Q:

Element of the group raise to power 1.

If I have a group with operation modulo 2 containing element 3. What would be the value of $3^1$?
Will it be 3 or 1 as 3 modulo 2=1 ?

A:

Formally, the group $\mathbb Z_2$ contains $2$ elements: 
$\overline 0 = 0 + 2\mathbb Z = \{ 0+2z: z \in \mathbb Z \}$ and
$\overline 1 = 1 + 2\mathbb Z = \{ 1+2z: z \in \mathbb Z \}$. In that sense, 
$\overline 1 = \overline 3 = \left(\overline 1 \right)^1 =
 \left(\overline 3 \right)^1$.
Edit due to @drhab's comments
$\mathbb Z_2$ is only a group with respect to addition. The structure
$[\mathbb Z_2, +]$ is a group and the structure $[\mathbb Z_2, \cdot]$ is not because, for example, $\overline 0$ does not have a multiplicative inverse.
For the group $[\mathbb Z_2, +]$, the notation $\overline x^1$ makes no sense.  But $1\overline x$ does and $1\overline x = \overline x$.
However, the structure $[\mathbb Z_2, +, \cdot]$ is a ring and, because $2$ is a prime number, it is even a field. In that sense,
$\overline 1^1 =\overline 3^1 = \overline 1$ but $\overline 0^1$ is still undefined.

