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.

작업 예제 · 3^100 mod 7

Number-Theory · Modular-Arithmetic

3^100 mod 7

3,\ 100,\ 7

단계별로

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.

답을 드러내세요

3^{100} \bmod 7 = 4

이제 너 문제를 선택하거나, 타이핑하거나, 자신만의 문제를 그려보세요. 모든 단계는 그림으로 표시되며, 답은 질문을 할 때까지 숨겨져 있습니다.

자신의 작업을 유지

무료 계정은 모든 수업에 노트를 추가, 당신이 완료 한 기록, 당신이 해결 한 문제를 한 곳에, 그리고 당신이이 페이지에 대해 물어 수업 지도자. 수학 자체는 모두에게 열려 있습니다, 로그인 여부.

가입하기 로그인

여기서 사용된 기호

기호를 탭하면 전체 정의, 이미지 및 각 문자의 의미를 확인할 수 있습니다.

어떻게: 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.

사람들이 묻는 질문

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.

에 더 Abstract Algebra

Subgroups, cosets and Lagrange's theorem Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula