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Cyclic group

 Def. Cyclic group : 

A group G is said to be cyclic if there exists an element aG such that every elementof G is of the form a^n, where n is an integer. The element ‘a’ is then , called a generator of G and we

write : G = < a > or (a) 

Results on cyclic groups : 

1. Every cyclic group is abelian.

2. Every subgroup of a cyclic group is cyclic.

3. If ‘a’ is a generator of a cyclic group G, then a^-1is also a generator of G.

4. The order of a cyclic group is equal to the order of its generator.

5. If a finite group of order n contains an element of order n then the group must be cyclic.

6. Every group of prime order is cyclic .

7. Every infinite cyclic group has exactly two generators.

8. The number of generators of a finite cyclic group of order n is. Ø(n ), where Ø denotes the Euler’s Ø function.

9. Let G be a finite cyclic group such that o(G)= n and G=<a> then a^m will be the generator of G if and only if gcd(n, m)

10. A group of prime order has no proper subgroups.

11. A non trivial group G which has no proper subgroups must be a group of prime order.

12.Converse of Lagrange’s theorem is true for finite cyclic group. In words, let G< a > be a finite cyclicgroup such that o(G) = n. Let d/ n then G has a subgroup of order d. Further the subgroup of order d is unique and this subgroup is given by <a^n/d> .

13 Total number of subgroups of a finite cyclic group of order n is t( n)  , the number of divisors of n.

14. Let G be cyclic group of order n and d be a positive integer which divides n, then G has Ø(d ) elements of order d.

15. Let G be a finite group and d be a positive integer which divides o(G) then number of cyclic subgroups of order d is number of elements of order d is (no of elements of order d)/Ø(d)

16 any two cyclic group of the same order are isomorphic

17 every group whose order is a prime p is necessary cyclic 2,3,4,5.......

18 v4 is not cyclic but abelian

19 v4 is smallest non cyclic group

20 A homorophic image of a cyçlic group is cyclic

21 Direct product of cyclic group is always a ablian group. Cm*cn= G

Cm*cn is cyclic if and only if(m, n)=1


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