maths.free › Abstract Algebra › Cyclic groups and permutation groups

Cyclic groups and permutation groups

Generators, orders, cycles, transpositions, parity.

A cyclic group is generated by one element; every permutation is a product of disjoint cycles, and its parity (even/odd) is well defined. Picture it: shuffling 5 cards as arrows between positions; a cycle is a loop of arrows. Think it: Cayley's theorem, every group is a permutation group, means permutations are the universal example.

Àwọn ààtò àwọn ìṣàmúlò-ètò · inverse of 5 mod 12

Number-Theory · Modular-Arithmetic

Inverse of 5 mod 12

5,\ 12

Àwọn ìṣàmúlò-ètò

5x \equiv 1 \pmod{12}
We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.
12 = 2 \times 5 + 2
Euclid step.
5 = 2 \times 2 + 1
Euclid step.
2 = 2 \times 1 + 0
Euclid step.
5 \times 5 = 25 \equiv 1 \pmod{12}
Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.

Fi àwọn àgbèwọlé hàn

5^{-1} \equiv 5 \pmod{12}

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: Cyclic groups and permutation groups

We want x with a·x ≡ 1 (mod m). It exists only when gcd(a, m) = 1.
Euclid step.
Euclid step.
Euclid step.
Back-substitute (extended Euclid) to express 1 as a combination of a and m; the coefficient of a is the inverse.

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

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