Examples and non-examples

Two fundamental examples

Before introducing the formal definition of a group, it’s worthwhile to look at two fundamental examples:

  • The integers modulo $n$ with the addition operation. video pdf
  • Symmetries of an equilateral triangle. video pdf
  • Symmetries and permutations. video pdf

Formal definition of a group

With our two examples at hand, we can give a set of axioms that define a group.

  • Axioms defining a group and first examples video pdf

More examples

Here are a suite of additional examples of groups. See pdf and videos:

  • The congruence classes of integers relatively prime to $N$, with the multiplication operation – called $U(N)$ in the book. video
  • The symmetries of a square. video. See also this animation.
  • The 2x2 matrices with real entries under addition $M_2(\mathbf{R})$. video
  • The invertible 2x2 matrices with real entries with multipliation ($\mathrm{GL}_2(\mathbf{R})$). video
  • The quaternion group with elements $\{\pm 1, \pm i, \pm j, \pm k\}$. video
  • The non-zero complex numbers, and the complex numbers of norm 1. video


  • The symmetric group of permutations on $n$ elements, defined as bijective maps from $\{1,\ldots, n\}$ to itself, with composition as the operation. video pdf


Three examples illustrating things that aren’t groups. video pdf

  • The integers are not a group under multiplication (no inverses)
  • Two by two integer matrices with non-zero determinant are not a group under multiplication (no inverse).
  • The cross product is an example of a non-associative operation.

Basic Theorems on Groups and Subgroups

Basic Theorems on Groups

See these notes and video

  • A group has exactly one identity element.
  • Every element of a group has exactly one inverse element.
  • Solving equations in groups.
  • A version of the laws of exponents that take account of non-commutativity hold in a group.


See these notes and video.

  • A subgroup is a subset $H$ of a group $G$ that is a group when the operation of $G$ is applied to elements of $H$.
  • Examples of subgroups.

Subgroup theorems

See these notes and video

  • Two tests for whether a subset is a subgroup.