Finite totally k-closed groups1

For a positive integer k, a group G is said to be totally k-closed if in each of its faithful permutation representations, say on a set , G is the largest subgroup of Sym() which leaves invariant each of the G-orbits in the induced action on × × = k. We prove that every finite abelian group G is totally (n(G) + 1)- closed, but is not totally n(G)-closed, where n(G) is the number of invariant factors in the invariant factor decomposition of G. In particular, we prove that for each k ≥ 2 and each prime p, there are infinitely many finite abelian p-groups which are totally k-closed but not totally (k - 1)-closed. This result in the special case k = 2 is due to Abdollahi and Arezoomand. We pose several open questions about total k-closure.