  because for n = 0 ∈ Z then = e ∈ H
2) H  G (from the definition of H)

3) Closed property

Take any , ∈ then according to the term of membership of H then ∃ m, n ∈ Z ∋ p =

and q =
It will be shown p ∗ q ∈ H
∗ = ∗

= + , m + n ∈ Z

= + ∈ H
4) Identity property (e ∈ G then e ∈ H)

Because G group then e ∈ G (identity element)
0
0
e = , ∈ then e = ∈ H
5) Inverse property

Take any p ∈ H then ∃ m ∈ Z ∋ p = , because −m ∈ Z so that p −1 = a −m ∈ H
Thus, the three conditions (3,4,5) are fulfilled, so it is proven that H ≤ G

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