## Computing Resolutions over Finite p-Groups (2000)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Grabmeier00computingresolutions,

author = {Johannes Grabmeier and Larry A. Lambe},

title = {Computing Resolutions over Finite p-Groups},

year = {2000}

}

### OpenURL

### Abstract

. A uniform and constructive approach for the computation of resolutions and for (co)homology computations for any nite p-group is detailed. The resolutions we construct ([32]) are, as vector spaces, as small as the minimal resolution of IFp over the elementary abelian p-group of the same order as the group under study. Our implementations are based on the development of sophisticated algebraic data structures. Applications to calculating functional cocycles are given and the possibility of constructing interesting codes using such methods is presented. 1 Introduction In this paper, we present a uniform constructive approach to calculating relatively small resolutions over nite p-groups. The algorithm we use comes from [32, 8.1.8 and the penultimate paragraph of 9.4]. There has been a massive amount of work done on the structure of p-groups since the beginning of group theory. A good introduction is [22]. We combine mathematical and computer methods to construct the uniform resolut...

### Citations

533 | Concrete Mathematics
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Citation Context ... [k = p − 1][η = 1] 1 ⊗ γi+1(y) ⊗ 1, (25) where we use the Kronecker-Iverson notation [b], which evaluates to 1 for Boolean expressions b having value true, and to 0 for those having value false, see =-=[11]-=-.sNote that IFp[t1, . . . , tn]/(t p 1 − 1, . . . , tpn − 1) ∼ = IFp[t1](t p 1 − 1) ⊗ . . . ⊗ IFp[tn]/(t p n − 1) ∼ = IFpG n +. Using tensor product formulae, we get a resolution C (n) over IFpG n + o... |

481 |
Homological Algebra
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Citation Context ...are concerned with the case that G is a finite p-group and R = IFp. 3.4 The Standard Resolution (Bar Construction) Let A be an R-Algebra with augmentation ɛ as above. The bar construction B(A) ([34], =-=[3]-=-) is a particular A-free resolution of R. It is ofsthe form B(A) = A ⊗R ¯ B(A) where ¯B(A) = � ⊗ n≥0 n RĀ, and Ā = A/R (thinking of R as a submodule of A via the unit). Following convention, we write ... |

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- 1991
(Show Context)
Citation Context ...ve new SDR data (X, ∂∞) ∇∞ ✲ ✛ f∞ ((Y, d ′ Y ), φ∞). (9)s10 Note that the limits will certainly exist if tφ is nilpotent in each degree. Examples are given in [2], [12], [28], [33], [31], [29], [30], =-=[13]-=-, [14], [21], [19], [20], for example. In particular, this paper discusses an implementation of the algorithm given at the end of section (9.4) in [32]. 3.3 Homology and Cohomology of Algebras Let A b... |

43 |
Finite groups
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- 1982
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Citation Context ... use comes from [32, 8.1.8 and the penultimate paragraph of 9.4]. There has been a massive amount of work done on the structure of p-groups since the beginning of group theory. A good introduction is =-=[22]-=-. We combine mathematical and computer methods to construct the uniform resolutions in this paper. These resolutions are much smaller than the bar construction [34, Chapter IV, §5] (or Sect. 3.4), but... |

41 |
On the chain-complex of a fibration
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Citation Context ...otopy φ ′′ . 3.2 The Perturbation Lemma Given the SDR (4) and in addition a second differential d ′ Y on Y , let t = d ′ Y − dY . The map t is called the initiator ([1]). The perturbation lemma, [2], =-=[12]-=-, [1] states that if we set tn = (tφ) n−1t, n ≥ 1 and, for each n, define new maps on X: On Y : ∂n = d + f(t1 + t2 + . . . + tn−1)∇ (5) ∇n = ∇ + φ(t1 + t2 + . . . + tn−1)∇. (6) fn = f + f(t1 + t2 + . ... |

