Page 3 - ALGEBRA STRUCTURE cyclic group BY MIFTAHUL JANNAH (4193311004) MESP2019

Definition B-2

Group H in theorem B-1 is called cyclic sub group with generator a and denoted < a >

Definition B-3

A group G is said to be cyclic group if there is a ∈ G such that < a > = G

Example 1

= {0,1,2,3} ; * = addition operation of modulo 4
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Is < ∗> a cyclic group? If yes, specify the generator.
4,

Solution :

≠ ∅ (from the definition)
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Because the member of are finite, the result of the operation can be seen in Cayley table below
4

+ 0 1 2 3
4
0 0 1 2 3

1 1 2 3 0

2 2 3 0 1

3 3 0 1 2

By looking at the table, it is obtained :

1) The first axiom (closed property) is fulfilled because all the results of the operation are on
the set .
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2) The second axiom (associative property) on the addition of modulo 4 is fulfilled in integers,
because on it is also fulfilled.
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3) The third axiom (identity element) is fulfilled :

∃ 0 ∈ as the identity element because ∀ a ∈ fulfilled a * 0 = 0 * a = a
4
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4) The fourth axiom (inverse element) is fulfilled, that is
0 the inverse is 0; 1 the inverse is 3; and 2 the inverse is 2

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