Cohomology tree probability distribution

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Cohomology tree probability distribution, all facts related to Cohomology tree probability distribution) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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 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 V2.