30 |
The structure of the group ring of a p-group over a modular field
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Citation Context ...presented for practical applications of the main theorems in Sect. 4. 2.1 The mod-p Lower Central Series For each finite p-group G, an n-dimensional mod-p restricted Lie algebra gr pG can be defined (=-=[25]-=-) using the mod-p lower central series G = Z1 ≥ Z2 ≥ . . .. Here Zi is defined by Zi = 〈(x1, (x2, (. . . (xj−1, xj) . . .)y pk |jp k ≥ i〉sor equivalently, for i > 1, Zi = (Zi−1, Z1)Zj p where j is the... |

24 | A fixed point approach to homological perturbation theory
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Citation Context ...ditions will hold for the chain homotopy φ ′′ . 3.2 The Perturbation Lemma Given the SDR (4) and in addition a second differential d ′ Y on Y , let t = d ′ Y − dY . The map t is called the initiator (=-=[1]-=-). The perturbation lemma, [2], [12], [1] states that if we set tn = (tφ) n−1t, n ≥ 1 and, for each n, define new maps on X: On Y : ∂n = d + f(t1 + t2 + . . . + tn−1)∇ (5) ∇n = ∇ + φ(t1 + t2 + . . . +... |

22 |
The Scientific Computation System
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Citation Context ...d computer science methods and techniques would be far beyond the scope of the current paper and hence will be developed and discussed elsewhere [5]. For now, we use the computer algebra system AXIOM =-=[24]-=- which consists of a language, compiler, interpreter and a user interface to accomplish our goals. The algorithm we present in Sect. 5 is of a recursive nature and can be applied naturally to p-groups... |

18 |
Perturbation theory and free resolutions for nilpotent groups of class 2
- HUEBSCHMANN
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Citation Context ...∇∞ ✲ ✛ f∞ ((Y, d ′ Y ), φ∞). (9)s10 Note that the limits will certainly exist if tφ is nilpotent in each degree. Examples are given in [2], [12], [28], [33], [31], [29], [30], [13], [14], [21], [19], =-=[20]-=-, for example. In particular, this paper discusses an implementation of the algorithm given at the end of section (9.4) in [32]. 3.3 Homology and Cohomology of Algebras Let A be an algebra over R. For... |

18 |
Classes of restricted Lie algebras of characteristic p
- Jacobson
- 1943
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Citation Context ...t for elements ρi(x) ∈ Zi/Zi+1 and ρj(y) ∈ Zj/Zj+1 is defined by while the definition [ρi(x), ρj(y)] = ρi+j((x, y)) ρi(x) p = ρip(x p ) satisfies the identities given on pages 91−93 in [25] (also see =-=[23]-=-), hence it is a p-restriction and this indeed yields a p-restricted Lie algebra. Let ɛ : IFpG ✲ IFp, � g agg ↦→ � g ag (cf. 3.3) be the augmentation of the group algebra IFpG and I = ker(ɛ) the augme... |

15 | Transferring algebra structures up to homology equivalence
- Johansson, Lambe
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Citation Context ...lse. There is however an algebra structure for which δ∞ is a derivation. The discussion of this and its consequences is beyond the scope of the current paper however. The interested reader should see =-=[26]-=-, [14]. We have the immediate Corollary 5. Let G be a finite p-group. If p = 2, let X ∗ = IF2[z1, . . . , zn] otherwise let X ∗ = IFp[z1, . . . , zn] ⊗ Λp[w1, . . . , wn]. There is a differential δ∞ o... |

11 |
Cocyclic development of designs
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- 1993
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Citation Context ...resenting 2-dimensional cohomology classes can be worked out in some cases. See [7], [8] and [17]. Also see [6]. Connections between combinatorial design theory and 2-cocycles has been pointed out in =-=[15]-=- (also see [16] for errata). Connections between coding theory and 2-cocycles have also been made in [18]. Evaluating functional 2-cocycles on all pairs of the group elements yields a matrix with entr... |

