Coding theory, group automorphisms and units of finite group algebras

Abstract

Group Algebras can be used to construct Low Density Parity Check Codes (LDPC) and Convolution Codes. These codes have applications within digital communication and storage, such as to improve the performance of digital radio, digital video, mobile phones, satellite and deep space communications, as well as bluetooth implementations. In this Masters Thesis, theoretical mathematical techniques are used to construct an atlas of finite group algebras. In particular, we find and list the automorphism group of abelian groups and the unit group of finite commutative group algebras. The aim of this atlas is to improve our understanding of group algebras and their applications to Coding Theory. Firstly, the basic concepts of coding theory are introduced. The next section of this thesis (Chapter 2- Automorphisms of Finite Abelian Groups) deals with various techniques for finding the automorphism group of different categories of abelian groups. In particular, where the group is an abelian p-group with 2 distinct direct factors, use is made of recent techniques by Bidwell and Curran (2010). Hillar and Rhea (2007) give a technique involving endomorphism rings which allows the calculation of the order of Aut(G) where G is abelian. Using these techniques and others a table is presented giving the structure and order of the automorphism group for many abelian groups. Chapter 3 (Automorphisms of Non-Abelian Groups) looks beyond abelian groups. Recent methods by Curran (2008) using crossed homomorphisms are used where G is a semidirect product. Dihedral groups and general linear groups are also examined. At the end of this section there are some conjectures relating to the automorphisms of groups in general and a table is presented showing the automorphism tower of small groups. The next section of the thesis (Chapter 4 - Finite Commutative Group Algebras) introduces the concepts of group algebras and unit groups. This Chapter contains many specific example of group algebras. In these examples the structure and order of the unit groups are examined. In Chapter 5 (U(FG) where F has char p and G is a p - group), a technique is presented for finding the structure of the unit group for non-Maschke cases. This technique involves counting the number of elements in the normalised unit group which have order dividing a particular power of p for group algebras of the form FG where G is a p - group and F is a field of characteristic p. This Chapter concludes with a Theorem which gives the unit group of all group algebras of the above form. There are also some examples illustrating this. Chapter 6 (Idempotents and the decomposition of FG) then looks at ways of finding the Artin Wedderburn decomposition where applicable. Here, recent techniques by Broche and Del Rio (2007) are used to find the decomposition and also to find the primitive central idempotents. Finally, in Chapter 7, The Perlis Walker Theorem (1950) is used and adapted to give more general results for all possible group algebras for abelian groups. This leads to a general table giving the decomposition and unit groups of the group algebras for all abelian groups of order up to 15. In doing this, we get a further insight into the isomorphism problem for group algebras. This includes the result that given two non-isomorphic abelian groups G and H each with order n, and a field F of order q such that q= 1(mod n), then FG~FH. Thus there is a whole class of isomorphic group algebras of this type and in each of these instances the decomposition is the direct product of n copies of the field F. We show that the minimal isomorphic pair of group algebras FG and FH with G and H not isomorphic which is not of this type is F5C12 and F5(C2 X C6). We also show that there is yet another class of isomorphic group algebras. Given two non-isomorphic abelian groups G and H each with order n and each containing m elements of order 2, and a field F of order q such that q= -1(mod e) where e is the exponent of the group, then FG~FH. In this case, FG~FH~Omi=1 FqoOmi [n-(m)]/2 i=1 Fq2 . An example of this is F7(C2 X C4 X C8) ~ F7(C3/4 ).