Exercise · ZW-LAGRANGES-THEOREM-010 · ★★★ Advanced
Let \(G\) be a finite group and \(H\) a proper subgroup. What can be guaranteed about \(|H|\)?
Choose one option.
Show answer
- C. \(|H|\le|G|/2\)
Explanation
If \(H\ne G\), the index satisfies \([G:H]\ge2\), and then \(|H|=|G|/[G:H]\le|G|/2\). Equality occurs only when the index is \(2\).
Review the concept: Lagrange's theorem
If \(H\) is a subgroup of a finite group \(G\), the order of \(H\) divides the order of \(G\), and the quotient is the index \([G:H]\).
\[|G|=[G:H]\,|H|\]
More practice problems: lagrange's theorem
- ZW-LAGRANGES-THEOREM-001 A group G has order 20. Which of these can NOT be the order of a subgroup of G?
- ZW-LAGRANGES-THEOREM-002 A group of order 15 has a subgroup H of order 5. What is the index [G:H]?
- ZW-LAGRANGES-THEOREM-003 How many subgroups does a group of order 11 have?
- ZW-LAGRANGES-THEOREM-004 The group S_4 has order 24. Which statement is correct?
- ZW-LAGRANGES-THEOREM-005 A group of order 36 has a subgroup of index 4. How many elements does that subgroup have?
- ZW-LAGRANGES-THEOREM-006 A group of order 48 has a subgroup of order 6. What is its index?
- ZW-LAGRANGES-THEOREM-007 A_4 has order 12. Which statement about its subgroups is true?
- ZW-LAGRANGES-THEOREM-008 Two subgroups H and K of a group have orders 12 and 18. Which can NOT be |H∩ K|?