Group Equation Explorer
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Rewrite expressions and solve equations using nothing but the group laws.
Chapter 1
Cancel what undoes itself
Recognize adjacent inverse pairs and clean up the identity they leave behind.
0% of the proof path complete
Course path
Five chapters of group reasoning
Each chapter moves from recognition to construction and then mixed practice. A locked problem always takes you to the earliest prerequisite still waiting.
1Identity and inverse movesLearn what may be removed, what may be inserted, and why both preserve a term.0/2
By the end: Recognize inverse pairs and deliberately create useful ones.
2Earn the inverse lawsFirst show why an inverse is unique, then turn that fact and the axioms into reusable theorems.0/8
By the end: Prove inverse uniqueness and apply it to the identity, double-inverse, and socks-and-shoes laws.
3Build the power lawsSeparate definitions from the properties they support, then compress the resulting proofs.0/7
By the end: Explain zero and negative exponents and justify the rules for inversion and exponent addition.
4Solve without commutingKeep track of which side you multiply on when order matters.0/6
By the end: Solve group equations and derive left and right cancellation.
5Mixed proof fluencyChoose among the laws without being told which family the next move belongs to.0/3
By the end: Combine inverse laws and solve equations with factors on both ends.
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What you have established
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No laws earned yet. The first is the identity inverting to itself, in challenge 05.
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Your own expression
Every law is available here, and a target is optional.
Write products by juxtaposition: ab, (ab)^-1 c, a^3, r2 s. A generator is one letter and any digits; e is the identity. One = makes it an equation.
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