Multiplying elements of a multiplicative cyclic group

Question

This is about correct operation in a multiplicative cyclic group. Say we have the group $\mathbb Z_4^{\times} = \{1,2,3\}$, if we multiply the elements, say $3\cdot 2\cdot 2 $ we get $12 \mod 4 = 0$ but $0\notin \mathbb Z_4^{\times}$ so that can't be right, so what is the correct result of this operation?

Answer 1

You're using $^\times$ in correctly. $S^\times$ does not mean "remove zero". It means "remove all non units". In some contexts this is the same, because your set is a field, where every nonzero element is invertible.

Definition: $x\in R$ is a unit (of $R$) if $xy=1_R$ has a solution. Another way of saying this is to ask for the invertible elements.

Definition: Let $R$ be a ring (with identity). $R^x\subset R$ is the set of all units of $R$. It forms a group under the multiplication operation of $R$.

Examples: $$\mathbb{Z}^\times=\{-1,1\}$$ $$\mathbb{Z}_m^\times=\{a\in\mathbb{Z}_m:(a,m)=1\}$$ $$\mathbb{Z}[i]^\times=\{1,i,-1,-i\}$$

In this specific case, $2$ is not an element of the group because it is not a unit, i.e. The equation $2x=1$ has no solutions. The correct unit group is $\{1,3\}\sim C_2$.

Answer 2

We have $\mathbb{Z}_4^\times = \{1,3\}$, since $2$ is not a unit in $\mathbb{Z}_4$.