A permutation group \(G\) on a set \(\Omega\) has type \(\{k\}\) (\(k\) a natural number) if every non-identity element has exactly \(k\) fixed points. It is an irredundant group of type \(\{k\}\) if in addition there are no orbits of size one, and no regular orbits. A partition of \(G\) is a set of non-trivial subgroups of \(G\) such that each non-identity element of \(G\) belongs to exactly one of these subgroups. It is non-trivial if it has more than one component, invariant if \(\pi\) is closed under conjugation, is called an f-partition if \(|N_G(X):X|\) is finite for each \(X\in\pi\), and is an f-partition of type \(m\) if this index divides \(m\) for each \(X\in\pi\). Extending to the infinite a result of Iwahori and Kondo, the author observes that if \(G\) is an irredundant group of type \(\{k\}\) then \(G\) has a non-trivial normal f-partition of type \(k!\). The elements of \(\pi\) are the non-trivial pointwise stabilisers of \(k\)-sets. Conversely, if \(G\) has a non-trivial normal f-partition of type \(m\), then \(G\) has a faithful irredundant permutation representation of type \(\{m\}\). A number of results about groups with partitions are proved. For example, if \(G\) is an infinite locally (soluble-by-finite) group with a non-trivial normal f-partition \(\pi\), then \(\pi\) has finite type \(m\) say, \(\pi\) contains the Fitting subgroup \(F(G)\), \(|G:F(G)|\) divides \(m\), and \(F(G)\) is nilpotent of class bounded in terms of \(m\) (and more is known). Also, infinite locally finite groups with a non-trivial normal f-partition are characterised. These results are combined to give very strong structural information on locally (soluble-by-finite) and on locally finite irredundant permutation groups of type \(\{k\}\). In particular, if \(G\) is a finite group of type \(\{k\}\) with no fixed set of size \(k\), then \(|G|\leq\max\{60,k!k(k!k-1)\}\). Results are also obtained about finite permutation groups of type \(\{k\}\) which are sharp, that is, have degree \(n\) and order \(n-k\). Finally, a permutation group is equidistant with Hamming distance \(d\) (\(d\) a positive integer) if, for all distinct \(x,y\in G\), \(xy^{-1}\) moves precisely \(d\) points. Extending results of Praeger, it is shown that if \(G\) is equidistant with Hamming distance \(d\), and \(G\) has no global fixed points, then the following hold: \(G\) has degree at most \(2d-2\), and \(|G|\) divides \(d\); \(G\) has at most \(d-1\) orbits; if \(G\) is transitive, then \(G\) is regular of degree \(d\).