This is the question:
Consider small groups.
(i) Show that a group of order 9 is isomorphic to Z9 or Z3 x Z3
(ii) List all groups of order at most 10 (up to isomorphism)
First, I show that G is abelian if the order of G is 9. Since |G|=9=3^2 and 3 is a prime number, then G is a p-group. Any p-group has nontrivial center Z(G). Then Z(G) has order 3 or 9. If the order of Z(G) has order 3, then Z(G) is a cyclic group. We can assume ...
This solution is comprised of a detailed explanation to show that a group of order 9 is isomorphic to Z9 or Z3 x Z3.