maths.free › Abstract Algebra › Groups

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.

Arbejdstxrd eksempel · 3^100 mod 7

Number-Theory · Modular-Arithmetic

3^100 mod 7

3,\ 100,\ 7

Trin for trin

  1. 3^{100} \bmod 7

    Never compute the huge power. Reduce after every multiplication (repeated squaring).

  2. 3 \equiv 3 \pmod{7}

    Reduce the base first.

  3. 100 = 1100100_2

    Write the exponent in binary: 7 squarings at most.

  4. 3^2 \equiv 2 \pmod{7}

    Square the running power.

  5. 2^2 \equiv 4 \pmod{7}

    Square the running power.

  6. r \leftarrow r \cdot 4 \equiv 4 \pmod{7}

    This bit is 1: multiply the result by the current power.

  7. 4^2 \equiv 2 \pmod{7}

    Square the running power.

  8. 2^2 \equiv 4 \pmod{7}

    Square the running power.

  9. 4^2 \equiv 2 \pmod{7}

    Square the running power.

  10. r \leftarrow r \cdot 2 \equiv 1 \pmod{7}

    This bit is 1: multiply the result by the current power.

  11. 2^2 \equiv 4 \pmod{7}

    Square the running power.

  12. r \leftarrow r \cdot 4 \equiv 4 \pmod{7}

    This bit is 1: multiply the result by the current power.

  13. 3^{100} \equiv 4 \pmod{7}

    Done.

Afslør svaret
3^{100} \bmod 7 = 4

Nu er du Vælg et problem, eller skriv eller tegne din egen. Hvert skridt, et billede, svaret skjult, indtil du spørger.

Fortsæt med at arbejde

En gratis konto tilføjer noter om hver lektion, en rekord over, hvad du har afsluttet, dine løste problemer på ét sted, og en vejleder, du kan spørge om denne side. Matematik selv er åben for alle, logget på eller ej.

Tilmeld dig Login

Symboler brugt her

Tryk på ethvert symbol for den fulde definition, et billede, og hvad hvert bogstav i det betyder.

Hvordan: Groups

  1. Check closure: the operation of any two elements stays in the set.
  2. Find the identity element.
  3. Find each element's inverse.
  4. Associativity usually comes for free from the underlying arithmetic; say why.

Spørgsmål folk spørger

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.

Mere i 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