S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 1: Sets and Groups / From my notebooks

N3 Laws of Composition (Artin 2.1–2.6)

2.1 Laws of composition

  • A law of composition is a function producing , where .
  • Note: cannot be recovered from ; i.e., without knowing , we cannot find them with .
  • We tend to reserve for commutative laws of composition, but can be either or.

Let be a set and and maps . Let . The rule is a law of composition on the maps .

Question N3.1. If composition is associative and any is a law of composition (?), then why is associativity of a group axiom?

Because a law of composition is not a binary operation; rather, it is a binary op.

Example N3.2. Let . Then the maps are 4 maps: (the identity), , ; , the “transposition”, defined by , ; and 2 constant functions , with , , and , . For on the maps we have the set , with a table (row , column , entry ):

Since and , composition is not commutative.

We can use the fact of associativity of composition to define a product of any given string of elements in . Observe .

Proposition N3.3 (2.1.4). For a law of composition on a set , there exists a unique way to define the product of elements , denoted temporarily by , with the following properties:

  1. , i.e. the product of any single is ;
  2. any product is given by our law of composition;
  3. for any with , .

Proof. By induction: take ; then for , but more formally lol. ∎

Identities, inverses and first examples

  • Often we use multiplicatively and additively for our identity element (the reason why is obvious).
  • If and , then (by associativity: ); is then invertible.
  • There may exist an element which is not invertible but has a right or a left inverse.
  • For the symmetric group , there are permutations of a set of elements.
  • All groups of order two are essentially the same group.