[Verse 1]
In the theater of groups, where symmetries dance
Lagrange whispers secrets of finite expanse
Order of G equals index times H's size
A fundamental truth that never lies
G over H times H's magnitude
Reveals the structure, crystal clear and crude
[Chorus]
Count the orbits, stabilize the core
Kernels map to images and more
Burnside burns bright, averaging the fixed
Sylow's theorem gets the p-groups mixed
Class equation balances the scale
Group theory's symphony will never fail
[Verse 2]
Orbit-stabilizer theorem calls the tune
Product equals G's order, sun and moon
Every element dances in its sphere
Stabilizer holds it crystal clear
Multiply these quantities together
Group order emerges, light as feather
[Chorus]
Count the orbits, stabilize the core
Kernels map to images and more
Burnside burns bright, averaging the fixed
Sylow's theorem gets the p-groups mixed
Class equation balances the scale
Group theory's symphony will never fail
[Verse 3]
First isomorphism cuts through the haze
G mod kernel equals image's maze
Homomorphisms build the bridge between
Quotient groups and subgroups unforeseen
Factor out the kernel's hidden space
Find the image in its rightful place
[Bridge]
Sylow subgroups congregate and count
One mod p, divisibility paramount
n-p congruent to unity's call
Dividing index, standing proud and tall
[Verse 4]
Burnside's lemma counts the patterns true
Average fixed points, orbits breaking through
One over G-size, sum across the board
Each element's fixes, perfectly scored
Class equation splits the center out
Conjugacy classes, removing doubt
[Outro]
Z of G plus centralizer's sum
Equals order when the counting's done
These six equations rule the abstract throne
Group theory's kingdom, carved in algebraic stone