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Cyclic groups and permutation groups
Generators, orders, cycles, transpositions, parity.
A cyclic group is generated by one element; every permutation is a product of disjoint cycles, and its parity (even/odd) is well defined. Picture it: shuffling 5 cards as arrows between positions; a cycle is a loop of arrows. Think it: Cayley's theorem, every group is a permutation group, means permutations are the universal example.
Ví dụ đã làm · inverse of 5 mod 12
Bước một bước
- 5x \equiv 1 \pmod{12}
We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.
- 12 = 2 \times 5 + 2
Euclid step.
- 5 = 2 \times 2 + 1
Euclid step.
- 2 = 2 \times 1 + 0
Euclid step.
- 5 \times 5 = 25 \equiv 1 \pmod{12}
Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.
Giải đáp
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Làm thế nào: Cyclic groups and permutation groups
- We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.
- Euclid step.
- Euclid step.
- Euclid step.
- Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.
Câu hỏi người ta hỏi
What is a group, in plain words?
A set with one operation that is associative, has an identity, and lets every element be undone. Symmetries of any object form a group. That is where the idea came from.
What is the difference between a ring and a field?
A ring has addition and multiplication that behave like the integers (you cannot always divide); a field is a ring where every non-zero element has a reciprocal, like the rationals or the reals.
More in Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula