Office_Shredder said:Try thinking about the automorphisms as mapping the generator of the group to another element
A cyclic group is a mathematical group that is generated by a single element. This means that the group is built by taking the powers of this element and combining them with the group operation. The resulting group is infinite if the element has infinite order, or finite if the element has finite order.
The number of automorphisms of a cyclic group is equal to the number of elements in the group. This is because every element in a cyclic group can be mapped to any other element by a unique automorphism. Therefore, the number of automorphisms is equal to the order of the group.
The order of a cyclic group is equal to the number of elements in the group. This is because a cyclic group is generated by a single element, and the order of this element determines the size of the group. For example, a cyclic group generated by an element of order 3 will have 3 elements.
Yes, a cyclic group can have an infinite number of automorphisms if the generating element has infinite order. This means that the group itself will also be infinite, and there will be no limit to the number of automorphisms that can be created.
Yes, there are a few special properties of the automorphisms of a cyclic group. One is that the identity automorphism is always included, as it maps every element to itself. Another is that the inverse of an automorphism is also an automorphism, as it can be seen as "undoing" the original automorphism. Additionally, the composition of two automorphisms is also an automorphism.