下面这个例子是怎么建立起群的概念的？

映射的复合我明白，平面平移的例子我也明白，但是中间M=N=P能说明什么？后面群的概念是如何建立的？？

Let$M$, $N$,$P$be sets and let

$f:N\rightarrow M$

$g:P\rightarrow N$

be maps between them. The product or composition of f and g is the map

$fg:P\rightarrow M$

defined as

$(fg)(a) = f(g(a)) \forall a \in P$

i.e., the result of successive application of, first, g and, then, f . In particular, when M = N = P, we obtain an operation on the set of all maps from M to itself. This operation provides many important examples of algebraic structures that are called groups. For example, according to the axioms of Euclidean geometry, the product of two motions of the plane is again a motion. When we consider the operation of multiplication on the set of all such motions, we obtain the algebraic structure called the group of motions of the plane.