Exercise · ZW-COSET-007 · ★★☆ Intermediate
Two left cosets \(aH\) and \(bH\) share an element. What follows?
Choose one option.
Show answer
- D. \(aH=bH\)
Explanation
If \(x=ah=bh'\), then \(a=bh'h^{-1}\) and \(aH=bH\). Cosets are either equal or disjoint, but \(a\) and \(b\) may differ.
Review the concept: Coset
The left coset of a subgroup \(H\) of a group \(G\) with representative \(a\in G\) is the set of products of \(a\) with the elements of \(H\).
\[aH=\{ah:\ h\in H\}\]
More practice problems: coset
- ZW-COSET-001 Let H=\0,3,6,9\ in (ℤ_12,+). What is the coset 1+H?
- ZW-COSET-002 Let H=\0,4,8\ in (ℤ_12,+). How many distinct cosets does H have?
- ZW-COSET-003 In (ℤ,+) let H=5ℤ. Which coset equals 17+H?
- ZW-COSET-004 Let H be a subgroup of order 4 of (ℤ_20,+). How many elements does the coset 7+H have?
- ZW-COSET-005 Let H=\0,3,6,9\ in (ℤ_12,+). Which of these elements lies in the same coset as 5?
- ZW-COSET-006 In the multiplicative group ℤ_7^* let H=\1,6\. What is the coset 3H?
- ZW-COSET-008 Let H=langle 12rangle in (ℤ_36,+). How many distinct cosets are there?
- ZW-COSET-009 In S_3 let H=\e,(1 2)\ and g=(1 3). How are gH and Hg related?