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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.
דוגמה עובדת · inverse of 5 mod 12
צעד אחר צעד
- 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.
גלה את התשובה
עכשיו אתה בחר בעיה, או הקלד או צייר בעצמך. כל צעד, תמונה, התשובה חבויה עד שתשאל.
סמלים המשמשים כאן
הקש כל סמל להגדרה המלאה, תמונה, ומה כל אות בזה אומר.
איך: 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.
שאלות שאנשים שואלים
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.
יותר בפנים. Abstract Algebra
GroupsSubgroups, cosets and Lagrange's theoremHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula