Exercise · ZW-NORMAL-SUBGROUP-009 · ★★★ Advanced
A subgroup \(H\) of \(G\) is the only subgroup of \(G\) with \(|H|\) elements. What follows?
Choose one option.
Show answer
- B. \(H\) is normal in \(G\)
Explanation
For each \(g\), the set \(gHg^{-1}\) is a subgroup with \(|H|\) elements (conjugation is bijective), so by uniqueness it equals \(H\). Hence \(H\) is normal; the rest does not follow (in \(S_3\), \(A_3\) is the only subgroup of order 3, yet \(S_3\) is not abelian and \(A_3\) is not central).
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-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?