**LAGUNA** -- **L**ie **A**l**G**ebras and **UN**its of group **A**lgebras -- is the new name of the **GAP**4 package **LAG**. The **LAG** package arose as a byproduct of the third author's PhD thesis [Ros97]. Its first version was ported to **GAP**4 and was brought into the standard **GAP**4 package format during his visit to St Andrews in September 1998.

The main objective of **LAG** is to deal with Lie algebras associated with some associative algebras, and, in particular, Lie algebras of group algebras. Using **LAG** it is possible to verify some properties or calculate certain Lie ideals of such Lie algebras very efficiently, due to their special structure. In the current version of **LAGUNA** the main part of the Lie algebra functionality is heavily built on the previous **LAG** releases.

The **GAP**4 package **LAGUNA** also extends the **GAP** functionality for calculations with units of modular group algebras. In particular, using this package, one can check whether an element of such a group algebra is invertible. **LAGUNA** also contains an implementation of an efficient algorithm to calculate the (normalized) unit group of the group algebra of a finite \(p\)-group over the field of \(p\) elements. Thus, the present version of **LAGUNA** provides a part of the functionality of the **SISYPHOS** program, which was developed by Martin Wursthorn to study the modular isomorphism problem; see [Wur93].

The corresponding functions of **LAGUNA** use the same algorithmic and theoretical approach as those in **SISYPHOS**. The reason why we reimplemented the normalised unit group algorithms in the **LAGUNA** package is that **SISYPHOS** has no interface to **GAP**4, and, even in **GAP**3, it is cumbersome to use the **SISYPHOS** output for further computation with the normalised unit group. For instance, using **SISYPHOS** with its **GAP**3 interface, it is difficult to embed a finite \(p\)-group into the normalized unit group of its group algebra over the field of \(p\) elements, but this can easily be done with **LAGUNA**.

The **LAGUNA** package provides a set of functions to carry out some basic computations with a group ring and its elements. Among other things, **LAGUNA** provides elementary functions to compute such basic notions as support, length, trace and augmentation of an element. For modular group algebras of finite \(p\)-groups **LAGUNA** is able to calculate the power-structure of the augmentation ideal, which is useful for the construction of the normalised unit group; see Sections 4.1--4.3 for more details.

One of the aims of the **LAGUNA** package is to carry out efficient computations in the normalised unit group of the group algebra \(FG\) of a finite \(p\)-group \(G\) over the field \(F\) of \(p\) elements. If \(U\) is the unit group of \(FG\) then it is easy to see that \(U\) is the direct product of \(F^*\) and \(V(FG)\), where \(F^*\) is the multiplicative group of \(F\), and \(V(FG)\) is the group of normalised units. A unit of \(FG\) of the form \(\alpha_1 \cdot g_1 + \alpha_2 \cdot g_2 + \cdots + \alpha_k \cdot g_k\) with \(\alpha_i \in F\) and \(g_i \in G\) is said to be normalised if the sum \(\alpha_1 + \alpha_2 + \cdots + \alpha_k\) is equal to \(1\).

It is well-known that the normalised unit group \(V\) has order \(|F|^{|G|-1}\), and so \(V\) is a finite \(p\)-group. Thus computing \(V\) efficiently means to compute a polycyclic presentation for \(V\). For the theory of polycyclic presentations refer to [Sim94, Chapter 9]. For this computation we use an algorithm that was also used in the **SISYPHOS** package. For a brief description see Chapter 3. The functions that compute the structure of the normalised unit group are described in Section 4.4.

The functions that are used to compute Lie properties of \(p\)-modular group algebras were already included in the previous versions of **LAG**. The bracket operation \([\cdot,\cdot]\) on a \(p\)-modular group algebra \(FG\) is defined by \([a,b]=ab-ba\). It is well-known and very easy to check that \((FG, +, [\cdot,\cdot])\) is a Lie algebra. Then we may ask what kind of Lie algebra properties are satisfied by \(FG\). The results in [LR86], [PPS73], and [Ros00] give fast, practical algorithms to check whether the Lie algebra \(FG\) is abelian, nilpotent, soluble, centre-by-metabelian, etc. The functions that implement these algorithms are described in Section 4.5.

**LAGUNA** does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is designed for **GAP**4.4 or later and no compatibility with previous releases of **GAP**4 is guaranteed.

To use the **LAGUNA** online help it is necessary to install the **GAP**4 package **GAPDoc** by Frank Lübeck and Max Neunhöffer, which is available from the **GAP** site or from https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/.

**LAGUNA** is distributed as a `tar.gz`

archive file and can be obtained from https://gap-packages.github.io/laguna/. To unpack the archive `laguna-X.X.X.tar.gz`

you need the program `tar`

. To install **LAGUNA**, copy this archive into the `pkg`

subdirectory of your **GAP**4 installation. The subdirectory `laguna`

will be created in the `pkg`

directory after the following command:

`tar -xf laguna-X.X.X.tar.gz`

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