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1. Finitely Generated abelian group   Math 402 Discussant : SHIELA JOY N. DECENA MAT- Mathematics

2.   A group is a set together with an operation that meets the following set of rules:

3. Closure: For any  a  and  b  in the group, the result of  a  *  b  should also be in the group. Associativity: For any  a ,  b , and  c  in the group, the operation * (multiplication) must be associative. That is, we must have the following: ( a  *  b ) *  c  =  a  * ( b  *  c )

4. Identity: There must exist an identity element  e  in the group that satisfies this requirement for all other elements  a  in the group: a  *  e  =  a  and  e  *  a  =  a Inverse: For every  a  in the group, there must exist another element  b  in the group such that satisfies this relation: a  *  b  =  e  and  b  *  a  =  e

5. Commutativity : For every  a  and  b  in the group, the following must hold: a  *  b  =  b  *  a

6. A  F initely G enerated A belian Group  is an abelian group,  G , for which there exists finitely many elements  g 1 ,  g 2 , ….,  g n  in  G , such that every  g  in  G  can be written in this form: g  =  a 1 g 1  +  a 2 g 2  +...+  a n g n where   a 1 ,...,  a n  are integers

7. A finitely generated abelian group is a special type of abelian group. it is an abelian group that has a finite number of generators.   A generator is defined as an element that can create every group element through repeated binary operation application.

8. Example of a Finitely Generated Abelian Group The set Z/12Z under addition is a finitely generated abelian group with two generators: 1 and 5 (mod 12).      

9. Properties of Finitely Generated Abelian Group Finitely generated abelian groups have the following properties: Every finitely generated A belian group has a generating set Abelian groups are not always finite There are infinitely many finitely generated A belian groups of size 8 Every finite group is A belian .