Determine the Residue group $ R_{10} $ whose binary operation is $\times \ mod \ 10$. Show that:
- $ R_{10} $ has only one element of order 2
- $ R_{10} \cong C_4 $
I found that the group $ R_{10} = \{1,3,7,9\} $ trivially. For no.1 I simply determined the order of each element and showed that the only element of order 2 is 9.
For no.2 I constructed Cayley table for $ C_4 $ and for $R_{10} $ . By observation I constructed a function $\psi C_4 \to R_{10} $ as follows: $$e \mapsto 1 \\ g \mapsto 3 \\ g^2 \mapsto 9 \\ g^3 \mapsto 7 $$
This function is clearly a bijection as every element is mapped to a unique element and the function is well-defined.
What is the easiest way to show that this is an homomorphism? I thought of taking any 2 elements $a,b \in C_4 $ and showing that $$\psi (ab)=\psi(a) \psi(b)$$ but this would leave me with $4C2=12$ combinations to consider.
Nevertheless is there an easier way to show that $C_4 \cong R_{10} $ althogether?