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

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SUMMARY

The discussion centers on the question of whether a group G of order n, divisible by k, must contain a subgroup of order k. The consensus is that this is not necessarily true, as evidenced by the counterexample of the alternating group \mathbb{A}_4, which has an order of 12 but lacks a subgroup of order 6. This conclusion highlights the limitations of Lagrange's theorem, which does not imply the existence of subgroups for all divisors of the group's order.

PREREQUISITES
  • Understanding of group theory concepts, specifically Lagrange's theorem.
  • Familiarity with the properties of alternating groups, particularly \mathbb{A}_4.
  • Knowledge of group orders and subgroup structures.
  • Basic mathematical proof techniques to analyze counterexamples.
NEXT STEPS
  • Study the implications of Lagrange's theorem in group theory.
  • Research the structure and properties of the alternating group \mathbb{A}_4.
  • Explore additional counterexamples in group theory that challenge subgroup existence.
  • Learn about Sylow theorems and their applications in subgroup analysis.
USEFUL FOR

Mathematicians, students of abstract algebra, and anyone studying group theory who seeks to understand subgroup structures and the limitations of Lagrange's theorem.

arshavin
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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?
 
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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.
 
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