Commutator subgroup
The commutator subgroup
Proof of normal subgroup
is a subgroup by construction. Let [ πΊ , πΊ ] . Then for any conjugate π β [ πΊ , πΊ ] it follows π¦ = π₯ π π₯ β 1 , so π¦ π β 1 = π₯ π π₯ β 1 π β 1 = [ π₯ , π ] and thus π¦ π β 1 β [ πΊ , πΊ ] . Therefore π¦ π β 1 π = π¦ β [ πΊ , πΊ ] is a normal subgroup. [ πΊ , πΊ ]
Wikipedia notes
[the commutator subgroup] is stable under every endomorphism ofΒ
: that is, πΊ is aΒ fully characteristic subgroupΒ ofΒ [ πΊ , πΊ ] , a property considerably stronger than normality. πΊ
Properties
- A quotient with the commutator subgroup of
πΊ πΊ