GAP (HAP commands) - Chapter 12: Poincare series

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12 Poincare series
12.1
12.1-1 EfficientNormalSubgroups
12.1-2 ExpansionOfRationalFunction
12.1-3 PoincareSeries
12.1-4 PoincareSeriesPrimePart
12.1-5 PoincareSeriesLHS
12.1-6 Prank

### 12 Poincare series

#### 12.1

##### 12.1-1 EfficientNormalSubgroups

`‣ EfficientNormalSubgroups`( G ) | ( function )

`‣ EfficientNormalSubgroups`( G, k ) | ( function )

Inputs a prime-power group G and, optionally, a positive integer k. The default is k=4. The function returns a list of normal subgroups N in G such that the Poincare series for G equals the Poincare series for the direct product (N × (G/N)) up to degree k.

Examples:

##### 12.1-2 ExpansionOfRationalFunction

`‣ ExpansionOfRationalFunction`( f, n ) | ( function )

Inputs a positive integer n and a rational function f(x)=p(x)/q(x) where the degree of the polynomial p(x) is less than that of q(x). It returns a list [a_0 , a_1 , a_2 , a_3 , ... ,a_n] of the first n+1 coefficients of the infinite expansion

f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + ... .

Examples:

##### 12.1-3 PoincareSeries

`‣ PoincareSeries`( G, n ) | ( function )

`‣ PoincareSeries`( R, n ) | ( function )

`‣ PoincareSeries`( L, n ) | ( function )

`‣ PoincareSeries`( G ) | ( function )

Inputs a finite p-group G and a positive integer n. It returns a quotient of polynomials f(x)=P(x)/Q(x) whose coefficient of x^k equals the rank of the vector space H_k(G,Z_p) for all k in the range k=1 to k=n. (The second input variable can be omitted, in which case the function tries to choose a "reasonable" value for n. For 2-groups the function PoincareSeriesLHS(G) can be used to produce an f(x) that is correct in all degrees.)

In place of the group G the function can also input (at least n terms of) a minimal mod p resolution R for G.

Alternatively, the first input variable can be a list L of integers. In this case the coefficient of x^k in f(x) is equal to the (k+1)st term in the list.

Examples: 1 , 2 , 3 , 4 , 5 , 6 , 7

##### 12.1-4 PoincareSeriesPrimePart

`‣ PoincareSeriesPrimePart`( G, p, n ) | ( function )

Inputs a finite group G, a prime p, and a positive integer n. It returns a quotient of polynomials f(x)=P(x)/Q(x) whose coefficient of x^k equals the rank of the vector space H_k(G,Z_p) for all k in the range k=1 to k=n.

The efficiency of this function needs to be improved.

Examples: 1 , 2

##### 12.1-5 PoincareSeriesLHS

`‣ PoincareSeriesLHS` | ( global variable )

Inputs a finite 2-group G and returns a quotient of polynomials f(x)=P(x)/Q(x) whose coefficient of x^k equals the rank of the vector space H_k(G,Z_2) for all k.

This function was written by Paul Smith. It use the Singular system for commutative algebra.

Examples:

##### 12.1-6 Prank

`‣ Prank`( G ) | ( function )

Inputs a p-group G and returns the rank of the largest elementary abelian subgroup.

Examples:

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