Elementary Group Theory - Cosets

Cosets

If S and T are subsets of G, and a is an element of G, we write "a*S" to refer to the subset of G made up of all elements of the form a*s, where s is an element of S; similarly, we write "S*a" to indicate the set of elements of the form s*a. We write S*T for the subset of G made up of elements of the form s*t, where s is an element of S and t is an element of T.

If H is a subgroup of G, then a left coset of H is a set of the form a*H, for some a in G. A right coset is a subset of the form H*a.

If H is a subgroup of G, the following useful theorems, stated without proof, hold for all cosets:

• Any x and y are elements of G, then either x*H = y*H, or x*H and y*H have empty intersection.

• Every left (right) coset of H in G contains the same number of elements.

• G is the disjoint union of the left (right) cosets of H.

• Then the number of distinct left cosets of H equals the number of distinct right cosets of H.

Define the index of a subgroup H of a group G (written "") to be the number of distinct left cosets of H in G.

From these theorems, we can deduce the important Lagrange's theorem, relating the order of a subgroup to the order of a group:

• Lagrange's theorem: If H is a subgroup of G, then |G| = |H|*.

For finite groups, this can be restated as:

• Lagrange's theorem: If H is a subgroup of a finite group G, then the order of H divides the order of G.

• If the order of group G is a prime number, G is cyclic.