[Nilmat] 2 Computing with nilpotent linear groups

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# 2 Computing with nilpotent linear groups

### Sections

1. Preliminaries
2. Testing nilpotency
3. Finiteness, Sylow subgroups, testing complete reducibility
4. A library of primitive nilpotent groups
5. Further examples of nilpotent matrix groups

This chapter contains the main functions of this package for computing with nilpotent matrix groups.

## 2.1 Preliminaries

We first describe some of the basic functions used in Nilmat for nilpotency testing of a group G input by a finite generating set of matrices.

- `JordanSplitting(` G`) A`

  For a subgroup G of GL(n,F), F=GF(q) or Q, returns a list of two groups [S,U], where S is the semisimple part of G (the group generated by the semisimple parts of the generators of G), and U is the unipotent part of G (the group generated by the unipotent parts of the generators of G). If G is nilpotent, then G @ S ×U, the group S is completely reducible and U is unipotent. This attribute relies on the GAP attribute `JordanDecomposition`.

- `IsUnipotentMatGroup(` G`) P`

  For a subgroup G of GL(n,F), F=GF(q) or Q, returns `true` if G is unipotent (i.e. conjugate to a group of upper unitriangular matrices) and `false` otherwise.

- `ClassLimit(` n`,` F`) F`

  returns an upper bound on the nilpotency class of nilpotent subgroups of GL(n,F), F=GF(q) or Q.

- `AbelianNormalSeries(` G`,` l`) F`

  Here G < GL(n,q) and l is a positive integer. If G is nilpotent of class at most l and the order of G is coprime to the characteristic of GF(q), then this function determines a normal series with abelian factors for G. Otherwise, the function may still return such a series or it may return `fail`. The function is based on recursively selecting non-central elements from the second centers of terms in the abelian series.

- `PiPrimarySplitting(` G`) A`

  For a subgroup G of GL(n,q), this function returns a list of two subgroups [B,C] with G = BC. If G is nilpotent, then G @ B ×C, the group C is the product of all Sylow p-subgroups with p > n and B is the product of all other Sylow subgroups of G.

  ## 2.2 Testing nilpotency

  The following is one of the main functions of the Nilmat package.

- `IsNilpotentMatGroup(` G`) F`

  For a subgroup G of GL(n,F), F=GF(q) or Q, returns `true` if G is nilpotent and `false` otherwise. This function is also installed as method for the property `IsNilpotentGroup`.

  We include a brief description of the algorithm behind this function. Let X be a generating set of the given group G. The first stage of testing nilpotency of G is reduction to the semisimple part S of G. The procedure for reducing to the semisimple case is based on the Nilmat functions `JordanSplitting` and `IsUnipotentGroup` described in the previous section. In the following, we assume that all elements of X are semisimple matrices.

  If F=GF(q), then we apply the function `PiPrimarySplitting` to the group S and thus reduce to a smaller group B. Next, we attempt to compute an abelian normal series for B using the function `AbelianNormalSeries`. If no such series exists, then G is not nilpotent. If such a series exists, then we use it to construct the Sylow subgroups of B and check that they commute pairwise. For details on this method, see DF06.

  If F=Q, then we first use a reduction mod p for a suitable prime p and check that the image of G under the corresponding congruence homomorphism is nilpotent using the finite field method above. If so, then we construct the kernel of the congruence homomorphism and test whether this is central in G. We refer to DF07 for details. Note that the construction of the congruence homomorphism and its kernel is based on the methods of the Package Polenta; see also AE05 for background.

  The nilpotency testing functions of the package Nilmat have advantages over the standard GAP methods for `IsNilpotentGroup`. When F is finite, the Nilmat functions have better runtimes for all input groups we tested. When F is infinite, the standard GAP functions frequently do not terminate at all in sensible time; on the other hand, the Nilmat functions always terminate, with comparatively small runtimes (see the examples in Chapter 3).

  ## 2.3 Finiteness, Sylow subgroups, testing complete reducibility

  The function `IsNilpotentMatGroup` determines various structural properties of the given group as by-products. The functions in this section have been designed to exploit these by-products.

- `IsFiniteNilpotentMatGroup(` G`) F`

  For a nilpotent subgroup G of GL(n,Q), returns `true` if G is finite and `false` otherwise. Note that the function assumes that G is nilpotent and may return an incorrect result if not. The function exploits the by-products of the nilpotency testing functions in Nilmat and hence runs particularly fast (and usually faster than the standard GAP method for testing finiteness) if they have been used to check nilpotency. This function is also installed as method for the property `IsFinite`.

- `SylowSubgroupsOfNilpotentFFMatGroup(` G`) F`

  For a nilpotent subgroup G of GL(n,F), F=GF(q), returns the list of all Sylow subgroups. The advantage of this function over the GAP function `SylowSubgroup` is that the former function returns all Sylow subgroups of G without first computing all prime divisors of the order of G. This function is installed as method for `SylowSystem` for nilpotent matrix groups.

- `SizeOfNilpotentMatGroup(` G`) F`

  For a finite nilpotent subgroup G of GL(n,F), F=GF(q) or Q, this function returns the order of G. The function is based on by-products of the nilpotency testing in Nilmat. Again, in some situations it is more efficient than the similar default GAP function; see the examples in Chapter 3.

- `IsCompletelyReducibleNilpotentMatGroup(` G`) F`

  For a nilpotent subgroup G of GL(n,F), F=GF(q) or Q, returns `true` if G is completely reducible and `false` otherwise.

  ## 2.4 A library of primitive nilpotent groups

  Another main part of Nilmat is a library of nilpotent primitive matrix groups over finite fields.

- `NilpotentPrimitiveMatGroups(` n`,` p`,` l`) F`

  returns a complete and irredundant list L of the conjugacy class representatives of the nilpotent primitive subgroups of GL(n,pl). The list L contains non-abelian (i.e. non-cyclic) subgroups only if n=2m, m is odd, and pl º 3 mod 4. Every non-abelian group in L is given by three generators. Note that the groups in L know their orders i.e. the attribute `Size` has been set for these groups.

- `SizesOfNilpotentPrimitiveMatGroups(` n`,` p`,` l`) F`

  returns the list of orders of groups in the list L output by `NilpotentPrimitiveMatGroups(` n`,` p`,` l`)`.

  ## 2.5 Further examples of nilpotent matrix groups

  In this section we describe various functions designed to produce interesting examples of nilpotent matrix groups.

- `MaximalAbsolutelyIrreducibleNilpotentMatGroup(` n`,` p`,` l`) F`

  constructs the unique (up to conjugacy) maximal absolutely irreducible nilpotent subgroup of GL(n,pl) if such a group exists. Note that such a group exists if and only if each prime divisor of n divides pl-1. Otherwise the function returns `fail`.

- `MonomialNilpotentMatGroup(` n`) F`

  constructs an example of a finite nilpotent monomial subgroup of GL(n,Q).

- `ReducibleNilpotentMatGroup(` m`,` k`, [`p`,` l`] ) F`

  constructs an example of a reducible but not completely reducible nilpotent subgroup of GL(mk, F), where F= Q if there are two arguments given and F= GF(pl) if there are four arguments given.

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