Cohomology tree probability distribution

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Definition

Let $p$ be a prime number and $n$ be a positive integer. The cohomology tree probability distribution is a probability distribution on the set of isomorphism classes of groups of order $p^{n}$ defined inductively as follows.

Base case of inductive definition: definition for prime order (n = 1)

There is only one group of order $p$, namely the cyclic group (see group of prime order). Therefore there is only one allowed probability distribution, that assigns a weight of 1 to this unique group.

Induction step: probability distribution for groups of order p^n based on probability distribution for groups of order p^{n - 1} for n > 1

Conceptually, the induction step is essentially a convolution product with the convolution process being the use of the second cohomology group for the trivial action of groups of order $p^{n-1}$ on the group of order $p$, to define group extensions. Let's go over this more specifically.

Denote by $C$ the cyclic group of order $p$.

For any group $G$ of order $p^{n-1}$, the elements of the second cohomology group for trivial group action $H^{2}(G,C)$ correspond to extensions with central subgroup $C$ and quotient group $G$. Each of these extensions is therefore a group of order $p^{n}$. For each element of $H^{2}(G,C)$, give the isomorphism class (as a group of order $p^{n}$) of the corresponding group extension, a weight that equals the probability distribution weight of $G$ divided by the size of $H^{2}(G,C)$.

Now, sum up these weights as $G$ varies over all isomorphism classes of groups of order $p^{n}$, to get a probability distribution for isomorphism classes of groups of order$p^{n}$.

Worked example

Let's work out the cohomology tree probability distribution for groups of prime-square order, i.e., groups of order $p^{2}$ where $p$ is a prime number.

Denote by $C$ the cyclic group of order $p$.

Since there's only one group of order $p$, namely $C$, the cohomology tree probability distribution for order $p^{2}$ is just based on the distribution of isomorphism classes of groups corresponding to the second cohomology group for trivial group action $H^{2}(C,C)$.