# Cohen-Lenstra measure

## Definition

Let $p$ be a prime number. The Cohen-Lenstra measure is a probability measure on the set of isomorphism classes of finite abelian $p$-groups (i.e., the set of abelian groups whose order is a power of the prime $p$), where the measure value on a particular isomorphism class of abelian $p$-group $P$ is:

$\frac{1}{\left|\operatorname{Aut}(P)\right|} \prod_{j=1}^\infty \left( 1 - \frac{1}{p^j} \right)$

The Cohen-Lenstra measure value for a group $P$ can be defined as the limit as $n \to \infty, d \to \infty$ of the probability that the kernel of the map:

$(\mathbb{Z}/p^n\mathbb{Z})^d \to (\mathbb{Z}/p^n\mathbb{Z})^d$

is isomorphic to $P$, where probability here simply means the quotient of the number of homomorphisms $(\mathbb{Z}/p^n\mathbb{Z})^d \to (\mathbb{Z}/p^n\mathbb{Z})^d$ with kernel isomorphic to $P$ to the total number of homomorphisms $(\mathbb{Z}/p^n\mathbb{Z})^d \to (\mathbb{Z}/p^n\mathbb{Z})^d$.

### Equivalence of definitions

Further information: equivalence of definitions of Cohen-Lenstra measure

## Particular cases

### Value of the product

The product $\prod_{j=1}^\infty \left( 1 - \frac{1}{p^j} \right)$ is the $q$-Pocchammer symbol with parameters $1/p$ and $1/p$. The approximate values are given below:

$p$ $\prod_{j=1}^\infty \left( 1 - \frac{1}{p^j} \right)$ Its reciprocal (equal to $\sum 1/|\operatorname{Aut}(P)|$)
2 $\! 0.2887880950866024$ $\! 3.462746619455064$
3 $\! 0.5601260779279490$ $\! 1.785312341998534$
5 $\! 0.7603327958712324$ $\! 1.315213555735345$
7 $\! 0.8367954070890379$ $\! 1.195035239830847$
11 $\! 0.900832706809715$ $\! 1.110084028300309$
13 $\! 0.917162472540942$ $\! 1.090319359916201$
17 $\! 0.937716969709066$ $\! 1.066419860472673$
19 $\! 0.944598742929940$ $\! 1.058650572515285$
23 $\! 0.954631535623623$ $\! 1.047524581666726$

For the prime $p = 2$, we list the Cohen-Lenstra measures of some small abelian $p$-groups:

$P$ prime-base logarithm of order partition corresponding to decomposition as product of cyclic groups order of automorphism group Cohen-Lenstra measure
trivial group 0 empty 1 $\! 0.288788$
cyclic group:Z2 1 1 1 $\! 0.288788$
cyclic group:Z4 2 2 2 $\! 0.144394$
Klein four-group 2 1 + 1 6 $\! 0.0481313$
cyclic group:Z8 3 3 4 $\! 0.072197$
direct product of Z4 and Z2 3 2 + 1 8 $\! 0.0360985$
elementary abelian group:E8 3 1 + 1 + 1 168 $\! 0.00171898$