group

Question

if (G,*) is a cyclic group of order 97 , then number of generator of G is equal to ___

Comments

Just find the number of co-primes of 97 which are less than 97, as 97 itself is a prime all the numbers less than that are co-prime to it, had the question given as a group of order 96 then it would have been {1,3,5,7,11....}

Co-Primes:Two numbers are coprime if their highest common factor (or greatest common divisor if you must) is 1.

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Number of generators present in a cyclic group is equal to euler totient of order of group.

Euler totient value for prime numbers (let n) is equals to (n - 1).

Here, 97 is the order of group given in the question which is prime number.

Hence, number of generators in cyclic group is equal to (97-1) = 96.

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If order of group is prime number then number of generators is n-1 so the number of generators is 96.

Answer 1

97 is a prime number. Hence, number of generators = n-1 = 97-1 = 96

Answer 2

Finding generators of a cyclic group depends upon the order of the group. If the order of a group is 88 then the total number of generators of group GG is equal to positive integers less than 88 and co-prime to 88. The numbers 11, 33, 55, 77 are less than 8 and co-prime to 88, therefore if a is the generator of GG, then a3,a5,a7a3,a5,a7 are also generators of G.G. Hence there are four generators of G.

Similarly generators in the given group are 1,2,3,4….96   i.e. total 96 generators