Let $(M,\omega)$ be a symplectic manifold. If $M$ is noncompact, require every time-dependent Hamiltonian used to define a Hamiltonian diffeomorphism to have compact support in $M$ for each $t\in[0,1]$, with support contained in a compact subset of $M$ independent of $t$. Then the set $\operatorname{Ham}(M,\omega)$ of time-one maps of Hamiltonian isotopies is a subgroup of the identity component $\operatorname{Symp}_0(M,\omega)$ of the symplectomorphism group.