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 \((N \times (G/N))\) up to degree \(k\).

Examples: 1

##### 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 \([a_0 , a_1 , a_2 , a_3 , \ldots ,a_n]\) of the first \(n+1\) coefficients of the infinite expansion

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

Examples: 1 , 2

##### 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 , 8 , 9

##### 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 , 3

##### 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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