groups
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How to install groups?
parcadei/continuous-claude-v3/groups
$npx -y skills add parcadei/Continuous-Claude-v3 --skill groupsProblem-solving strategies for groups in abstract algebra
SKILL.md
| 1 | # Groups |
| 2 | |
| 3 | ## When to Use |
| 4 | |
| 5 | Use this skill when working on groups problems in abstract algebra. |
| 6 | |
| 7 | ## Decision Tree |
| 8 | |
| 9 | |
| 10 | 1. **Is G a group under operation *?** |
| 11 | - Check closure: a,b in G implies a*b in G? |
| 12 | - Check associativity: (a*b)*c = a*(b*c)? |
| 13 | - Check identity: exists e such that e*a = a*e = a? |
| 14 | - Check inverses: for all a exists a^(-1) such that a*a^(-1) = e? |
| 15 | - Verify with `z3_solve.py prove "group_axioms"` |
| 16 | |
| 17 | 2. **Subgroup Test** |
| 18 | - Show H is non-empty (usually by showing e in H) |
| 19 | - Show that for all a, b in H: ab^(-1) in H |
| 20 | - `z3_solve.py prove "subgroup_criterion"` |
| 21 | |
| 22 | 3. **Homomorphism Proof** |
| 23 | - Verify phi(ab) = phi(a)phi(b) for all a, b in G1 |
| 24 | - Note: phi(e1) = e2 and phi(a^(-1)) = phi(a)^(-1) follow automatically |
| 25 | - `sympy_compute.py simplify "phi(a*b) - phi(a)*phi(b)"` |
| 26 | |
| 27 | 4. **Order and Structure** |
| 28 | - Element order: smallest n where a^n = e |
| 29 | - Group order: |G| = number of elements |
| 30 | - Lagrange: |H| divides |G| for subgroup H |
| 31 | |
| 32 | |
| 33 | ## Tool Commands |
| 34 | |
| 35 | ### Z3_Group_Axioms |
| 36 | ```bash |
| 37 | 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)))" |
| 38 | ``` |
| 39 | |
| 40 | ### Z3_Subgroup |
| 41 | ```bash |
| 42 | uv run python -m runtime.harness scripts/z3_solve.py prove "subgroup_criterion" |
| 43 | ``` |
| 44 | |
| 45 | ### Sympy_Simplify |
| 46 | ```bash |
| 47 | uv run python -m runtime.harness scripts/sympy_compute.py simplify "phi(a*b) - phi(a)*phi(b)" |
| 48 | ``` |
| 49 | |
| 50 | ## Key Techniques |
| 51 | |
| 52 | *From indexed textbooks:* |
| 53 | |
| 54 | - [Abstract Algebra] Write a computer program to add and multiply mod n, for any n given as input. The output of these operations should be the least residues of the sums and products of two integers. Also include the feature that if (a,n) = 1, an integer c between 1 and n — 1 such that a-c = | may be printed on request. |
| 55 | - [Abstract Algebra] With a certain amount of elementary argument (calculations in A7, for example see Exercise 27) it can be shown that there is, up to isomorphism, a unique simple group of order 168 (it is not always the case that there is at most one simple group of a given order: there are 2 nonisomorphic simple groups of order +8! We could further show that such a G would have no elements of order pg, p and q distinct primes, no elements of order 9, and that distinct Sylow subgroups would intersect in the identity. We could then count the elements in Sylow p-subgroups for all primes p and we would find that these would total to exactly |G|. |
| 56 | - [Abstract Algebra] Some Techniques Before listing some techniques for producing normal subgroups in groups of a given (“medium”) order we note that in all the problems where one deals with groups of order n, for some specific n, it is first necessary to factor n into prime powers and then to compute the permissible values of np, for all primes p dividing n. We emphasize the need to be comfortable computing mod p when carrying out the last step. The techniques we describe may be listed as follows: (1) Counting elements. |
| 57 | - [Abstract Algebra] Composition Series and the Hélder Program Sec. This proof takes 255 pages of hard mathematics. Part (2) of the Hélder Program, sometimes called the extension problem, was rather vaguely formulated. |
| 58 | - [Abstract Algebra] APPLICATIONS IN GROUPS OF MEDIUM ORDER The purpose of this section is to work through a number of examples which illustrate many of the techniques we have developed. These examples use Sylow’s Theorems ex- tensively and demonstrate how they are applied in the study of finite groups. Motivated by the Holder Program we address primarily the problem of showing that for certain n every group of order n has a proper, nontrivial normal subgroup (i. |
| 59 | |
| 60 | ## Cognitive Tools Reference |
| 61 | |
| 62 | See `.claude/skills/math-mode/SKILL.md` for full tool documentation. |
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