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Consider the following statements:
S1: Every cyclic group is Abelian group.
S2: Every Abelian group is cyclic group.
S3: Cyclic group of order 10 have 4 generators.
Which of the following is true?

in Graph Theory by Loyal (5.7k points) | 87 views

1 Answer

+2 votes
s1 is true

s2 is false

s3 is true
by Junior (663 points)
+1
S3 is true 10 has 4 generator

10=2*5

phi(10)=1*4

phi(5)=5^1-5^0

=4

phi(2)=1
+1
thanks @akshat sharma actually i thought there is only one statement true
0

@akshat sharma

How you have applied the above things. I do not really get.

For finding the number of generators of the cyclic group, I know that we need to find the number of prime factors but what's after that?

+1

this is the theorem @

if a cyclic group Gis generated by an element a of order n,then $a^{m}$ is a generator of G if and only if the greatest common divisor of m  and n is 1 that is m & n are relatively prime

 cyclic group of order 10 

now in this 1,3,7,9 are the nos which are relatively prime to 10

hence 4 generators!!

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