GAP (HAP commands) - Chapter 14: Cohomology rings of \(p\)-groups (mainly \(p=2)\)

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14 Cohomology rings of \(p\)-groups (mainly \(p=2)\)
14.1
14.1-1 Mod2CohomologyRingPresentation
14.1-2 PoincareSeriesLHS

### 14 Cohomology rings of \(p\)-groups (mainly \(p=2)\)

The functions on this page were written by Paul Smith. (They are included in HAP but they are also independently included in Paul Smiths HAPprime package.)

#### 14.1

##### 14.1-1 Mod2CohomologyRingPresentation

`‣ Mod2CohomologyRingPresentation`( G ) | ( function )

`‣ Mod2CohomologyRingPresentation`( G, n ) | ( function )

`‣ Mod2CohomologyRingPresentation`( A ) | ( function )

`‣ Mod2CohomologyRingPresentation`( R ) | ( function )

When applied to a finite \(2\)-group \(G\) this function returns a presentation for the mod 2 cohomology ring \(H^*(G,Z_2)\). The Lyndon-Hochschild-Serre spectral sequence is used to prove that the presentation is correct.

When the function is applied to a \(2\)-group \(G\) and positive integer \(n\) the function first constructs \(n\) terms of a free \(Z_2G\)-resolution \(R\), then constructs the finite-dimensional graded algebra \(A=H^(*\le n)(G,Z_2)\), and finally uses \(A\) to approximate a presentation for \(H^*(G,Z_2)\). For "sufficiently large" the approximation will be a correct presentation for \(H^*(G,Z_2)\).

Alternatively, the function can be applied directly to either the resolution \(R\) or graded algebra \(A\).

This function was written by Paul Smith. It uses the Singular commutative algebra package to handle the Lyndon-Hochschild-Serre spectral sequence.

Examples: 1 , 2

##### 14.1-2 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:

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