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.
