【正文】 (2) whenever any 。 關(guān)鍵詞：基本同態(tài)定理；直積；內(nèi)同構(gòu)。（2）If is a normal in, then is a normal in。 Direct Products。（2） if and only if .By using the above lemmas we can obtain the following mainly results.Theorem 1. Let G and H be two groups. Suppose JG and KH, then and .Proof. First we will prove . For any and every . We have:.Since and , we can get , . .Thus . We make use of the FHT to prove that is isomorphic to. Therefore we must look for a group homomorphism from onto and determine the kernel of it. In fact one can define correspondencedefined by . Clearly, , there must be to satisfy. Thus, is onto.Because of JG, we have for, similarly, for .When , there are .For any , we have ====.Hence . Therefore is group a homomorphism from ontoand is the identity of. For any , then, according to the property of coset, we can get: if and only if and , . =. Now let we look at our proof: , is a group homomorphism from onto and the kernel of is . According to the FHT, we can get .Theorem 2. Let is a group homomorphism from onto .If and , then where .Proof: According to Lemma 2.[2] (2), we know .To establish , we firstly need to construct a mapping and prove is a group homomorphism from onto . We give the mapping defined by where =.For , since is a surjection from to , we must be found such that .Thus is onto.For arbitrary , Therefore is a group homomorphism.We will now show , in fa