This chapter contains the main functions of this package for computing with nilpotent matrix groups.
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.
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 congS timesU, the group S is completely reducible and U
is unipotent. This attribute relies on the GAP attribute
For a subgroup G of GL(n,F), F=GF(q) or Q, returns
if G is unipotent (i.e. conjugate to a group of upper unitriangular
returns an upper bound on the nilpotency class of nilpotent subgroups of GL(n,F), F=GF(q) or Q.
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.
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 congB timesC, 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.
The following is one of the main functions of the Nilmat package.
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
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
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
the group S and thus reduce to a smaller group B. Next, we
attempt to compute an abelian normal series for B using the
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).
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.
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
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
SylowSystem for nilpotent matrix groups.
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.
For a nilpotent subgroup G of GL(n,F), F=GF(q) or Q, returns
true if G is completely reducible and
Another main part of Nilmat is a library of nilpotent primitive matrix groups over finite fields.
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 equiv3 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
returns the list of orders of groups in the list L output by
In this section we describe various functions designed to produce interesting examples of nilpotent matrix groups.
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
constructs an example of a finite nilpotent monomial subgroup of GL(n,Q).
] ) 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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