# 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.