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【正文】 1. Let be a subgroup of a group with symbol ≤, we say is the normal subgroup of if one of the following conditions hold. To simplify matters, we write . (1) for any 。(2) whenever any 。(3) for every and any .Definition 2. The kernel of a group homomorphism from to a group with identity is the set . The kernel of is denoted by .Definition 3. Let be a collection of groups. The external direct product of , 廣西自然科學(xué)基金(0447038)資助項(xiàng)目written as , is the set of all mtuples for which the its ponent is an element of , and the operation is ponentwise. In symbols =,where is defined to be Notice that it is easily to verify that the external direct product of groups is itself a group.[4] Definition 4. Let be a group and be a subgroup of . For any , the set is called the left coset of in containing . Analogously is called the right coset of H in containing .Lemma 1.[1] ( The fundamental homomorphism theorem) Let be a group homomorphism from to . Then the = is the normal subgroup of , and . To simplify matters, we call the theorem as the FHT.Lemma 2.[2] Let be a group homomorphism from to . Then we have the following properties:（1）If is a subgroup of , then is a subgroup of 。（2）If is a normal in, then is a normal in。（3）If is a subgroup of , then is a subgroup of 。（4）If is a normal subgroup of , then is a normal subgroup of Lemma 3.[3] Let be a homomorphism from a gr
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