Exercise · ZW-NORMAL-SUBGROUP-005 · ★★☆ Intermediate
In \(S_3\), how many distinct subgroups of the form \(g\langle(1\,2)\rangle g^{-1}\) are there as \(g\) ranges over \(S_3\)?
Choose one option.
Show answer
- B. 3
Explanation
The conjugates of \(\langle(1\,2)\rangle\) are \(\langle(1\,2)\rangle\), \(\langle(1\,3)\rangle\) and \(\langle(2\,3)\rangle\): three distinct subgroups, which confirms it is not normal (a normal subgroup has a single conjugate).
Review the concept: Normal subgroup
A normal subgroup is a subgroup \(N\) of a group \(G\) that equals all of its conjugates by elements \(g\in G\).
\[gNg^{-1}=N\ \ \forall g\in G\]
More practice problems: normal subgroup
- ZW-NORMAL-SUBGROUP-001 Let G be an abelian group and N a subgroup of G. What can be said about N?
- ZW-NORMAL-SUBGROUP-002 In S_3, which of these subgroups is normal?
- ZW-NORMAL-SUBGROUP-003 In S_3, with g=(1 3), what is g (1 2) g^-1?
- ZW-NORMAL-SUBGROUP-004 If N is a subgroup of G with [G:N]=2, what can be said?
- ZW-NORMAL-SUBGROUP-006 Which of these conditions is equivalent to N being a normal subgroup of G?
- ZW-NORMAL-SUBGROUP-007 Let H=langle(1 2)rangle in S_3. Which g shows that H is not normal?
- ZW-NORMAL-SUBGROUP-008 Let N be normal in G and H a subgroup with N⊆ H⊆ G. What always holds?
- ZW-NORMAL-SUBGROUP-009 A subgroup H of G is the only subgroup of G with |H| elements. What follows?