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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.

Mfano wenye matokeo: divisors of 12

Number-Theory · Divisors

Divisors of 12

12

Hatua kwa hatua

  1. 12 = 2^{2} \times 3

    Prime-factorise first.

  2. 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.

  3. 1, 2, 3, 4, 6, 12

    List them in increasing order.

Lafunua jibu
1, 2, 3, 4, 6, 12

Symbols used here

\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 \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
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: Subgroups, cosets and Lagrange's theorem

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

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.

Jaribu kufanya mambo yako mwenyewe

Mengi zaidi katika Abstract Algebra

GroupsCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula