Exercise · ZW-NORMAL-SUBGROUP-007 · ★★☆ Intermediate
Let \(H=\langle(1\,2)\rangle\) in \(S_3\). Which \(g\) shows that \(H\) is not normal?
Choose one option.
Show answer
- C. \(g=(1\,3)\)
Explanation
With \(g=e\) or \(g=(1\,2)\) the conjugate of \((1\,2)\) is \((1\,2)\), which stays in \(H\). With \(g=(1\,3)\) we get \((2\,3)\notin H\), so \(gHg^{-1}\ne H\).
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-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-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?