One of the first bits of ``math'' you learned as a student is ordinary
arithmetic: how to add and subtract two numbers, how to multiply and
divide two numbers. Although you may not have realized it at the time,
you were learning not only your first arithmetic, but your first
*group theory!* However, group theory is a lot more general than
``just'' arithmetic.

A group
is a set of elements that is *closed*
with respect to an operation of *composition* (think
``multiplication'', although it often isn't, so people use a
multiplication-like symbol,
, instead of
when discussing
abstract groups) that turns two elements into one (not necessarily
different) element:

(5.1) |

The set of elements has to contain one special element, the
*identity* element
, such that:

Every element must have a corresponding *inverse* element in the
group:

Finally, the composition rule has to be *associative*:

The simplest, and smallest, group consists of only one element, the identity element, which is its own inverse, represented by a single line:

where we see that the identity element is always its own inverse and forms all by itself a special group called the trivial group. The trivial group is denoted (or sometimes ).

You are familiar with a number of groups already, even though you may
not have thought of them as such. The set of positive and negative
integers, with the *addition* symbol used for composition, forms a
group, with *zero* being the identity and a negative number being
the inverse of a positive one and vice versa. The set of integers
together with multiplication used as a composition rule is **not a
group!** It is associate, it is closed, and it has an identity (the
integer one) but the *inverse* of almost all elements is not in the
group. The set of all *rational numbers* excluding zero forms a
group with respect to multiplication (why must we exclude zero?).
Mathematicians notationally write this exclusion with the
symbol, for example the general multiplicative group over the
set (field) of all complex numbers
is denoted
.