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<div class="ChapSects"><a href="chap3_mj.html#X7D9FCE3A8526ACBE">3 The basic theory behind LAGUNA</a>
<div class="ContSect"> <a href="chap3_mj.html#X85C2DE1486FB45FD">3.1 Notation and definitions</a>

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<div class="ContSect"> <a href="chap3_mj.html#X7E26AE6C807D7C07">3.2 \(p\)-modular group algebras</a>

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<div class="ContSect"> <a href="chap3_mj.html#X7B74767A849D921E">3.3 Polycyclic generating set for \(V\)</a>

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<div class="ContSect"> <a href="chap3_mj.html#X7DCB12F48752C17E">3.4 Computing the canonical form</a>

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<div class="ContSect"> <a href="chap3_mj.html#X873F4F068759DF5D">3.5 Computing a power commutator presentation for \(V\)</a>

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<div class="ContSect"> <a href="chap3_mj.html#X79183196837B7CC1">3.6 Verifying Lie properties of \(FG\)</a>

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<h3>3 The basic theory behind LAGUNA</h3>

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

<a id="X85C2DE1486FB45FD" name="X85C2DE1486FB45FD"></a>

<h4>3.1 Notation and definitions</h4>

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 id="X7E26AE6C807D7C07" name="X7E26AE6C807D7C07"></a>

<h4>3.2 \(p\)-modular group algebras</h4>

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 <a href="chapBib_mj.html#biBHB">[HB82, Chapter VIII]</a>. 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.

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</pan>

<h4>3.3 Polycyclic generating set for \(V\)<java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 66

pjava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 20

\[
A\rhd A^2\rhd\cdots\rhd A^{c}\rhd A^{c+1}=0.
\]

It is clear that a quotient \(A^i/A^{i+1}\)of this chain has trivial multiplication, that is, such a quotient is a java.lang.StringIndexOutOfBoundsException: Index 152 out of bounds for length 0

\[
V=1+A\rhd 1+A^2\rhd\cdots\rhd 1+A^c\rhd 1+A^{c+1}=1.
\]

<

Now we show how to compute a basis for \[
G=G_1 \rhd G_2\rhd\cdots\rhd G_{k}\rhd G_{
\]

be the Jennings series of Oneof the aims the LAGUNA package is to compute a power-commutator presentationjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

A non-empty product

\[
u=(x_{1,1}-1)^{\alpha_{1,1}}\cdots(x_{1,java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 0
({1-1^\alpha_{,}\dots(x_{,_}-1){\alpha_kl_k}}
\]

where \(0\leq \alpha_{i,j}\leq p-1\) is said to java.lang.StringIndexOutOfBoundsException: Range [0, 85) out of bounds for length 0

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\]</< =""\[

<> number of products <spanclass=">(G|1\./>

Lemma (<a href="chapBib_mj.html#biBHB">[HB82, Theorem VIII\

A basis for \(A\) consisting of java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 0

Let \(x_1,\ldots,x_{{|G|}-1}\) denote the standard products where we choose the indices java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

Jennings </m> of\G\)/>.That , ({i+}[_i,G]_j^p}\) where\(\ is the smallest -negative integer that <pan class=SimpleMath">(j\eq /\)/> select elements <pan class="SimpleMath">\(x_{i,1},\ldots,x_{i,l_i}\) of \(G_i\) such that \(\{x_{i,1}G_{i+1},\ldots,x_{java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
v=y_1^{\alpha_1}\cdots]/>
\alpha_1,\ldots,\alpha_{|G|-1} \in \{0,\ldots,p-1\}.
\]

This expression is called the canonical form of

<d=X7DCB12F48752C17E ="7DCB12F48752C17E<a>/>

<h4>3<pThetotalnumber standard products <pan =SimpleMath"(G|1)/> .<pjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91

We show how to compute the canonical form of a normalised unit with respect to the polycyclic generating set \[
w\equiv \alpha_{s_i}x_{s_i}+\cdots+\alpha_{r_i}x_{r_i}\quad \bmodjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
\<

<p\lpha_1,ldots,\alpha_{|G|-1} \in \{0,\ldots,p-1\}.

<pclass"">[
1+w\equivpT expression is called the canonical form of \(v\)<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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\<pjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

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 \(1+w\).

<a id="X873F4F068759DF5D" name="X873F4F068759DF5D"></a>

<h4>3.5 Computing a power commutator presentation for \(V\)</h4>

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 <a href="chap3_mj.html#X7B74767A849D921E">3.3</a>. 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 <a href="chap3_mj.html#X7DCB12F48752C17E">3.4</a>.

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

<a id="X79183196837B7CC1" name="X79183196837B7CC1"></a>

<h4>3.6 Verifying Lie properties of \(FG\)</h4>

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 <a href="chapBib_mj.html#biBLR86">[LR86]</a>, <a href="chapBib_mj.html#biBPPS73">[PPS73]</a>, and <a href="chapBib_mj.html#biBRos00">[Ros00]</a>.

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