maths.free › Abstract Algebra › 6. Cosets and Lagrange's Theorem › Lagrange's Theorem

Lagrange's Theorem

Proposition Let H be a subgroup of G with g \in G and define a map \phi:H \rightarrow gH by \phi(h) = gh. The map \phi is bijective; hence, the number of elements in H is the same as the number of elements in gH.

Lagrange's Theorem

suggests that groups of prime order \(p\) must somehow look like \({\mathbb Z}_p\).

In fact, we can say more about when two cycles have the same length.

Condensed — the full section is in Judson, Abstract Algebra: Theory and Applications.

Symbols used here

x \in A,\ A \subseteq B
element of, subset
x belongs to A; every element of A is in B.
\neq
not equal
The two sides are different.
\sup,\ \inf
supremum, infimum
Least upper bound, greatest lower bound.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
\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.

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.

Zama ngokwakho

Parts of this page are adapted from Judson, Abstract Algebra: Theory and Applications (GFDL 1.3). Condensed and re-explained here; errors are ours.

IiNkqubo Abstract Algebra

GroupsSubgroups, cosets and Lagrange's theoremCyclic groups and permutation groupsHomomorphisms, normal subgroups and quotient groupsRings and fieldsGalois theory: why the quintic has no formula