Exercise · ZW-NORMAL-SUBGROUP-002 · ★☆☆ Basic
In \(S_3\), which of these subgroups is normal?
Choose one option.
Show answer
- D. \(\langle(1\,2\,3)\rangle\)
Explanation
The subgroup \(\langle(1\,2\,3)\rangle=\{e,(1\,2\,3),(1\,3\,2)\}\) has index 2, and conjugating a 3-cycle always gives another 3-cycle in that same subgroup. Subgroups generated by transpositions are not normal: conjugation moves a transposition to another one.
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-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-005 In S_3, how many distinct subgroups of the form glangle(1 2)rangle g^-1 are there as g ranges over…
- 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?