Pfanzagl (1972) has shown that under suitable regularity conditions a family of probability measures which is generated by a transformation group and which for some sample size greater than one admits a sufficient statistic which is continuous, real-valued, and equivariant, is equivalent to the location parameter family of normal distributions or to a scale parameter family of Gamma distributions. This was proved under the assumption that the transformation group is Abelian. In this not commutativity of the group is replaced by local compactness.
"Note on the Paper "Transformation Groups and Sufficient Statistics" by J. Pfanzagl." Ann. Statist. 3 (2) 478 - 482, March, 1975. https://doi.org/10.1214/aos/1176343075