S. D. V. Stephens

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

N2 Introduction to Groups

Course content

  1. (A bit of) group theory: Artin Ch. 2 (2nd ed.).
  2. Field theory + vector spaces + linear algebra + multilinear algebra: Axler (4th ed.).
  3. More group theory: Artin Ch. 6–7.
  4. Intro to representation theory: Artin + Fulton–Harris.

Groups

A group is a binary operation plus the group axioms:

  1. identity element;
  2. inverse element;
  3. associativity.

It is an abstract concept for operations on things. Operations: addition, composition, …; “things”: numbers, functions, ….

Definition N2.1 (Group). A group consists of a set (or ) with an operation (law of composition), i.e. a map written or or , …, satisfying the group axioms:

  1. there exists an identity element such that for all ;
  2. for all there exists such that ;
  3. associativity: for all , .

Remark.

  1. The axioms imply the cancellation law, i.e. for all , .
  2. We write .
  3. Generalizations: if we omit the second axiom (inverses), it is a monoid.
  4. If we omit the first axiom (identity), it is a semigroup.
  5. If we have a group which is commutative, it is an abelian group. When this is the case we use instead of , , etc., by convention.

Examples

Example N2.2.

  1. The trivial group , with operation .
  2. , , , .
  3. Groups with two elements. If , let be the identity; there exists , and is its own inverse, since , , , . Think (even, odd) with , or under multiplication (). All of these groups are isomorphic to each other.
  4. with operation (addition mod ): Similarly , with if and else, also under addition.
  5. Nonzero numbers with multiplication: with multiplication (identity , inverse ). The unit circle is a subgroup of .

(Margin: quaternions?)

Symmetries and permutations (the big boy)

Definition N2.3 (Injective, surjective, bijective).

  • is injective (one-to-one) if for all , ; or, (each element of is hit by at most one ).
  • is surjective (onto) if for all there exists such that .
  • is bijective (one-to-one correspondence) if it is injective and surjective.

Definition N2.4 (Permutation). A permutation of is a bijection . The set of permutations of , with composition, is a group, .

  • Identity: , .
  • Inverses come from the inverse mapping?
  • Note: composition is associative.

Definition N2.5 (Symmetric group). The symmetric group is the symmetric group on elements.

Example N2.6. : has the symmetries of an equilateral triangle. These are the identity, the rotations by and , and the reflections across the three axes.

An equilateral triangle with vertices a, b, c, its three reflection axes drawn dashed, and an arrow showing rotation by 120 degrees about the centre.

As arrangements of :

Example N2.7 (Groups of invertible matrices). the general linear group; likewise , …, , … etc.