  
  [1m[4m[31m3. The basic theory behind [1mLAGUNA[1m[4m[31m[0m
  
  In this chapter we describe the theory that is behind the algorithms used by
  [1mLAGUNA[0m.
  
  
  [1m[4m[31m3.1 Notation and definitions[0m
  
  Let  G  be  a group and F a field. Then the [22m[36mgroup algebra[0m FG consists of the
  set of formal linear combinations of the form
  
  \[
       \sum_{g \in G}\alpha_g g,\qquad \alpha_g \in F
  \]
  
  where all but finitely many of the alpha_g are zero. The group algebra FG is
  an F-algebra with the obvious operations. Clearly, dim FG=|G|.
  
  The [22m[36maugmentation homomorphism[0m chi : FG -> F is defined by
  
  \[
       \chi\left(\sum_{g \in G}\alpha_g g\right)=\sum_{g \in G}\alpha_g.
  \]
  
  It  is  easy  to see that chi is indeed a homomorphism onto F. The kernel of
  chi  is  called  the  [22m[36maugmentation  ideal[0m  of  FG. The augmentation ideal is
  denoted  A(FG), or simply A when there is no danger of confusion. It follows
  from  the isomorphism theorems that dim A(FG)=dim FG-1=|G|-1. Another way to
  write the augmentation ideal is
  
  \[
       A(FG)=\left\{\sum_{g \in G}\alpha_g g\ |\ \sum_{g \in
       G}\alpha_g=0\right\}.
  \]
  
  An  invertible element of FG is said to be a [22m[36munit[0m. Clearly the elements of G
  and  the non-zero elements of F are units. The set of units in FG is a group
  with  respect  to  the multiplication of FG. The [22m[36munit group[0m of FG is denoted
  U(FG) or simply U when there is no risk of confusion. A unit u is said to be
  [22m[36mnormalised[0m  if chi(u)=1. The set of normalised units forms a subgroup of the
  unit  group, and is referred to as the [22m[36mnormalised unit group[0m. The normalised
  unit  group  of  FG  is denoted V(FG), or simply V. It is easy to prove that
  U(FG) = F^* times V(FG) where F^* denotes the multiplicative group of F.
  
  
  [1m[4m[31m3.2 p-modular group algebras[0m
  
  A  group  algebra  FG  is  said  to  be  p-modular  if  F  is  the  field of
  characteristic  p, and G is a finite p-group. A lot of information about the
  structure  of p-modular group algebras can be found in [HB82, Chapter VIII].
  In a p-modular group algebra we have that an element u is a unit if and only
  if chi(u)<> 0. Hence the normalised unit group V consists of all elements of
  FG  with  augmentation  1.  In  other words V is a coset of the augmentation
  ideal, namely V=1+A. This also implies that |V|=|A|=|F|^|G|-1, and so V is a
  finite p-group.
  
  One  of  the  aims  of  the  [1mLAGUNA[0m package is to compute a power-commutator
  presentation  for  the  normalised unit group in the case when G is a finite
  p-group  and  F  is  a  field of p elements. Such a presentation is given by
  generators    y_1,    ...,    y_|G|-1    and   two   types   of   relations:
  y_i^p=(y_i+1)^alpha_i,i+1}  cdots  (y_|G|-1)^alpha_i,|G|-1}  for  1  <= i <=
  |G|-1, and [y_j,y_i]=(y_j+1)^alpha_j,i,j+1} cdots (y_|G|-1)^alpha_j,i,|G|-1}
  for  1  <=  i<j  <= |G|-1, where the exponents alpha_i,k and alpha_i,j,k are
  elements of the set 0,...,p-1. Having such a presentation, it is possible to
  carry  out efficient computations in the finite p-group V; see [S94, Chapter
  9].
  