7 | The twisted Eilenberg-Zilber theorem
- Brown
- 1964
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Citation Context ...n homotopy φ ′′ . 3.2 The Perturbation Lemma Given the SDR (4) and in addition a second differential d ′ Y on Y , let t = d ′ Y − dY . The map t is called the initiator ([1]). The perturbation lemma, =-=[2]-=-, [12], [1] states that if we set tn = (tφ) n−1t, n ≥ 1 and, for each n, define new maps on X: On Y : ∂n = d + f(t1 + t2 + . . . + tn−1)∇ (5) ∇n = ∇ + φ(t1 + t2 + . . . + tn−1)∇. (6) fn = f + f(t1 + t... |

7 | Next generation computer algebra systems AXIOM and the scratchpad concept: applications to research in algebra
- Lambe
- 1994
(Show Context)
Citation Context ...s (provided they exist), we have new SDR data (X, ∂∞) ∇∞ ✲ ✛ f∞ ((Y, d ′ Y ), φ∞). (9)s10 Note that the limits will certainly exist if tφ is nilpotent in each degree. Examples are given in [2], [12], =-=[28]-=-, [33], [31], [29], [30], [13], [14], [21], [19], [20], for example. In particular, this paper discusses an implementation of the algorithm given at the end of section (9.4) in [32]. 3.3 Homology and ... |

7 | Perturbation theory in di®erential homological algebra ii - Gugenheim, Lambe, et al. - 1991 |

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Codes from cocycles
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Citation Context ...ee [6]. Connections between combinatorial design theory and 2-cocycles has been pointed out in [15] (also see [16] for errata). Connections between coding theory and 2-cocycles have also been made in =-=[18]-=-. Evaluating functional 2-cocycles on all pairs of the group elements yields a matrix with entries from IF2 in case of p = 2. In light of the connection between cocycles and combinatorics just mention... |

5 |
Algèbras d’Eilenberg–MacLane et homotopie. 2ième ed., revue et corrigée. Secrétariat mathématique (Hektograph
- Cartan, Moore, et al.
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Citation Context ...maps f : X ✲ Y and g : Y ✲ X such that fg and gf are both chain homotopy equivalent to the identity map. We will need a constructive version of this for free resolutions that essentially goes back to =-=[4]-=-. In fact, we are interested only in the case when Y = B(A) and fg = 1X, so that we actually obtain an SDR. The explicit formulae and discussion were given in [33]. We will simply repeat the formulas ... |

5 |
The homotopy type of F Ψ q . The complex and symplectic cases, in: Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Contemp
- Huebschmann
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Citation Context ..., ∂∞) ∇∞ ✲ ✛ f∞ ((Y, d ′ Y ), φ∞). (9)s10 Note that the limits will certainly exist if tφ is nilpotent in each degree. Examples are given in [2], [12], [28], [33], [31], [29], [30], [13], [14], [21], =-=[19]-=-, [20], for example. In particular, this paper discusses an implementation of the algorithm given at the end of section (9.4) in [32]. 3.3 Homology and Cohomology of Algebras Let A be an algebra over ... |

3 | Transgression and the calculation of cocyclic matrices
- Flannery
- 1995
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Citation Context ...ds involving the universal coefficient theorem, Schur multipliers, and transgression, methods for finding 2-cocycles representing 2-dimensional cohomology classes can be worked out in some cases. See =-=[7]-=-, [8] and [17]. Also see [6]. Connections between combinatorial design theory and 2-cocycles has been pointed out in [15] (also see [16] for errata). Connections between coding theory and 2-cocycles h... |

3 |
Cocyclic Hadamard matrices and difference sets
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Citation Context ...homology. Having functional cocycles in hand allows one to examine interesting combinatorial properties. Ins2 this way we mention briefly how certain codes arise from some explicit cocycles. Also see =-=[9]-=- and [10]. We note that, in general, in order to increase practicality further, one needs to devise “reduction strategies” along the lines of [27] to reduce the size of the resolution for a general gr... |

