In which of the following decay processes does the element remain...

In $\gamma$-decay, an excited nucleus transitions to a lower energy state by emitting a gamma-ray photon.

The atomic number $Z$ and mass number $A$ of the nucleus remain unchanged during this process.

Since the atomic number $Z$ determines the identity of the element, the element remains unchanged.

In $\alpha$-decay, $Z$ decreases by $2$.

In $\beta^-$-decay, $Z$ increases by $1$.

In $\beta^+$-decay, $Z$ decreases by $1$.

Therefore, the element remains unchanged only in $\gamma$-decay.