[Verse 1]
Count the elements, that's the order of your group
Single generator spinning, makes a cyclic loop
Take one element, let it multiply around
Every power that it makes is what gets found
[Chorus]
Cyclic groups are always abelian, commutative and clean
Order n looks just like Z mod n, that's the scene
Infinite cyclic matches Z, the integers we know
From one generator, watch the whole structure grow
[Verse 2]
Order of an element, smallest power back to e
If no such power exists, then infinity's the key
Subgroups hiding in the cycles, all are cyclic too
Every divisor of n gives a subgroup true
[Chorus]
Cyclic groups are always abelian, commutative and clean
Order n looks just like Z mod n, that's the scene
Infinite cyclic matches Z, the integers we know
From one generator, watch the whole structure grow
[Bridge]
Take a to the power d, where d divides your n
Subgroup order equals n divided by d again
Exactly one subgroup for each divisor found
Lattice of subgroups, perfectly arranged and sound
[Verse 3]
Generator a creates the group G with its might
Every element's a power, spinning left and right
Isomorphic to the modular integers we see
Cyclic beauty, algebraic harmony
[Chorus]
Cyclic groups are always abelian, commutative and clean
Order n looks just like Z mod n, that's the scene
Infinite cyclic matches Z, the integers we know
From one generator, watch the whole structure grow
[Outro]
Single element commanding, building all the rest
Cyclic groups, the simplest forms, algebra's blessed