[Verse 1] Take a group and pick a subset H Add an element g from what's left Multiply it through, left or right gH creates a coset sight Every element finds its place In exactly one coset space [Chorus] Cosets tile the floor so neat Equal pieces, no repeat Size of H in every part That's the theorem's beating heart Lagrange tells us crystal clear Only certain sizes here [Verse 2] Left coset takes g times H Right coset flips the way Like colored tiles across the ground Perfect patterns can be found No overlap and no gaps wide Partition rules the group inside [Chorus] Cosets tile the floor so neat Equal pieces, no repeat Size of H in every part That's the theorem's beating heart Lagrange tells us crystal clear Only certain sizes here [Bridge] Order of H must divide Order of G, there's no place to hide Index times the subgroup size Equals parent group's surprise Prime order means just two Trivial or cyclic through [Verse 3] Element order splits the whole Every power plays its role G raised to order gives identity Symmetry's constraint, you see Subgroups can't be any size Mathematical compromise [Chorus] Cosets tile the floor so neat Equal pieces, no repeat Size of H in every part That's the theorem's beating heart Lagrange tells us crystal clear Only certain sizes here [Outro] Tiling theorem holds the key Divisibility decree Structure carved from group alone Mathematical stepping stone
← 3 Cyclic Groups and Order | 5 Normal Subgroups and Quotient Groups →