Sylow's theorem with operators
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This article states and (possibly) proves a fact that involves two finite groups of relatively prime order, requiring the additional datum that at least one of them is solvable. Due to the Feit-Thompson theorem, we know that for two finite groups of relatively prime orders, one of them is solvable. Hence, the additional datum of solvability can be dropped. However, the proof of the Feit-Thompson theorem is considered heavy machinery.
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This article states and (possibly) proves a fact about a finite group and a Coprime automorphism group (?): a subgroup of the automorphism group whose order is relatively prime to the order of the group itself.
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Contents
Statement
Suppose is a finite group and is a coprime automorphism group of : is a subgroup of such that the orders of and are relatively prime. A subgroup of is termed -invariant if it equals its image under any element of .
Suppose further that either or is solvable.
Let be any prime.
- Existence (E): The set of -invariant -Sylow subgroups of is nonempty.
- Conjugacy (C): Any two -invariant -Sylow subgroups of are conjugate by an element in .
- Domination (D): Any -invariant -subgroup of is contained in a -invariant -Sylow subgroup of .
Note that since given two groups of coprime order, one of them is solvable, the assumption that either or is solvable is redundant.
Facts used
- Sylow subgroups exist
- Frattini's argument
- Normal Hall implies permutably complemented
- Hall retract implies order-conjugate: This states that any two complements to a normal Hall subgroup are conjugate. The proof of this is what requires the assumption that either the normal Hall subgroup or the quotient group is solvable.
- Normal Hall satisfies transfer condition
Proof
Proof of existence
Given: A finite group , a subgroup of such that and have relatively prime orders. A prime .
To prove: There exists a -invariant -Sylow subgroup of .
Proof: Let be the semidirect product of with . is a normal Hall subgroup of , and .
- By fact (1), has a -Sylow subgroup, say .
- Let . Then, : This follows from fact (2), and the fact that is normal in .
- There exists a permutable complement to in : Since is normal Hall in , is normal Hall in . Fact (3) thus applies.
- is a permutable complement to in : Since , we get . Since , this yields . Further, , so , which is trivial from the previous step.
- is conjugate to by an element of . Since , and are conjugate via an element of : This follows from fact (4), and the previous step, which shows that is also a permutable complement to in .
- Let be the element of conjugating to . Let be the conjugate of by . Then, : This follows from the fact that conjugation, being an automorphism, preserves normalizers.
- is a -invariant -Sylow subgroup of : This is an immediate corollary of the preceding step.
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 224, Theorem 2.2 (i)-(iii), Section 6.2 (The Schur-Zassenhaus theorem), ^{More info}