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

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:

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