  Most measurements are designed to tell you which of several alternatives have
occurred, but it is also possible to make measurements that eliminate
possibilities and tell you an alternative that did not occur. Measurements of
this type have proven useful in quantum foundations and in quantum
cryptography. Here we show how group theory can be used to design such
measurements. After some general considerations, we focus on the case of
measurements on two-qubit states that eliminate one state. We then move on to
construct measurements that eliminate two three-qubit states and four
four-qubit states. A condition that constrains the construction of elimination
measurements is then presented. Finally, in an appendix, we briefly consider
the case of elimination measurements with failure probabilities and an
elimination measurement on $n$-qubit states.
