a subgroup of index 2 is normal


Let (G,) be a group and let H be a subgroupMathworldPlanetmathPlanetmath of G of index 2. Then H is normal in G.


Let G be a group and let H be an index 2 subgroup of G. By definition of index, there are only two left cosetsMathworldPlanetmath of H in G, namely:


where g1 is any element of G which is not in H. Notice that if g1,g2 are two elements in G which are not in H then g1g2 belongs to H. Indeed, the coset g1g2Hg1H (because g1g2=g1h would immediately yield g2=hH) and so g1g2H=H and g1g2H.

Let hH be an arbitrary element of H and let gG. If gH then ghg-1H and we are done. Otherwise, assume that gH. Thus ghH and by the remark above ghg-1=(gh)g-1H, as desired. ∎

Title a subgroup of index 2 is normal
Canonical name ASubgroupOfIndex2IsNormal
Date of creation 2013-03-22 15:09:25
Last modified on 2013-03-22 15:09:25
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Theorem
Classification msc 20A05
Related topic Coset
Related topic QuotientGroup
Related topic NormalityOfSubgroupsOfPrimeIndex