# Cohen-Lenstra measure

## Definition

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

The Cohen-Lenstra measure value for a group can be defined as the limit as of the probability that the kernel of the map:

is isomorphic to , where *probability* here simply means the quotient of the number of homomorphisms with kernel isomorphic to to the total number of homomorphisms .

### Equivalence of definitions

`Further information: equivalence of definitions of Cohen-Lenstra measure`

## Particular cases

### Value of the product

The product is the -Pocchammer symbol with parameters and . The approximate values are given below:

Its reciprocal (equal to ) | ||
---|---|---|

2 | ||

3 | ||

5 | ||

7 | ||

11 | ||

13 | ||

17 | ||

19 | ||

23 |

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

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 | |

cyclic group:Z2 | 1 | 1 | 1 | |

cyclic group:Z4 | 2 | 2 | 2 | |

Klein four-group | 2 | 1 + 1 | 6 | |

cyclic group:Z8 | 3 | 3 | 4 | |

direct product of Z4 and Z2 | 3 | 2 + 1 | 8 | |

elementary abelian group:E8 | 3 | 1 + 1 + 1 | 168 |