For an Euclidean groupG acting freely on the parameter space, we derive, among several noninformative priors, the reference priors of Berger-Bernardo and Chang-Eaves for our parameter of interest θ₁, a scalar maximal invariant parametric function. Identifying the nuisance parameter vector with the group element, we derive a simple structure of the information matrix which is used to obtain different noninformative priors. We compare these priors using the marginalization paradox and the probability-matching criteria. The Chang-Eaves and the Berger-Bernardo reference priors appear to be the most attractive choice. Several illustrative examples are considered.