The subgroup generated by {4,6} in Z12, denoted Z12, is the smallest subgroup of the group Z12 that contains the elements 4 and 6. It is formed by taking the multiples of 4 and 6, and then taking the common elements between the two sets.
The subgroup generated by {4,6} in Z12 contains 3 elements: 0, 4, and 8. This is because the common elements between the multiples of 4 and 6 in Z12 are 0, 4, 8, 12, 16, 20, etc. However, since we are working in Z12, all elements greater than 12 are equivalent to a number less than 12, so only 0, 4, and 8 are distinct elements in the subgroup.
The order of the subgroup generated by {4,6} in Z12 is 3. This is because there are 3 distinct elements in the subgroup: 0, 4, and 8.
Yes, the subgroup generated by {4,6} in Z12 is a cyclic subgroup. This is because it is generated by a single element, 4, which can be used to generate all other elements in the subgroup through its powers.
The subgroup generated by {4,6} in Z12 is a subset of the group Z12. This means that all elements in the subgroup are also elements of the group Z12. Additionally, the subgroup inherits the group operation (addition) from the group Z12, and is closed under this operation.