Groups
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.
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
Permutations
- 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
Non-examples
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
- 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.
Subgroups
- 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
- Two tests for whether a subset is a subgroup.