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<div class="ChapSects"><a href="chap3_mj.html#X7D9FCE3A8526ACBE">3 The basic theory behind LAGUNA</a>
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<s
</java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
<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

< class="center">\[
\chi\left(\sum_{g \in G}\alpha_g g\right)=\sum_{g \in G}\alpha_g.
\]

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 java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

\[
A(FG)=\left\{\sum_{g \in G}\alpha_g g\ |\ \sum_{g \in G}\alpha_g=0\right\}.
\]

An invertible java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

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

<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

A group algebra <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<> of java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 1295

<a id="X7B74767A849D921E" name="X7B74767A849D921E"></a>

<h4>3.3 Polycyclic generating set for \(Vjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

Let \(G\) be a finite \(p\)</x_{,}){alpha_{,}c(x_{lk1^\{,}java.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62

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

< is clear <panclass="impleMath"(^i/A^{+}\<>of chainhastrivial multiplication thatis,such quotientis a -ing. The A)/>gives a of normal subgroups in span class=SimpleMath>(V)/pan>

[
V=1+A<>Thetotal numberstandard productsis < =SimpleMath\|-)/ <pjava.lang.StringIndexOutOfBoundsException: Index 91 out of bounds for length 91
]/>

It is easy to java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0

Now we show how to compute a basis for \(A^i\) that gives a polycyclic java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 0

\[
G=G_1 \rhd G_2\rhd\cdots\rhd G_{k}\rhd G_{k+1}=1
\]

e<series<e spanclassS"\\<span.Thatis,span""\G_1=Gi]{p\<> spanclass=SimpleMath>\j)/>is the non-integer such<"">jg ip<span.Forall<class=S>(\leqk)A non-empty product

\[
u=(x_{1,1}-1)^{\alpha_{1,1}}\cdots(x_{1,l_1}-1)^{\alpha_{1,l_1}}\cdots
(x_{k,1}-1)^{\alpha_{k,1}}\cdots(x_{k,l_k}-1)^{\alpha_{k,l_k}}
\]/java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6

java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0

\[
\sum_{i"X7DCB12F48752C17E"name=X">/>/>
\]<

he total of productsis s class"SimpleMath"\|-\ .</>

Lemma (<a href="chapBib_mj.html#biBHB">[java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

A basis for \(A\)Let \(x_1,\ldots,x_{{|G|}-1}\) java.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 0

\[
v=y_1^{\alpha_1}\cdots (y_{|G|-1})^{\]
\lpha_1\java.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for length 52
\]</pp class=center">[

hisjava.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 349

<a id="X7DCB12F48752C17E" name="X7DCB12F48752C17E"></a>

<h4quad 1+^i+.

We show how to compute the canonical form of \]/>

Lemma.\[
w\equiv \alpha_{s_i}x_{s_i}+\cdots+\alpha_{r_i}x_{r_i}\quad \bmod \quad A^{i+1}
\]

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 \(1+w\).

<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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