the derived subgroup is normal

We are going to prove:
”The derived subgroup (or commutator subgroup) [G,G] is normal in G

We have to show that for each x[G,G], gxg-1 it is also in [G,G].

Since [G,G] is the subgroupMathworldPlanetmathPlanetmath generated by the all commutators in G, then for each x[G,G] we have x=c1c2cm –a word of commutators– so ci=[ai,bi] for all i.

Now taking any element of gG we can see that

g[ai,bi]g-1 = gaibiai-1bi-1g-1
= gaig-1gbig-1gai-1g-1gbi-1g-1
= (gaig-1)(gbig-1)(gaig-1)-1(gbig-1)-1
= [gaig-1,gbig-1],

that is


so a conjugationMathworldPlanetmath of a commutator is another commutator, then for the conjugation

gxg-1 = gc1c2cmg-1
= gc1g-1gc2g-1gg-1gcmg-1
= (gc1g-1)(gc2g-1)(gcmg-1)

is another word of commutators, hence gxg-1 is in [G,G] which in turn implies that [G,G] is normal in G, QED.

Title the derived subgroup is normal
Canonical name TheDerivedSubgroupIsNormal
Date of creation 2013-03-22 16:04:39
Last modified on 2013-03-22 16:04:39
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 8
Author juanman (12619)
Entry type Proof
Classification msc 20A05
Classification msc 20E15
Classification msc 20F14