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This paper provides a bridge between two active areas of research, thespectrum (set of element orders) and the power graph of a finite group.  The order sequence of a finite group $G$ is the list of orders of elements ofthe group, arranged in non-decreasing order. Order sequences of groups of order$n$ are ordered by elementwise domination, forming a partially ordered set. Weprove a number of results about this poset, among them the following.  Abelian groups are uniquely determined by their order sequences, and theposet of order sequences of abelian groups of order $p^n$ is naturallyisomorphic to the (well-studied) poset of partitions of $n$ with its naturalpartial order.  If there exists a non-nilpotent group of order $n$, then there exists such agroup whose order sequence is dominated by the order sequence of any nilpotentgroup of order $n$.  There is a product operation on finite ordered sequences, defined by formingall products and sorting them into non-decreasing order. The product of ordersequences of groups $G$ and $H$ is the order sequence of a group if and only if$|G|$ and $|H|$ are coprime.  The paper concludes with a number of open problems.

On the order sequence of a group