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Groups
The axioms, examples (integers mod n, symmetries, permutations), order of an element.
A group is a set with an associative operation, an identity and inverses. The integers mod 7 under addition; the non-zero ones under multiplication; the six symmetries of a triangle. Picture it: the rotations and reflections of a triangle composing. That is the group S₃ acting. Think it: "3¹⁰⁰ mod 7" is a computation in the cyclic group (ℤ/7)*, whose order 6 is why Fermat's little theorem works.
Exemple résolu · 3^100 mod 7
Étape par étape
- 3^{100} \bmod 7
Never compute the huge power. Reduce after every multiplication (repeated squaring).
- 3 \equiv 3 \pmod{7}
Reduce the base first.
- 100 = 1100100_2
Write the exponent in binary: 7 squarings at most.
- 3^2 \equiv 2 \pmod{7}
Square the running power.
- 2^2 \equiv 4 \pmod{7}
Square the running power.
- r \leftarrow r \cdot 4 \equiv 4 \pmod{7}
This bit is 1: multiply the result by the current power.
- 4^2 \equiv 2 \pmod{7}
Square the running power.
- 2^2 \equiv 4 \pmod{7}
Square the running power.
- 4^2 \equiv 2 \pmod{7}
Square the running power.
- r \leftarrow r \cdot 2 \equiv 1 \pmod{7}
This bit is 1: multiply the result by the current power.
- 2^2 \equiv 4 \pmod{7}
Square the running power.
- r \leftarrow r \cdot 4 \equiv 4 \pmod{7}
This bit is 1: multiply the result by the current power.
- 3^{100} \equiv 4 \pmod{7}
Done.
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Comment faire: Groups
- Check closure: the operation of any two elements stays in the set.
- Find the identity element.
- Find each element's inverse.
- Associativity usually comes for free from the underlying arithmetic; say why.
Questions posées par les gens
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.
Plus en Abstract Algebra
Subgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula