maths.free › Abstract Algebra › Subgroups, cosets and Lagrange's theorem

Subgroups, cosets and Lagrange's theorem

Why the order of a subgroup divides the order of the group.

Cosets of a subgroup H tile the group in equal-sized pieces, so |H| divides |G|. Picture it: the integers mod 12 as a clock; the subgroup {0, 4, 8} and its three cosets colour the clock in three colours. Think it: Lagrange is the reason element orders divide group orders, hence the reason RSA works.

Àwọn ààtò àwọn ìṣàmúlò-ètò · divisors of 12

Number-Theory · Divisors

Divisors of 12

Àwọn ìṣàmúlò-ètò

12 = 2^{2} \times 3
Prime-factorise first.
d(n) = (2+1) \cdot (1+1) = 6
Each divisor picks an exponent from 0 up to the prime's exponent, so multiply (exponent + 1) for each prime.
1, 2, 3, 4, 6, 12
List them in increasing order.

Fi àwọn àgbèwọlé hàn

1, 2, 3, 4, 6, 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: Subgroups, cosets and Lagrange's theorem

Prime-factorise first.
Each divisor picks an exponent from 0 up to the prime's exponent, so multiply (exponent + 1) for each prime.
List them in increasing order.

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 Cyclic groups and permutation groups Homomorphisms, normal subgroups and quotient groups Rings and fields Galois theory: why the quintic has no formula