Exercise · ZW-INDEX-OF-A-SUBGROUP-010 · ★★★ Advanced
Compute \([\mathbb{Z}:(4\mathbb{Z}\cap6\mathbb{Z})]\).
Choose one option.
Show answer
- C. \(12\)
Explanation
An integer lies in both subgroups if it is a multiple of \(4\) and of \(6\), that is, of \(\operatorname{lcm}(4,6)=12\). The intersection is \(12\mathbb{Z}\), of index \(12\). The product \(24\) is not the least common multiple.
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]?
- ZW-INDEX-OF-A-SUBGROUP-002 In ℤ_12, the subgroup H=\0,3,6,9\ has index:
- ZW-INDEX-OF-A-SUBGROUP-003 The alternating group A_4 has 12 elements and sits in S_4, which has 24. What is [S_4:A_4]?
- 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?