3 |
Huebschmann and Tornike Kadeishvili. Small models for chain algebras
- Johannes
- 1991
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Citation Context ...ata (X, ∂∞) ∇∞ ✲ ✛ f∞ ((Y, d ′ Y ), φ∞). (9)s10 Note that the limits will certainly exist if tφ is nilpotent in each degree. Examples are given in [2], [12], [28], [33], [31], [29], [30], [13], [14], =-=[21]-=-, [19], [20], for example. In particular, this paper discusses an implementation of the algorithm given at the end of section (9.4) in [32]. 3.3 Homology and Cohomology of Algebras Let A be an algebra... |

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Citation Context ...se fundamental considerations combining mathematical and computer science methods and techniques would be far beyond the scope of the current paper and hence will be developed and discussed elsewhere =-=[5]-=-. For now, we use the computer algebra system AXIOM [24] which consists of a language, compiler, interpreter and a user interface to accomplish our goals. The algorithm we present in Sect. 5 is of a r... |

2 |
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Citation Context ...rturbation lemma, [2], [12], [1] states that if we set tn = (tφ) n−1t, n ≥ 1 and, for each n, define new maps on X: On Y : ∂n = d + f(t1 + t2 + . . . + tn−1)∇ (5) ∇n = ∇ + φ(t1 + t2 + . . . + tn−1)∇. =-=(6)-=- fn = f + f(t1 + t2 + . . . + tn−1)φ (7) φn = φ + φ(t1 + t2 + . . . + tn−1)φ. (8) then in the limits (provided they exist), we have new SDR data (X, ∂∞) ∇∞ ✲ ✛ f∞ ((Y, d ′ Y ), φ∞). (9)s10 Note that t... |

2 |
Computing 2-cocycles for central extensions and relative difference sets
- Flannery, O’Brien
- 1939
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Citation Context .... Having functional cocycles in hand allows one to examine interesting combinatorial properties. Ins2 this way we mention briefly how certain codes arise from some explicit cocycles. Also see [9] and =-=[10]-=-. We note that, in general, in order to increase practicality further, one needs to devise “reduction strategies” along the lines of [27] to reduce the size of the resolution for a general group. Our ... |

2 |
Erratum: \Cocylic development of designs
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- 1994
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Citation Context ...ensional cohomology classes can be worked out in some cases. See [7], [8] and [17]. Also see [6]. Connections between combinatorial design theory and 2-cocycles has been pointed out in [15] (also see =-=[16]-=- for errata). Connections between coding theory and 2-cocycles have also been made in [18]. Evaluating functional 2-cocycles on all pairs of the group elements yields a matrix with entries from IF2 in... |

2 |
Generation of cocyclic Hadamard matrices
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- 1992
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Citation Context ...the universal coefficient theorem, Schur multipliers, and transgression, methods for finding 2-cocycles representing 2-dimensional cohomology classes can be worked out in some cases. See [7], [8] and =-=[17]-=-. Also see [6]. Connections between combinatorial design theory and 2-cocycles has been pointed out in [15] (also see [16] for errata). Connections between coding theory and 2-cocycles have also been ... |

2 |
Emil Skoldberg. On constructing resolutions over the polynomial algebra
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- 2000
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Citation Context ...tain codes arise from some explicit cocycles. Also see [9] and [10]. We note that, in general, in order to increase practicality further, one needs to devise “reduction strategies” along the lines of =-=[27]-=- to reduce the size of the resolution for a general group. Our moderate sized resolutions are a good starting point for these methods, but such reductions are not within the scope of the current paper... |

2 | Cocyclic Hadamard matrices and dierence sets - Horadam, Flannery, et al. - 1997 |

2 | The homotopy type of F q . The complex and symplectic cases. In Applications of algebraic K-theory to algebraic geometry and number theory, Part I - Huebschmann - 1983 |

1 | Second cohomology of groups with trivial coecients - Ellis, Kholodna - 1999 |