Final answer:
The cyclic subgroup generated by an element a and the cyclic subgroup generated by its inverse a' in a group are the same because they consist of the same set of elements; any power of a can be represented as a corresponding inverse power of a' and vice versa.
Step-by-step explanation:
The question seems to have a typo and lacks clarity in the statement regarding the cyclic subgroup generated by a'. Assuming that we are discussing the cyclic subgroup generated by an element a in a group and comparing it to the cyclic subgroup generated by its inverse a', the cyclic subgroups are indeed the same due to the properties of group elements and their inverses.
Let's denote the cyclic subgroup generated by a as <a> and the cyclic subgroup generated by a' as <a'>. By definition, <a> includes all integral powers of a, that is, all elements of the form a^n for n being an integer. Similarly, <a'> will include all elements of the form (a')^n since a' is the inverse of a, (a')^n is equivalent to a^-n, which are also elements of the original subgroup <a>.
Thus, the cyclic subgroups generated by a and a' are equal because they consist of the same set of elements. This shows that in any group, the cyclic subgroup generated by any element and its inverse are the same, demonstrating an important property of group theory.