**<The Detail of This Chapter.pdf>**

operation, like set, is the most fundamental notion in mathematics, so we have to see how about the structure of a set when we define an operation in it.

1. the most general operation, a composition +: a+b=c

At first, we can make the composition is associative in the sense that: a+b=c, c+d=e, b+d=f, a+f=h, then e=h. This constraint brings a kind of order to the set, and as soon as we endue a set with an associative operation, we will see the law's meaning from the structure of that set.

if we define an associative composition + in a set A, then (A, +) is a
*semi-group*.

associative law means: if a+b+...+n=z, there is only one z in the semi-group.

if the operation is non-associative, such z is NOT only one at sometimes.

we can always use a multiplication table to express the composition in a set, then if the composition holds associative law, what we can say about the table?

2. for the completeness of operation itself, we need introduce an element e to the set that for any element x: x+e=x or e+x=x. this means that composition can change an element, or can not change an element.

if there is e_{1}: e_{1}+x=x, for any x; and there is e: e_{2}+x=x,
for any x. does e_{1}=e_{2}?

3. unit

if we want to investigate the structure of a group, let's begin from: when we can say two groups have the same construction? and when they don't? this question introduce the notion of

isomorphism: when we can get a 1-to-1 correspondence between two groups, and this correspondence retain composition