**Notation. **For a group and a subgroup of the centralizer and the normalizer of in are denoted by and respectively.

**Lemma.** Let be a finite group and let be a Sylow subgroup of Suppose that and for some There exists such that

*Proof*. Let the centralizer of in Clearly because and so commutes with every element of Also if then

Thus So both and are Sylow subgroups of and so, since there exists such that Then clearly and, since we have

**Burnside’s Normal Complement Theorem.** (Burnside, 1900) Let be a finite group and let be a Sylow subgroup of If then there exists a normal subgroup of such that and

The normal subgroup in the theorem is called the normal complement of

*Proof*. Clearly is abelian because Let and Note that, since is a Sylow subgroup, and hence there exist integers such that

Now let The theorem in part (2) gives us elements and integers such that and where is the group homomorphism defined in the theorem in part (1). Note that, by the equation (7) in the proof of the theorem in part (2), for all Thus, by the Lemma, for every there exists some such that Hence for all because Thus

Now, (1) and (2) give us

because and so So (3) shows that is onto. Let Then

We can now easily prove that is the required subgroup in the theorem:

1) Obviously is normal because is the kernel of the group homomorphism To see why let Then and But then using (1) we will have

2) To see why we first note that because as we just showed. We also have by (4). Thus and hence