# Commuting fraction more than half implies nilpotent

This article is about a result whose hypothesis or conclusion has to do with the fraction of group elements or tuples of group elements satisfying a particular condition.The fraction involved in this case is 1/2

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## Contents

## Statement

Suppose the commuting fraction of a finite group is strictly greater than . In other words, the probability that two elements, picked independently uniformly at random, commute, is more than half. Then, the finite group is a nilpotent group, and in particular, a finite nilpotent group.

In other words, if the number of conjugacy classes is more than half the order of the group, the group is nilpotent.

## Related facts

### Stronger facts

- Commuting fraction more than half implies direct product of class two 2-group and odd-order abelian group
- Commuting fraction more than five-eighths implies abelian

### Tightness

The group symmetric group:S3 is an example of a non-nilpotent group (in fact, a centerless group) whose commuting fraction is exactly .

### Converse

The converse is not true -- a nilpotent group, even a group of nilpotency class two, can have an arbitrarily low commuting fraction.

## Facts used

- Equivalence of definitions of commuting fraction: We use the definition that characterizes it as the quotient of the number of conjugacy classes to the order of the group.
- Commuting fraction of quotient group is at least as much as that of whole group

## Proof

### Showing that the group has a nontrivial center

We show that there have to be at least two conjugacy classes of size .**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

### Using induction

We use the fact that the quotient of the group by its center has a commuting fraction at least as large (fact (2)) to argue, by induction on the order, that the quotient group is nilpotent, hence so is the original group.