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 $p^{n-1}$. This gives the cohomology tree probability distribution for isomorphism classes of groups of order$p^n$.

Notes

• Prime power order implies not centerless can be used to show that every group of order $p^n$ has a positive weight. Specifically, the center is nontrivial, so it has a subgroup of prime order, and the whole group can be obtained as an extension with normal subgroup as this group of prime order, and quotient of order $p^{n-1}$.
• The term "cohomology tree" refers to the idea that this process of generating extensions can be depicted using a tree, whose root is the group of order $p$, the next layer is the groups of order $p^2$, and so on. Each layer of the tree is groups of order $p^n$ for some $n$. Each node of the tree can be assigned a numeric value which is its weight in its probability distribution; the sum of the numeric values in each layer is 1. The value for a given node is the sum of the values of all its children.
• Note that in the above tree, a single isomorphism class of groups of order $p^n$ may appear as multiple nodes, based on different ways of building the group (through different choices of series of subgroups with each successive quotient of order $p$). If we were to collapse all nodes that are isomorphic groups, the graph would no longer be a tree. If we gave the edges direction (going from the group of order $p^{n-1}$ to the group of order $p^n$) then we would get a "graded graph" -- a directed acyclic graph rooted at the group of order $p$ where the distance from the root to any isomorphism class of groups of order $p^n$ is $n-1$, though there could be multiple such paths.

Worked example for groups of prime-square order

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 $H^2(C,C)$.

This cohomology group is worked out at second cohomology group for trivial group action of group of prime order on group of prime order. The group $H^2(C,C)$ has order $p$, with the identity element corresponding to the extension that is the elementary abelian group of order $p^2$, and the remaining $p-1$ non-identity elements corresponding to the extension that is the cyclic group of order $p^2$.

The cohomology tree probability distribution therefore works out to the following:

• Cyclic group of order $p^2$: This appears $p-1$ out of $p$ times, so it gets weight $(p-1)/p$ or equivalently $1-1/p$.
• Elementary abelian group of order $p^2$: This appears 1 out of $p$ times, so it gets weight $1/p$.

Sketch of worked example for groups of order 8

Let's work out the cohomology tree probability distribution for groups of order 8.

For groups of order 4, we have, per the above distribution for groups of prime-square order, that cyclic group:Z4 and Klein four-group (the elementary abelian group of order 4) both have weight 1/2.

To get the probability distribution for groups of order 8, we need to look at two cohomology groups.

Second cohomology group for trivial group action of cyclic group of order 4 on cyclic group of order 2

This is covered in second cohomology group for trivial group action of Z4 on Z2. The cyclic group of order 4 itself has weight 1/2 in the cohomology tree probability distribution on groups of order 4, so the weights need to be multiplied by 1/2.

The result is that the second cohomology group is cyclic of order two, and produces two isomorphism classes:

• Direct product of Z4 and Z2 occurs once, as the identity element, so it gets weight 1/2 times 1/2 = 1/4.
• Cyclic group:Z8 occurs once, as the non-identity element, so it gets weight 1/2 times 1/2 = 1/4.

Second cohomology group for trivial group action of Klein four-group on cyclic group of order 2

This is covered in second cohomology group for trivial group action of V4 on Z2. The Klein four-group itself has weight 1/2 in the cohomology tree probability distribution on groups of order 4, so the weights need to be multiplied by 1/2.

The result is that the second cohomology group is cyclic of order 8, and produces four isomorphism classes:

• Elementary abelian group:E8 occurs once, as the identity element, so it gets weight 1/2 times 1/8 = 1/16.
• Direct product of Z4 and Z2 occurs 3 times, so it gets weight 1/2 times 3/8 = 3/16.
• Dihedral group:D8 occurs 3 times, so it gets weight 1/2 times 3/8 = 3/16.
• Quaternion group occurs once, so it gest weight 1/2 times 1/8 = 1/16.

Summing up

• Cyclic group:Z8 is in only the first list, with a total weight of 1/4.
• Direct product of Z4 and Z2 is in both lists, with weights of 1/4 and 3/16, totaling to 7/16.
• Dihedral group:D8 is in only the second list, with a total weight of 3/16.
• Quaternion group is in only the second list, with a total weight of 1/16.
• Elementary abelian group:E8 is in only the second list, with a total weight of 1/16.

We can verify that these weights add up to 1, confirming that we have obtained a probability distribution on the groups of order 8.