Exercise · ZW-INDEX-OF-A-SUBGROUP-006 · ★★☆ Intermediate
In \(\mathbb{Z}_{30}\), the subgroup generated by \(6\) has \(5\) elements. Its index is:
Choose one option.
Show answer
- B. \(6\)
Explanation
The subgroup is \(\{0,6,12,18,24\}\), and there are \(30/5=6\) cosets, one for each residue \(0,\dots,5\).
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
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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:
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- 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?