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.

Exemplo trabalhado: `3^100 mod 7`

Number-Theory · Modular-Arithmetic

3^100 mod 7

3,\ 100,\ 7

Passo a passo

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.

Revelar a resposta

3^{100} \bmod 7 = 4

Symbols used here

a \equiv b \pmod n

congruent modulo n
n divides a − b; a and b have the same remainder.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

a \bmod n

remainder
What is left after dividing a by n.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

x \in A,\ A \subseteq B

element of, subset
x belongs to A; every element of A is in B.

\blacksquare\ \text{or}\ \square

end of proof (halmos)
Marks the point where the statement has been established.

a \mid b,\ \gcd(a,b)

divides, greatest common divisor
b is a multiple of a; the largest number dividing both.

(G, \cdot),\ e,\ g^{-1}

group, identity, inverse
A set with an operation; the do-nothing element; the element that undoes g.

G \cong H,\ G / N

isomorphic, quotient group
Same structure; the group of cosets of a normal subgroup N.

\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n

integers modulo n
The remainders 0…n−1 with clock arithmetic.

\operatorname{Hom}(A, B),\ f \circ g

arrows from A to B, composition
The set of morphisms; do g then f.

How to: 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 people ask

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.

Tente o seu próprio

Mais em 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

Exemplo trabalhado: 3^100 mod 7