  
  [1m[4m[31m3.3 Polycyclic generating set for V[0m
  
  Let  G  be  a  finite  p-group  and F the field of p elements. Our aim is to
  construct a power-commutator presentation for V=V(FG). We noted earlier that
  V=1+A,  where  A is the augmentation ideal. We use this piece of information
  and  construct  a polycyclic generating set for V using a suitable basis for
  A.  Before  constructing  this generating set, we note that A is a nilpotent
  ideal  in  FG. In other words there is some c such that A^c<> 0 but A^c+1=0.
  Hence we can consider the following series of ideals in A:
  
  \[
       A\rhd A^2\rhd\cdots\rhd A^{c}\rhd A^{c+1}=0.
  \]
  
  It   is   clear   that   a  quotient  A^i/A^i+1of  this  chain  has  trivial
  multiplication,  that is, such a quotient is a nil-ring. The chain A^i gives
  rise to a series of normal subgroups in V:
  
  \[
       V=1+A\rhd 1+A^2\rhd\cdots\rhd 1+A^c\rhd 1+A^{c+1}=1.
  \]
  
  It   is   easy   to   see   that  the  chain  1+A^i  is  central,  that  is,
  (1+A^i)/(1+A^i+1)<= Z((1+A)/(1+A^i+1)).
  
  Now  we  show  how  to  compute  a  basis  for  A^i  that gives a polycyclic
  generating set for 1+A^i. Let
  
  \[
       G=G_1 \rhd G_2\rhd\cdots\rhd G_{k}\rhd G_{k+1}=1
  \]
  
  be  the  [22m[36mJennings  series[0m  of  G. That is, G_i+1=[G_i,G]G_j^p where j is the
  smallest  non-negative  integer  such  that  j>=  i/p.  For all i<= k select
  elements x_i,1,...,x_i,l_i of G_i such that x_i,1G_i+1,...,x_i,l_iG_i+1 is a
  minimal  generating  set for the elementary abelian group G_i/G_i+1. For the
  Jennings  series  it  may  happen that G_i=G_i+1 for some i. In this case we
  choose  an  empty  generating set for the quotient G_i/G_i+1 and l_i=0. Then
  the  set  x_1,1,...,x_1,l_1,...,x_k,1,...,x_k,l_k  is said to be a [22m[36mdimension
  basis[0m for G. The [22m[36mweight[0m of a dimension basis element x_i,j is i.
  
  A non-empty product
  
  \[
       u=(x_{1,1}-1)^{\alpha_{1,1}}\cdots(x_{1,l_1}-1)^{\alpha_{1,l_1}}\cdots
       (x_{k,1}-1)^{\alpha_{k,1}}\cdots(x_{k,l_k}-1)^{\alpha_{k,l_k}}
  \]
  
  where  0<=  alpha_i,j<=  p-1  is  said  to  be [22m[36mstandard[0m. Clearly, a standard
  product  is  an  element  of  the  augmentation  ideal  A. The weight of the
  standard product u is
  
  \[
       \sum_{i=1}^k i(\alpha_{i,1}+\cdots+\alpha_{i,l_i}).
  \]
  
  The total number of standard products is |G|-1 .
  
  [1m[46mLemma  ([0m[HB82,  Theorem  VIII.2.6][1m[46m).[0m  For  i<=  c,  the  set S_i of standard
  products  of  weight  at  least  i  forms a basis for A^i. Moreover, the set
  1+S_i=1+s | s in S_i is a polycyclic generating set for 1+A^i. In particular
  1+S_1 is a polycyclic generating set for V.
  
  A  basis  for  A  consisting  of  the  standard products is referred to as a
  [22m[36mweighted  basis[0m.  Note that a weighted basis is a basis for the augmentation
  ideal, and not for the whole group algebra.
  
  Let  x_1,...,x_{|G|-1  denote  the  standard  products  where  we choose the
  indices so that the weight of x_i is not larger than the weight of x_i+1 for
  all  i, and set y_i=1+x_i. Then every element v of V can be uniquely written
  in the form
  
  \[
       v=y_1^{\alpha_1}\cdots (y_{|G|-1})^{\alpha_{|G|-1}}, \quad
       \alpha_1,\ldots,\alpha_{|G|-1} \in \{0,\ldots,p-1\}.
  \]
  
  This  expression is called the [22m[36mcanonical form[0m of v. We note that by adding a
  generator  of  F^*  to  the  set y_1,...,y_|G|-1| we can obtain a polycyclic
  generating set for the unit group U.
  
  
  [1m[4m[31m3.4 Computing the canonical form[0m
  
  We  show how to compute the canonical form of a normalised unit with respect
  to  the  polycyclic  generating  set  y_1,...,y_|G|-1.  We use the following
  elementary lemma.
  
