The case k = 1 is the Erdős-Ginzburg-Ziv Theorem. Take a look at this Wikipedia article which has links to some surveys of the large literature of similar results. (The particular generalizations I'm aware of ask for a set of vectors summing to 0 whose size is the cardinality of the group, not its exponent.)

The case k = 2 is the Kemnitz conjecture, as pointed out by Kristal, and as Ricky mentioned (and also as mentioned on the Wikipedia page I linked to) it was proved by Reiher in 2003.

The case k = 1 is the Erdős-Ginzburg-Ziv Theorem. Take a look at this Wikipedia article which has links to some surveys of the large literature of similar results. (The particular generalizations I'm aware of ask for a set of vectors summing to 0 whose size is the cardinality of the group, not its exponent.)

The case k = 2 is the Kemnitz conjecture, as pointed out by Kristal, and as Ricky mentioned (and also as mentioned on the Wikipedia page I linked to) it was proved by Reiher in 2003.

The case k = 1 is the Erdős-Ginzburg-Ziv Theorem. Take a look at this Wikipedia article which has links to some surveys of the large literature of similar results. (The particular generalizations I'm aware of ask for a set of vectors summing to 0 whose size is the cardinality of the group, not its exponent.)

The case k = 1 is the Erdős-Ginzburg-Ziv Theorem. Take a look at this Wikipedia article which has links to some surveys of the large literature of similar results. (The particular generalizations I'm aware of ask for a set of vectors summing to 0 whose size is the cardinality of the group, not its exponent.)

The case k = 2 is the Kemnitz conjecture, as pointed out by Kristal, and as Ricky mentioned (and also as mentioned on the Wikipedia page I linked to) it was proved by Reiher in 2003.