In this chapter we describe the theory that is behind the algorithms used by **LAGUNA**.

Let \(G\) be a group and \(F\) a field. Then the *group algebra* \(FG\) 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 *augmentation homomorphism* \( \chi : FG \rightarrow F\) 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 *augmentation ideal* 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 *unit*. 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 *unit group* of \(FG\) is denoted \(U(FG)\) or simply \(U\) when there is no risk of confusion. A unit \(u\) is said to be *normalised* if \(\chi(u)=1\). The set of normalised units forms a subgroup of the unit group, and is referred to as the *normalised unit group*. 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 \(F\).

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 \(\chi(u)\neq 0\). 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 **LAGUNA** 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 \(y_1, \ldots, y_{|G|-1} \) and two types of relations: \(y_i^p=(y_{i+1})^{\alpha_{i,i+1}} \cdots (y_{|G|-1})^{\alpha_{i,|G|-1}}\) for \( 1 \leq i \leq |G|-1 \), and \( [y_j,y_i]=(y_{j+1})^{\alpha_{j,i,j+1}} \cdots (y_{|G|-1})^{\alpha_{j,i,|G|-1}} \) for \( 1 \leq i < j \leq |G|-1\), where the exponents \(\alpha_{i,k}\) and \(\alpha_{i,j,k}\) are elements of the set \(\{0,\ldots,p-1\}\). Having such a presentation, it is possible to carry out efficient computations in the finite \(p\)-group \(V\); see [Sim94, Chapter 9].

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 \(A^c\neq 0\) 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 \(A^i/A^{i+1}\)of 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, \((1+A^i)/(1+A^{i+1})\leq Z((1+A)/(1+A^{i+1}))\).

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 *Jennings series* of \(G\). That is, \(G_{i+1}=[G_i,G]G_{j^p}\) where \(j\) is the smallest non-negative integer such that \(j\geq i/p\). For all \(i\leq k\) select elements \(x_{i,1},\ldots,x_{i,l_i}\) of \(G_i\) such that \(\{x_{i,1}G_{i+1},\ldots,x_{i,l_i}G_{i+1}\}\) 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 \(x_{1,1},\ldots,x_{1,l_1},\ldots,x_{k,1},\ldots,x_{k,l_k}\) is said to be a *dimension basis* for \(G\). The *weight* 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 \(0\leq \alpha_{i,j}\leq p-1\) is said to be *standard*. 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 \(|G|-1\) .

**Lemma (**[HB82, Theorem VIII.2.6]**).** For \(i\leq c\), 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 *weighted basis*. Note that a weighted basis is a basis for the augmentation ideal, and not for the whole group algebra.

Let \(x_1,\ldots,x_{{|G|}-1}\) 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 *canonical form* of \(v\). We note that by adding a generator of \(F^*\) to the set \(y_1,\ldots,y_{|G|-1|}\) we can obtain a polycyclic generating set for the unit group \(U\).

We show how to compute the canonical form of a normalised unit with respect to the polycyclic generating set \(y_1,\ldots,y_{|G|-1}\). We use the following elementary lemma.

**Lemma.** Let \(i\leq c\) and suppose that \(w \in A^i\). Assume that \(x_{s_i},x_{s_i+1}\ldots,x_{r_i}\) are the standard products with weight \(i\) and for \(s_i\leq j\leq r_i\) set \(y_j=1+x_j\). Then for all \(\alpha_{s_i},\ldots,\alpha_{r_i}\in\{0,\ldots,p-1\}\) 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 \(x_1,\ldots,x_{r_1}\) be the standard products of weight 1, and let \(y_1,\ldots,y_{r_1}\) be the corresponding elements in the polycyclic generating set. Then using the previous lemma, we find \(\alpha_1,\ldots,\alpha_{r_1}\) 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 \(x_{s_2},x_{s_2+1},\ldots,x_{r_2}\) are the standard products of weight 2. We find \(\alpha_{s_2},\ldots,\alpha_{r_2}\) 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 \(1+w\).

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 \(y_1,\ldots,y_{|G|-1}\) as in Section 3.3. 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 3.4.

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 \(y^{p-1}=1\).

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 [Ros00].

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