order of subgroup/

[arshavin]

If G is a group of order n, and n is divisible by k. Then must G have a subgroup of order k?

proof or counterexamples?

[Zorba]

No - one reason to see immediately why, is because if it was true, then Lagrange's theorem should be a two way implication.

Simplest example via wikipedia is \mathbb{A}_4 with has order 12 and no subgroup of order 6.