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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.

ឧទាហរណ៍​ដែល​បាន​ធ្វើ · 3^100 mod 7

Number-Theory · Modular-Arithmetic

3^100 mod 7

3,\ 100,\ 7

ជំហាន​ដោយ​ជំហាន

  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.

បង្ហាញ​ចម្លើយ
3^{100} \bmod 7 = 4

ឥឡូវ​អ្នក ជ្រើស​បញ្ហា ឬ​វាយ​បញ្ចូល ឬ​គូរ​របស់​អ្នក​ផ្ទាល់ ។ ជំហាន​នីមួយៗ រូបភាព ចម្លើយ​ត្រូវ​បាន​លាក់​រហូត​ដល់​អ្នក​សួរ ។

រក្សា​ការ​ធ្វើការ​របស់​អ្នក​ផ្ទាល់

គណនី​ឥត​គិត​ថ្លៃ​បន្ថែម​ចំណាំ​លើ​មេរៀន​នីមួយៗ កំណត់​ហេតុ​នៃ​អ្វី​ដែល​អ្នក​បាន​បញ្ចប់ បញ្ហា​ដែល​អ្នក​បាន​ដោះស្រាយ​នៅ​កន្លែង​មួយ និង​គ្រូ​ដែល​អ្នក​អាច​សួរ​អំពី​ទំព័រ​នេះ ។ គណិតវិទ្យា​ដោយ​ខ្លួន​វា​គឺ​បើក​ចំហ​សម្រាប់​អ្នក​ទាំងអស់​គ្នា ចុះឈ្មោះ​ចូល ឬ​មិន​ចុះឈ្មោះ​ចូល ។

ចុះឈ្មោះ ចូល

និមិត្ត​សញ្ញា​ដែល​បាន​ប្រើ​នៅ​ទីនេះ

ចុច​និមិត្ត​សញ្ញា​ណាមួយ​សម្រាប់​ការ​កំណត់​ពេញលេញ រូបភាព និង​តួអក្សរ​នីមួយៗ​មាន​ន័យ​អ្វី ។

របៀប​: 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.

សំណួរ​ដែល​មនុស្ស​សួរ

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.

បន្ថែម​ទៀត​ក្នុង 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