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^(*≤ 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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