Equivalence of definitions of Sylow subgroup of normal subgroup

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This article gives a proof/explanation of the equivalence of multiple definitions for the term Sylow subgroup of normal subgroup
View a complete list of pages giving proofs of equivalence of definitions

The definitions that we have to prove as equivalent

The following are equivalent for a subgroup of a finite group:

  1. It is a Sylow subgroup of a normal subgroup of the whole group.
  2. It is the intersection of a normal subgroup of the whole group with a Sylow subgroup of the whole group.
  3. It is a Sylow subgroup inside its normal closure.

Related facts

Similar facts

Facts involving normal Sylow subgroups instead

Facts used

  1. Sylow subgroups exist
  2. Sylow implies order-dominating
  3. Product formula
  4. Sylow satisfies intermediate subgroup condition

Proof

(1) implies (2)

Given: A group G, a normal subgroup N of G, a p-Sylow subgroup P of N.

To prove: There exists a p-Sylow subgroup S of G, such that S \cap N = P.

Proof: By fact (1), there exists a p-Sylow subgroup T of G, and by fact (2), there exists g \in G such that P \le gTg^{-1}. Let S = gTg^{-1}. Then, P \le S. Also, P \le N, so P \le N \cap S.

On the other hand, since S is a p-group, so is N \cap S. But P is a p-subgroup of the largest possible order in N, forcing P = N \cap S.

(2) implies (1)

Given: A group G, a normal subgroup N of G, a Sylow subgroup S of G.

To prove: S \cap N is a Sylow subgroup of N.

Proof: By the product formula (fact (3)), we have:

\frac{|NS|}{|S|} = \frac{|N|}{|N \cap S|}

Since N is a normal subgroup, N and S are permuting subgroups, so NS is a subgroup. In particular, |NS|/|S| is a divisor of |G|/|S|, and is hence relatively prime to p. Thus, the right side is also relatively prime to p. So, N \cap S is a p-subgroup of N whose index is relatively prime to p, and hence must be a p-Sylow subgroup.

Equivalence of (1) and (3)

Clearly (3) implies (1). For the converse, use fact (4) (a Sylow subgroup in the whole group is also a Sylow subgroup in any intermediate subgroup). For a general statement to this effect, refer: Composition of subgroup property satisfying intermediate subgroup condition with normality equals property in normal closure.

References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 101, Exercise 9, Section 3.3, and Page 147, Exercise 34, Section 4.5 (Sylow's theorem)