Commuting of non-identity elements defines an equivalence relation between prime divisors of the order of a finite CN-group
Suppose is a finite CN-group (i.e., a finite group that is also a CN-group). Let be the set of prime divisors of the order of . Define a relation on as follows: for (possibly equal, possibly distinct), if and only if either or there exists a -Sylow subgroup of and a -Sylow subgroup of such that the following equivalent conditions hold:
- There exists a non-identity element of and a non-identity element of such that and commute.
- Every element of commutes with every element of .
Note that the conditions are equivalent because Sylow subgroups for distinct primes in CN-group centralize each other iff they have non-identity elements that centralize each other. Note also that these conditions are not as strong as the statement that every element of -power order in commutes with every element of -power order. It is simply a statement about being able to find two specific Sylow subgroups that centralize each other.
The claim is that is an equivalence relation on .
- Sylow subgroups for distinct primes in CN-group centralize each other iff they have non-identity elements that centralize each other is a pre-fact that is necessary to make sense of this.
- Equivalence classes of primes under commuting equivalence relation on finite CN-group give Hall subgroups that are nilpotent and TI
To prove that a relation is an equivalence relation, we need to prove that it is reflexive, symmetric, and transitive.
This follows by definition.
This follows by definition.
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We want to show that, for (possibly equal, possibly distinct) primes, implies . Note that if , the conclusion follows. If or , it again follows. Thus, we can assume that are all distinct.
Given: A finite CN-group , three distinct primes such that .
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||There is a -Sylow subgroup and a -Sylow subgroup of such that every element in centralizes every element in .||plus the interpretation of||from given|
|2||There is a -Sylow subgroup and a -Sylow subgroup of such that every element in centralizes every element in .||plus the interpretation of||from given|
|3||There exists a -Sylow subgroup of such that every element of commutes with every element of .||Fact (1)||Steps (1), (2)||The -Sylow subgroups of are conjugate in by Fact (1). Let be a conjugation operation in such that . Let . Since is an automorphism, it preserves centralizing, so the fact that centralizes implies that centralizes . Finally, by order considerations, is also a -Sylow subgroup of .|
|4||Let be a non-identity element of .||divides the order of , is -Sylow||From the given, is nontrivial.|
|5||is a nilpotent subgroup of .||is a CN-group||Step (4)||Direct from the given and being non-identity by Step (4).|
|6||and are respectively the -Sylow and -Sylow subgroups of .||Fact (2)||Steps (1), (3), (4)||By Steps (1) and (3), and centralize , hence are both in . Note that since they are already Sylow in , Fact (2) gives that they are Sylow in .|
|7||and are respectively -Sylow and -Sylow subgroups of that centralize each other.||Fact (3)||Steps (1), (3), (5), (6)||By Fact (3) and Step (5), the Sylow subgroups in form an internal direct product, hence centralize each other. Apply to Step (6) to get that and centralize each other. Note that they are Sylow by Steps (1) and (3).|
|8||Definition of||Step (7)||Step-direct|