# Frobenius' normal p-complement theorem

This article gives the statement, and possibly proof, of a normal p-complement theorem: necessary and/or sufficient conditions for the existence of a Normal p-complement (?). In other words, it gives necessary and/or sufficient conditions for a given finite group to be a P-nilpotent group (?) for some prime number .
View other normal p-complement theorems

## Statement

Let  be a finite group and  be a prime number. Then, the following are equivalent:

1.  has a -Sylow subgroup  that is conjugacy-closed in : any two elements of  that are conjugate in  are conjugate in .
2. There is a normal p-complement in : a normal subgroup whose index is a power of  and whose order is relatively prime to . In other words, any -Sylow subgroup is a retract of .
3. For every non-identity -subgroup  of , the subgroup  has a normal p-complement
4. For every non-identity -subgroup , the quotient  is a -group.

## Proof

### (2) implies (3)

(Note: This implication uses nothing about the special nature of normalizers of -subgroups, and works for all subgroups).

Given: A finite group , a prime , a -Sylow subgroup  of . A normal subgroup  of  such that  and  is trivial.

To prove: If  is a non-identity -subgroup of , then  has a normal -complement.

Proof: Let . By the second isomorphism theorem, we have:

.

Since  divides , and  equals the order of ,  is a power of . Thus, so is the order of the right side, .

Thus,  is a normal subgroup of  whose order is relatively prime to  (since it is also a subgroup of ) and whose index is a power of . Thus, it is a normal -complement in .

### (3) implies (4)

Given: A finite group , a prime , a -Sylow subgroup  of . For any non-identity -subgroup ,  has a normal -complement.

To prove: If  is a non-identity -subgroup of , then  is a -group.

Proof: Proof: Let . Let  be a normal -complement in .

Now,  and  are both normal subgroups of  and since the order of  is relatively prime to ,  is trivial. Thus, . Thus, by fact (1):

.

Since the left side is a power of , so are both terms of the right side. In particular,  is a -group.

### (2) implies (1)

This follows from fact (3).

### (1) implies (2)

This follows from fact (4).

## References

### Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page Theorem 4.5, Chapter 7 (Fusion, transfer and p-factor groups), warning.png",Chapter7(Fusion,transferandp-factorgroups)" is not declared as a valid unit of measurement for this property.warning.png",Chapter7(Fusion,transferandp-factorgroups)" is not declared as a valid unit of measurement for this property.warning.png",Chapter7(Fusion,transferandp-factorgroups)" is not declared as a valid unit of measurement for this property.More info