SECTION 4.2 Basics of Group Theory 211
Definition of Binary Operation on a Set. Abinary operation
on a non-empty setSis a mapping∗:S×S→S.
No more, no less! We usually writes∗s′in place of the more formal
∗(s,s′).
We have a wealth of examples available; we’ll review just a few of
them here.
- The familiar operations + and· are binary operations on our fa-
vorite number systems: Z,Q,R,C. - Note that ifS is the set of irrational numbers then neither + nor
·defines a binary operation onS. (Why not?) - Note that subtraction−defines a binary operation onR.
- Let Matn(R) denote then×nmatrices with real coefficients. Then
both + (matrix addition) and·(matrix multiplication) define bi-
nary operations on Matn(R). - Let S be any set and let F(S) = {functions : S → S}. Then
function composition◦defines a binary operation on F(S). (This
is a particularly important example.) - Let Vect 3 (R) denote the vectors in 3-space. Then the vector
cross product×is a binary operation on Vect 3 (R). Note that the
scalar product·does not define a binary operation on Vect 3 (R). - LetAbe a set and let 2A be its power set. The operations∩,∪,
and + (symmetric difference) are all important binary operations
on 2A. - Let Sbe a set and let Sym(S) be the set of all permutations on
S. Then function composition ◦ defines a binary operation on
Sym(S). We really should prove this. Thus letσ, τ : S → S
be permutations; thus they are one-to-one and onto. We need to
show thatσ◦τ :S→S is also one-to-one and onto.
σ◦τ is one-to-one: Assume that s, s′ ∈S and that σ◦τ(s) =
σ◦τ(s′). Sinceσis one-to-one, we conclude thatτ(s) =τ(s′).