Exercise · ZW-COSET-009 · ★★★ Advanced
In \(S_3\) let \(H=\{e,(1\,2)\}\) and \(g=(1\,3)\). How are \(gH\) and \(Hg\) related?
Choose one option.
Show answer
- B. Different because \(g\) does not commute with \((1\,2)\)
Explanation
Here \(gH=\{g,g(1\,2)\}\) and \(Hg=\{g,(1\,2)g\}\), and \(g(1\,2)\neq(1\,2)g\) because \(S_3\) is not abelian. Left and right cosets can 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-007 Two left cosets aH and bH share an element. What follows?
- ZW-COSET-008 Let H=langle 12rangle in (ℤ_36,+). How many distinct cosets are there?