groups

# Groups ## When to Use Use this skill when working on groups problems in abstract algebra. ## Decision Tree 1. **Is G a group under operation *?** - Check closure: a,b in G implies a*b in G? - Check associativity: (a*b)*c = a*(b*c)? - Check identity: exists e such that e*a = a*e = a? - Check inverses: for all a exists a^(-1) such that a*a^(-1) = e? - Verify with `z3_solve.py prove "group_axioms"` 2. **Subgroup Test** - Show H is non-empty (usually by showing e in H) - Show that for all a, b in H: ab^(-1) in H - `z3_solve.py prove "subgroup_criterion"` 3. **Homomorphism Proof** - Verify phi(ab) = phi(a)phi(b) for all a, b in G1 - Note: phi(e1) = e2 and phi(a^(-1)) = phi(a)^(-1) follow automatically - `sympy_compute.py simplify "phi(a*b) - phi(a)*phi(b)"` 4. **Order and Structure** - Element order: smallest n where a^n = e - Group order: |G| = number of elements - Lagrange: |H| divides |G| for subgroup H ## Tool Commands ### Z3_Group_Axioms ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "ForAll([a,b,c], op(op(a,b),c) == op(a,op(b,c)))" ``` ### Z3_Subgroup ```bash uv run python -m runtime.harness scripts/z3_solve.py prove "subgroup_criterion" ``` ### Sympy_Simplify ```bash uv run python -m runtime.harness scripts/sympy_compute.py simplify "phi(a*b) - phi(a)*phi(b)" ``` ## Key Techniques *From indexed textbooks:* - [Abstract Algebra] Write a computer program to add and multiply mod n, for any n given a...