Equivalence of definitions of Sylow subgroup of normal subgroup
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
Contents
The definitions that we have to prove as equivalent
The following are equivalent for a subgroup of a finite group:
- It is a Sylow subgroup of a normal subgroup of the whole group.
- It is the intersection of a normal subgroup of the whole group with a Sylow subgroup of the whole group.
- It is a Sylow subgroup inside its normal closure.
Related facts
Similar facts
- Intersection of Hall subgroup and permuting subgroup is Hall in that subgroup
- Intersection of Hall subgroup and normal subgroup implies Hall subgroup of normal subgroup
- Equivalence of definitions of Sylow subgroup of permutable subgroup
Facts involving normal Sylow subgroups instead
- Normal Sylow satisfies transfer condition: This states that the intersection of a normal Sylow subgroup with any subgroup is normal Sylow in the other subgroup.
- Normal Hall satisfies transfer condition
Facts used
- Sylow subgroups exist
- Sylow implies order-dominating
- Product formula
- Sylow satisfies intermediate subgroup condition
Proof
(1) implies (2)
Given: A group , a normal subgroup of , a -Sylow subgroup of .
To prove: There exists a -Sylow subgroup of , such that .
Proof: By fact (1), there exists a -Sylow subgroup of , and by fact (2), there exists such that . Let . Then, . Also, , so .
On the other hand, since is a -group, so is . But is a -subgroup of the largest possible order in , forcing .
(2) implies (1)
Given: A group , a normal subgroup of , a Sylow subgroup of .
To prove: is a Sylow subgroup of .
Proof: By the product formula (fact (3)), we have:
Since is a normal subgroup, and are permuting subgroups, so is a subgroup. In particular, is a divisor of , and is hence relatively prime to . Thus, the right side is also relatively prime to . So, is a -subgroup of whose index is relatively prime to , and hence must be a -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)