Frobenius' normal p-complement theorem

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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.

Related facts

Facts used

  1. Index is multiplicative
  2. Second isomorphism theorem
  3. Retract implies conjugacy-closed
  4. Conjugacy-closed and Sylow implies retract

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