A group automorphism is a function that maps a group onto itself while preserving the group operation. This means that the elements of the group are rearranged in a way that the group operation remains unchanged.
A subgroup is a subset of a group that also forms a group under the same operation. A group automorphism is a function that preserves the group operation, but it does not necessarily form a subgroup. In other words, a group automorphism is a rearrangement of the elements in a group, while a subgroup is a subset of the original group.
A group automorphism is not always a subgroup because it may not satisfy the closure property, which is necessary for a subset to be a subgroup. This means that the result of applying the automorphism to two elements in the group may not always be an element of the subset.
No, a group automorphism cannot be a proper subgroup. A proper subgroup is a subset of a group that is also a subgroup, meaning that it contains all the necessary elements for the operation to be closed. However, a group automorphism only rearranges the elements in a group and does not necessarily form a subgroup.
An example of a group automorphism that is not a subgroup is the identity automorphism, which maps each element in a group to itself. This is because it does not change the group operation, but it does not form a subgroup because it does not satisfy the closure property.