{"id":"groups","name":"groups","summary":"抽象代数における群の問題解決戦略","body":"# Groups\n\n## When to Use\n\nUse this skill when working on groups problems in abstract algebra.\n\n## Decision Tree\n\n\n1. **Is G a group under operation *?**\n   - Check closure: a,b in G implies a*b in G?\n   - Check associativity: (a*b)*c = a*(b*c)?\n   - Check identity: exists e such that e*a = a*e = a?\n   - Check inverses: for all a exists a^(-1) such that a*a^(-1) = e?\n   - Verify with `z3_solve.py prove \"group_axioms\"`\n\n2. **Subgroup Test**\n   - Show H is non-empty (usually by showing e in H)\n   - Show that for all a, b in H: ab^(-1) in H\n   - `z3_solve.py prove \"subgroup_criterion\"`\n\n3. **Homomorphism Proof**\n   - Verify phi(ab) = phi(a)phi(b) for all a, b in G1\n   - Note: phi(e1) = e2 and phi(a^(-1)) = phi(a)^(-1) follow automatically\n   - `sympy_compute.py simplify \"phi(a*b) - phi(a)*phi(b)\"`\n\n4. **Order and Structure**\n   - Element order: smallest n where a^n = e\n   - Group order: |G| = number of elements\n   - Lagrange: |H| divides |G| for subgroup H\n\n\n## Tool Commands\n\n### Z3_Group_Axioms\n```bash\nuv run python -m runtime.harness scripts/z3_solve.py prove \"ForAll([a,b,c], op(op(a,b),c) == op(a,op(b,c)))\"\n```\n\n### Z3_Subgroup\n```bash\nuv run python -m runtime.harness scripts/z3_solve.py prove \"subgroup_criterion\"\n```\n\n### Sympy_Simplify\n```bash\nuv run python -m runtime.harness scripts/sympy_compute.py simplify \"phi(a*b) - phi(a)*phi(b)\"\n```\n\n## Key Techniques\n\n*From indexed textbooks:*\n\n- [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.\n- [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|.\n- [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.\n- [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.\n- [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.\n\n## Cognitive Tools Reference\n\nSee `.claude/skills/math-mode/SKILL.md` for full tool documentation.","author":"@parcadei","ownerProfile":null,"authorContacts":null,"sourceUrl":"https://github.com/parcadei/Continuous-Claude-v3/tree/main/.claude/skills/math/abstract-algebra/groups","license":"MIT","category":"writing","lang":"en","tokens":955,"stars":0,"calls30d":1,"claimed":false,"visibility":"public","origin":"crawler","version":"0.1.0","createdAt":"2026-08-22","updatedAt":"2026-08-22","files":[],"requires":{"mcp":[],"tools":["Bash","Read"]},"safety":{"flags":[],"scannedAt":"2026-08-22","hasScripts":false,"networkEndpoints":[]}}