[Verse 1]
In the algebra garden where groups grow wild
There's a special breed, both fierce and mild
Subgroups live inside their parent's frame
Playing by the rules of the same old game
Take a subset H from group G's domain
Check if it can stand through joy and pain
Not empty handed, that's the first demand
Then test the magic with your steady hand
[Chorus]
One-step test, one-step test
Take two elements, put them to rest
Multiply a with b-inverse
If it stays inside, you've broken the curse
Subgroups dancing in formation
Smaller circles, same equation
One-step test will tell you true
Is this family really group too
[Verse 2]
Center stage stands Z of G so proud
Elements that whisper, never too loud
Commuting with everything in sight
Left times right equals right times left tonight
Centralizer holds a different key
C sub G of a, that's the family
All the elements that shake hands nice
With your chosen a, no rolling dice
[Chorus]
One-step test, one-step test
Take two elements, put them to rest
Multiply a with b-inverse
If it stays inside, you've broken the curse
Subgroups dancing in formation
Smaller circles, same equation
One-step test will tell you true
Is this family really group too
[Bridge]
Generated subgroups spin around
Take one element, let powers abound
a to the n where n's an integer
Positive, negative, zero's the trigger
Brackets holding a create the smallest cage
That keeps it trapped upon algebra's stage
[Verse 3]
Three important breeds roam free
Center, centralizer, generated spree
Each one carved from parent group's flesh
Following laws both old and fresh
When you find H sitting inside G
Check the subset's pedigree
Non-empty vessel, closure tight
One-step test reveals the light
[Chorus]
One-step test, one-step test
Take two elements, put them to rest
Multiply a with b-inverse
If it stays inside, you've broken the curse
Subgroups dancing in formation
Smaller circles, same equation
One-step test will tell you true
Is this family really group too
[Outro]
H less than or equal G
That's the symbol's decree
Subgroups nested, structures blessed
One-step test puts you to rest