Cohen-Lenstra measure

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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