Exercise · ZW-INDEX-OF-A-SUBGROUP-002 · ★☆☆ Basic
In \(\mathbb{Z}_{12}\), the subgroup \(H=\{0,3,6,9\}\) has index:
Choose one option.
Show answer
- C. \(3\)
Explanation
The subgroup has \(4\) elements and the group has \(12\); the cosets \(H,\ 1+H,\ 2+H\) cover everything, so there are \(3\). The number \(4\) is the order of \(H\), not its index.
Review the concept: Index of a subgroup
The index of a subgroup \(H\) of a group \(G\) is the number of distinct left cosets of \(H\) in \(G\).
\[[G:H]\]
More practice problems: index of a subgroup
- ZW-INDEX-OF-A-SUBGROUP-001 What is the index [ℤ:4ℤ]?
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- ZW-INDEX-OF-A-SUBGROUP-004 A group G of order 60 has a subgroup H of order 12. The index [G:H] is:
- ZW-INDEX-OF-A-SUBGROUP-005 Which subgroup H of a group G has index 1?
- ZW-INDEX-OF-A-SUBGROUP-006 In ℤ_30, the subgroup generated by 6 has 5 elements. Its index is:
- ZW-INDEX-OF-A-SUBGROUP-007 In the dihedral group D_4 of order 8, the rotations form a subgroup of order 4. Its index is:
- ZW-INDEX-OF-A-SUBGROUP-008 Let H=langle(1\ 2\ 3)rangle, of order 3, in S_4. How many distinct left cosets does H have?
- ZW-INDEX-OF-A-SUBGROUP-009 If [G:H]=2, what are the left cosets of H?