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 Ω, G is the largest subgroup of Sym(Ω) which leaves invariant each of the Gorbits in the induced action on Ω × • • • × Ω = Ω k . We prove that every 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.

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