  [1m[46mLemma.[0m   Let   i<=   c   and   suppose   that   w   in   A^i.   Assume  that
  x_s_i,x_s_i+1...,x_r_i are the standard products with weight i and for s_i<=
  j<=  r_i  set  y_j=1+x_j. Then for all alpha_s_i,...,alpha_r_iin0,...,p-1 we
  have that
  
  \[
       w\equiv \alpha_{s_i}x_{s_i}+\cdots+\alpha_{r_i}x_{r_i}\quad \bmod
       \quad A^{i+1}
  \]
  
  if an only if
  
  \[
       1+w\equiv (y_{s_i})^{\alpha_{s_i}}\cdots
       (y_{r_i})^{\alpha_{r_i}}\quad \bmod \quad 1+A^{i+1}.
  \]
  
  Suppose  that  w  is  an  element  of  the augmentation ideal A and 1+w is a
  normalised unit. Let x_1,...,x_r_1 be the standard products of weight 1, and
  let y_1,...,y_r_1 be the corresponding elements in the polycyclic generating
  set. Then using the previous lemma, we find alpha_1,...,alpha_r_1 such that
  
  \[
       w\equiv \alpha_{1}x_{1}+\cdots+\alpha_{r_1}x_{r_1}\quad \bmod
       \quad A^{2},
  \]
  
  and so
  
  \[
       1+w\equiv (y_{1})^{\alpha_{1}}\cdots (y_{r_1})^{\alpha_{r_1}}\quad
       \bmod \quad 1+A^{2}.
  \]
  
  Now  we have that 1+w=(y_1)^alpha_1}cdots (y_r_1)^alpha_r_1}(1+w_2) for some
  w_2  in  A^2.  Then  suppose  that  x_s_2,x_s_2+1,...,x_r_2 are the standard
  products of weight 2. We find alpha_s_2,...,alpha_r_2 such that
  
  \[
       w_2\equiv \alpha_{s_2}x_{s_2}+\cdots+\alpha_{r_2}x_{r_2}\quad
       \bmod \quad A^{3}.
  \]
  
  Then the lemma above implies that
  
  \[
       1+w_2\equiv (y_{s_2})^{\alpha_{s_2}}\cdots
       (y_{r_2})^{\alpha_{r_2}}\quad \bmod \quad 1+A^{3}.
  \]
  
  Thus 1+w_2=(y_s_2)^alpha_s_2}cdots (y_r_2)^alpha_r_2}(1+w_3) for some w_3 in
  A^3,               and               so              1+w=(y_1)^alpha_1}cdots
  (y_r_1)^alpha_r_1}(y_s_2)^alpha_s_2}cdots    (y_r_2)^alpha_r_2}(1+w_3).   We
  repeat  this process, and after c steps we obtain the canonical form for the
  element 1+w.
  
  
  [1m[4m[31m3.5 Computing a power commutator presentation for V[0m
  
  Using  the  procedure in the previous section, it is easy to compute a power
  commutator presentation for the normalized unit group V of a p-modular group
  algebra  over  the  field  of  p  elements.  First we compute the polycyclic
  generating sequence y_1,...,y_|G|-1 as in Section [1m3.3[0m. Then for each y_i and
  for  each y_j, y_i such that i<j we compute the canonical form for y_i^p and
  [y_j,y_i] as described in Section [1m3.4[0m.
  
  Once  a  power-commutator  presentation  for V is constructed, it is easy to
  obtain  a  polycyclic  presentation  for the unit group U by adding an extra
  central generator y corresponding to a generator of the cyclic group F^* and
  enforcing that y^p-1=1.
  
  
  [1m[4m[31m3.6 Verifying Lie properties of FG[0m
  
  If  FG  is  a  group algebra then one can consider the Lie bracket operation
  defined  by  [a,b]=ab-ba.  Then it is well-known that FG with respect to the
  scalar multiplication, the addition, and the bracket operation becomes a Lie
  algebra  over F. This Lie algebra is also denoted FG. Some Lie properties of
  such Lie algebras can be computed very efficiently. In particular, it can be
  verified  whether  the  Lie  algebra  FG  is nilpotent, soluble, metabelian,
  centre-by-metabelian. Fast algorithms that achieve these goals are described
  in [LR86], [PPS73], and [R00].
  
