Cohomology tree probability distribution: Difference between revisions

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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 $p^{n}$ .

@@ Line 5: / Line 5: @@
 Let <math>p</math> be a [[prime number]] and <math>n</math> be a positive integer. The '''cohomology tree probability distribution''' is a probability distribution on the set of isomorphism classes of groups of order <math>p^n</math> defined inductively as follows.
-===Base case of inductive definition: definition for prime order (<math>n = 1</math>)===
+===Base case of inductive definition: definition for prime order (n = 1)===
 There is only one group of order <math>p</math>, 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 <math>p^n</math> based on probability distribution for groups of order <math>p^{n-1}</math> for <math>n > 1</math>===
+===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 <math>p^{n-1}</math> on the group of order <math>p</math>, to define group extensions. Let's go over this more specifically.