= Solution
Work in the initial rest frame of the free <electron>. If it absorbed a <photon> of positive <energy> $E_\gamma$, <conservation of momentum> would give the final electron <momentum> magnitude $p_f=E_\gamma/c$, while <conservation of energy> would give $E_f=m_ec^2+E_\gamma$. But the final <electron> must obey the relativistic <energy–momentum relation>,
$$
E_f^2=m_e^2c^4+p_f^2c^2=m_e^2c^4+E_\gamma^2.
$$
Combining these statements gives
$$
(m_ec^2+E_\gamma)^2=m_e^2c^4+E_\gamma^2
\quad\Longrightarrow\quad 2m_ec^2E_\gamma=0,
$$
which contradicts $E_\gamma>0$. Thus
$$
\boxed{\text{An isolated free electron cannot absorb a single real photon.}}
$$
The initial rest frame always exists for an <electron>, so this excludes the process in every inertial frame. <Compton scattering> is allowed because an outgoing <photon> carries additional <energy> and <momentum>. Absorption by an <electron> bound in matter is also possible because the rest of the system can share recoil.
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