The number of generators in group ({1,2,3,4,5,6} x7) area

To find the number of generators in the group ({1,2,3,4,5,6} x 7), we need to understand what a generator is in the context of group theory.

Group Theory and Generators

In group theory, a group is a set of elements along with an operation that combines any two elements to form a third element. The operation must satisfy certain properties such as closure, associativity, identity element, and inverse element.

A generator in a group is an element that, when combined with itself multiple times using the group operation, can generate all the elements of the group.

The Group ({1,2,3,4,5,6} x 7)

The group ({1,2,3,4,5,6} x 7) represents the Cartesian product of the set {1,2,3,4,5,6} with the group of integers modulo 7. This means that each element in the Cartesian product is a pair consisting of an element from the set {1,2,3,4,5,6} and an element from the group of integers modulo 7.

Calculating the Generators

To calculate the number of generators in the group ({1,2,3,4,5,6} x 7), we can use the following formula:

Number of generators = φ(n)

where φ(n) represents Euler's totient function, which gives the number of positive integers less than or equal to n that are relatively prime to n.

In this case, n = 7 since we are considering the group of integers modulo 7. Therefore, we need to calculate φ(7).

The Euler's totient function φ(n) for a prime number n is simply n-1. Since 7 is a prime number, φ(7) = 7-1 = 6.

Therefore, the number of generators in the group ({1,2,3,4,5,6} x 7) is 6.

Conclusion

The correct answer is option 'C', which states that there are 2 generators in the group ({1,2,3,4,5,6} x 7). This is determined by calculating the Euler's totient function φ(7), which is equal to 6. The number of generators in the group is equal to φ(7), so there are 6 generators.