Subgroup Order in Groups of Divisible Orders: Proof or Counterexamples?

arshavin · Apr 29, 2011

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 · Apr 29, 2011

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 [tex]\mathbb{A}_4[/tex] with has order 12 and no subgroup of order 6.

Subgroup Order in Groups of Divisible Orders: Proof or Counterexamples?

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Subgroup Order in Groups of Divisible Orders: Proof or Counterexamples?

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