# Commuting of non-identity elements defines an equivalence relation between prime divisors of the order of a finite CN-group

## Statement

Suppose $G$ is a finite CN-group (i.e., a finite group that is also a CN-group). Let $\pi$ be the set of prime divisors of the order of $G$. Define a relation $\sim$ on $\pi$ as follows: for $p,q \in \pi$ (possibly equal, possibly distinct), $p \sim q$ if and only if either $p = q$ or there exists a $p$-Sylow subgroup $P$ of $G$ and a $q$-Sylow subgroup $Q$ of $G$ such that the following equivalent conditions hold:

1. There exists a non-identity element $x$ of $P$ and a non-identity element $y$ of $Q$ such that $x$ and $y$ commute.
2. Every element of $P$ commutes with every element of $Q$.

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 $p$-power order in $G$ commutes with every element of $q$-power order. It is simply a statement about being able to find two specific Sylow subgroups that centralize each other.

The claim is that $\sim$ is an equivalence relation on $\pi$.

## Proof

To prove that a relation is an equivalence relation, we need to prove that it is reflexive, symmetric, and transitive.

### Reflexivity

This follows by definition.

### Symmetry

This follows by definition.

### Transitivity

This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format

We want to show that, for $p,q,r$ (possibly equal, possibly distinct) primes, $p \sim q, q \sim r$ implies $p \sim r$. Note that if $p = r$, the conclusion follows. If $p = q$ or $q = r$, it again follows. Thus, we can assume that $p,q,r$ are all distinct.

Given: A finite CN-group $G$, three distinct primes $p,q,r$ such that $p \sim q, q \sim r$.

To prove: $p \sim r$

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 There is a $p$-Sylow subgroup $P$ and a $q$-Sylow subgroup $Q$ of $G$ such that every element in $P$ centralizes every element in $Q$. $p \sim q$ plus the interpretation of $\sim$ from given
2 There is a $q$-Sylow subgroup $Q_1$ and a $r$-Sylow subgroup $R_1$ of $G$ such that every element in $Q_1$ centralizes every element in $R_1$. $q \sim r$ plus the interpretation of $\sim$ from given
3 There exists a $r$-Sylow subgroup $R$ of $G$ such that every element of $Q$ commutes with every element of $R$. Fact (1) Steps (1), (2) The $q$-Sylow subgroups $Q,Q_1$ of $G$ are conjugate in $G$ by Fact (1). Let $\varphi$ be a conjugation operation in $G$ such that $\varphi(Q_1) = Q$. Let $R = \varphi(R_1)$. Since $\varphi$ is an automorphism, it preserves centralizing, so the fact that $Q_1$ centralizes $R_1$ implies that $Q = \varphi(Q_1)$ centralizes $R = \varphi(R_1)$. Finally, by order considerations, $R$ is also a $r$-Sylow subgroup of $G$.
4 Let $x$ be a non-identity element of $Q$. $q$ divides the order of $G$, $Q$ is $q$-Sylow From the given, $Q$ is nontrivial.
5 $C_G(x)$ is a nilpotent subgroup of $G$. $G$ is a CN-group Step (4) Direct from the given and $x$ being non-identity by Step (4).
6 $P$ and $R$ are respectively the $p$-Sylow and $r$-Sylow subgroups of $C_G(x)$. Fact (2) Steps (1), (3), (4) By Steps (1) and (3), $P$ and $R$ centralize $Q$, hence are both in $C_G(x)$. Note that since they are already Sylow in $G$, Fact (2) gives that they are Sylow in $C_G(x)$.
7 $P$ and $R$ are respectively $p$-Sylow and $r$-Sylow subgroups of $G$ that centralize each other. Fact (3) Steps (1), (3), (5), (6) By Fact (3) and Step (5), the Sylow subgroups in $C_G(x)$ form an internal direct product, hence centralize each other. Apply to Step (6) to get that $P$ and $R$ centralize each other. Note that they are Sylow by Steps (1) and (3).
8 $p \sim r$ Definition of $\sim$ Step (7) Step-direct

## References

### Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 408, Theorem 14.2.5(i), (The theorem is stated for minimal simple CN-groups of odd order, but this part of the theorem does not require those assumptions.)More info