About primary decomposition of $\mathbb{Z}_7$

Question

I want to find the primary decomposition of $\mathbb{Z}_7^*$ under multiplication group action. So I know 7 is a prime number so it is a group of order 6. So possibilities are $\mathbb{Z}_7^*\cong \mathbb{Z}_2\oplus \mathbb{Z}_3$. However when I check the element $[a]\in \mathbb{Z}_7$ has order 6 by fermat's little theorem. I do not see how to decompose $\mathbb{Z}_7$ into $\mathbb{Z}_2\oplus \mathbb{Z}_3$ though I am expecting it is supposed to happen.

Answer 1

So you found a primitive root in $\Bbb Z_7$, i.e. which generates its multiplicative group.
Therefore, the multiplicative group $\Bbb Z_7^*$ is cylcic, hence $\cong\Bbb Z_6$.

Finally, the isomorphism $\Bbb Z_2\oplus\Bbb Z_3\ \to\ \Bbb Z_6$ can be given by $$(a,b)\mapsto 3a+2b\,.$$