How did you get that? 25+ 25= 50= 20 (mod 30). 25+ 25+ 25= 75= 15 (mod 30). 25+ 25+ 25+ 25= 100= 10 (mod 30). 25+ 25+ 25+ 25+ 25= 125= 5 (mod 30). 25+25+ 25+ 25+ 25+ 25= 150= 0 (mod 50). 25+25+ 25+ 25+ 25+ 25+ 25= 175= 25 (mod 30). Those are your 6 elements.hitmeoff said:How do we go about finding the number of elements of a cyclic subgroup that's generated by an element in the main group. For example:
The subgroup Z30 generated by 25.
I would think this subgroup would be {0,1,5,25} but there's supposed to be 6 elements and not four. Whats going on?
A cyclic group is a mathematical structure that consists of a set of elements and a binary operation that combines any two elements to produce a third element. The group is said to be cyclic if there exists an element, called a generator, that can generate all the other elements in the group by repeated application of the operation.
The number of elements in a cyclic group is equal to the order of the group, which is the smallest positive integer n such that the generator raised to the power n gives the identity element of the group. In other words, the order of a cyclic group is the number of times the generator needs to be multiplied by itself to obtain the identity element.
No, the number of elements in a cyclic group depends on the order of the group, which can vary for different groups. However, if two cyclic groups have the same order, they are isomorphic, meaning they have the same structure and behave in the same way.
Yes, for a cyclic group of order n, the number of elements is given by Euler's totient function, φ(n). This function counts the number of positive integers less than n that are relatively prime to n, and is given by the formula φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk), where p1, p2, ..., pk are the distinct prime factors of n.
No, the number of elements in a cyclic group is always finite. This is because the order of the group, and therefore the number of elements, is determined by the generator and the binary operation, both of which are finite. However, the order of a cyclic group can be arbitrarily large.