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.

Àwọn ààtò àwọn ìṣàmúlò-ètò · 3^100 mod 7

Number-Theory · Modular-Arithmetic

3^100 mod 7

3,\ 100,\ 7

Àwọn ìṣàmúlò-ètò

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.

Fi àwọn àgbèwọlé hàn

3^{100} \bmod 7 = 4

Nígbà yìí Yan àwọn àkóónú kan, tàbí tẹ̀ àtí kọ́ àwọn àkóónú rẹ̀. Àwọn ìṣàmúlò-ètò gbogbo, àwòrán, àwọn à

Fi iṣẹ́ rẹ̀ pamọ́

A free account adds notes on every lesson, a record of what you have finished, your solved problems in one place, and a tutor you can ask about this page. The maths itself is open to everyone, signed in or not.

Fikún Àwọn Ìṣàmúlò-ètò

Àwọn àmì-ìwé tí a lo níbẹ̀

Tẹ àwọn àmì-ìwé fún ìṣàfarawé fún àwọn ìṣàmúlò-ètò, àwòrán, àti àwọn àmì-ìwé àwọn àmì-ìwé.

Bii o ṣe le: 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.

Àwọn Àtòjọ-ẹ̀yàn

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.

Diẹ̀ nínú 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