在本論文中，利用semidirect product定理去建構一些可以做的個數小的群。但是，這並不是所有的群都可以用semidirect product去建構，所以最後做一個群的建構，不是用semidirect product這種方法建構的。首先，利用semidirect product做一些可以建構的群，之後推廣出order為p^3(p an odd prime);pq(with p and q are primes,p is smaller than q)和4p(p:prime)的群，這些特別的order的群所建構出來的形式是固定的。最後建構一個不能用semidirect product這個定理做的16個元素的群，所以我們利用Sylow's Thorem直接去建構出來。

We study the "semidirect product" of two groups H and K, which is a generalization of the direct product of H and K obtained by relaxing the requirement that both H and K be normal.

    Semidirect product construction will enable us to build a "larger" group from the groups H and K.

    If G contains subgroups isomorphic to H and K, in this case the subgroup H will be normal in G but the subgroup K will not necessarily be normal.

    Thus, we shall be able to construct non-abelian groups even if H and K are abelian.

1.Background.................................	1

2.Constructions with semidirect product......	3

Example1.	............................................ .3
(The classification of groups of order p^3, p is odd prime)

Example2..............................................	6
(The classification of groups of order pq(p,q:prime;p is smaller than q))


Example3..............................................	7
(The classification of groups of order 4p(p:prime;p is greater than 3))

Example4.	.............................................10
(The classification of groups of order 30)

3.Constructions without semidirect product............	13

Example:...............................................13
(The classification of groups of order 16)

4.References..........................................	49

[1]  S. Dummit and M. Foote, Abstract Algebra, 3th Ed., John Wiley and Sons, Inc., 2004

[2]  W. Hungerford, Algebra, Springer Science+Business Media, LLC.,1974

[3]  W. Nicholson, Introduction to Abstract Algebra, 2th Ed.,John Wiley and Sons, Inc., 1999