Cyclic group | Group Theory | sinhala explanation

#Studies_Plus_DotLK #cyclic group #group theory
Cyclic group | Group Theory | sinhala explanation

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Cyclic group

A group G is said to be cyclic if there exists a ∈ G s.t
G = {𝒂^𝒏;n ∈Z}.
In this case we say that G is generated by a. i.e ”a” is a generator of G
The elements of G will be of the form
..., 𝒂^(−𝟑), 𝒂^(−𝟐), 𝒂^(−𝟏), 𝒂^𝟎, 𝒂^𝟏, 𝒂^𝟐, 𝒂^𝟑,...

Suppose that G is a group w.r.t addition.
If we can find an element a ∈ G s.t
G = {ra;r ∈Z} then G is cyclic .
The elements of G will be of the form
...,−3a ,−2a ,−1a , 0 , 1a , 2a ,3a,...

Example 01:
The multiplicative group G={ 1,-1,i,-i }

Example 02:
(Z,+)

Example 03:
G = ({ 1,3,7,9 }, "⊗" _𝟏𝟎)

Example 04:
G = ({ 0,1,2,3,4,5 }, "⊕" _𝟔)

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