Normal Sylow satisfies transfer condition
From Groupprops
This article gives the statement, and possibly proof, of a subgroup property (i.e., normal Sylow subgroup) satisfying a subgroup metaproperty (i.e., transfer condition)
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Contents
Statement
Statement with symbols
Suppose is a normal Sylow subgroup of a finite group and is any subgroup. Then is a normal Sylow subgroup of .
Related facts
More on transfer condition for normal and Sylow subgroups
- Normality satisfies transfer condition
- Sylow does not satisfy transfer condition
- Hall does not satisfy transfer condition
- Normal Hall satisfies transfer condition
Intersecting a normal subgroup with a Sylow subgroup
- Equivalence of definitions of Sylow subgroup of normal subgroup: This states that a subgroup that is the intersection of a Sylow subgroup and a normal subgroup is a Sylow subgroup of the normal subgroup. Conversely, a subgroup expressible as a Sylow subgroup of a normal subgroup is expressible as an intersection of the normal subgroup with a Sylow subgroup of the whole group.
Facts used
Proof
Given: A finite group , a normal Sylow subgroup , and a subgroup .
To prove: is a normal Sylow subgroup of .
Proof: By the second isomorphism theorem, is normal in , and we have:
.
Here, is a subgroup of . Since has order relatively prime to , so does (by fact (2)). Thus, has order relatively prime to , so is relatively prime to . Thus, is a normal -subgroup of with index relatively prime to , and is thus a normal -Sylow subgroup of .
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 147, Exercise 32, Section 4.5 (Sylow's theorem)