Groups

Definition and Examples of Groups

Let G be a set and let * be a binary operation defined on G. Then we say that (G,*) is a group, or that G is a group with respect to *, iff the four axioms are satisfied:

• If a,b∈G, then a*b∈G;
• If a,b,c∈G, then a*(b*c)=(a*b)*c;
• There exists e∈G such that for any a∈G, a*e = a = e*a;
• If a∈G then there exists b∈G such that a*b = e = b*a.

Examples of groups: Z, Q and R are groups with respect to addition; Q\{0} and R\{0} are groups with respect to multiplication.

Lagrange's Theorem

Let G be a group. We say that H is a subgroup of G and write H≤G iff H⊆G and H satisfies the group axioms. The identity of H must be the identity of G. Associativity is inherited, being a property of the operation.

Let G be a group and let H be a subgroup of G. Then a left coset gH of H in G is the set {gh | h∈H}. A right coset is defined similarly.

Let G be a group and let H be any subgroup of G. Lagrange's Theorem states that |H| divides |G|. This follows from the following results:

• The cosets of H in G partition G.
• Any coset of H has the same size as H.

It also follows that if g∈G, then o(g) divides |G|. For if H = <g>, then |H| = o(g).

Groups of small order

Let G be a group and let p be a prime.

If |G| = p, then G≅C_p. This concerns groups of order 2, 3, 5 and 7.

If |G| = 4, then G≅C₄ or G≅C₂×C₂.

If |G| = 6, then G≅C₆≅C₃×C₂, or G≅D₆.

If |G| = 8, then G is isomorphic to one of C₈, C₄×C₂, C₂×C₂×C₂, D₈, or Q₈.

Group Actions