Show that $C_3 \times C_3$ is not isomorphic to $C_9$ I am trying to show that $C_3 \times C_3$ is not isomorphic to $C_9$. I am new to group-theory so forgive my foolish intuitions. One natural thought I had was to show that they are of differing cardinality but intuitively, why can't we map the elements $c_1,c_2,c_3 \in C_3$ to $C_9$ like this:
$f: C_3 \times C_3 \rightarrow C_9$ defined by $f(c_i, c_j) = c_{i+j} \in C_9$.

Each isomorphism preserves the orders of elements of the respective groups. But $C_9$ has an element of order $9$ whereas $C_3\times C_3$ does not. Hence they are not isomorphic.
