INSTITUTE OF TECHNOLOGY SLIGO Coding Theory, Group Automorphisms and Units of Finite Group Algebras Fergal Gallagher This project is submitted in fulfilment of the requirements for the Degree of Master of Engineering Supervisor: Dr. Leo Creedon Submitted to the Institute of Technology, Sligo July 2014 Declaration I certify that the material which I submit for assessment leading to the award of MEng is entirely my own work and has not been taken from others, save the extent that such work has been cited and acknowledged within the text of the study. .................................................. Fergal Gallagher 2 Contents Abstract ii Acknowledgements iv Frequently Used Notation v 1 Introduction 1 1.1 Cyclic Codes . 7 1.2 Group Ring Matrices . 9 1.3 Zero Divisor Codes . 11 2 Automorphisms of Finite Abelian Groups 14 2.1 Automorphisms of Cyclic Groups Cp and Cpn . 17 2.2 Automorphisms of Elementary Abelian groups . 22 n 2.3 Automorphisms of Cpα ...................... 24 2.4 Automorphisms of Cpm × Cpn . 25 2.5 Automorphisms of abelian p − groups with repeating factors . 29 2.6 Finding jAut(Hp)j using Endomorphism Rings . 31 2.7 Table of Aut(G) for Small Abelian Groups . 40 3 Automorphisms of Non-Abelian Groups 44 3.1 Semidirect products . 49 3.2 Automorphisms of Dihedral Groups . 52 3.3 Automorphisms of General Linear Groups . 55 3.4 Automorphism Groups of Automorphism Groups . 59 3.5 Automorphism Tower for Small Groups . 61 4 Finite Commutative Group Algebras 62 4.1 The Unit Group of FG ...................... 63 5 U(FG) where F has char p and G is an abelian p − group 83 5.1 U(FG) where char(F ) = 2 and G is an abelian 2 − group . 86 i 5.2 U(FG) where char(F ) = p and G is an abelian p − group . 89 6 Idempotents and the Decomposition of Semisimple Group Algebras 102 6.1 Field Automorphisms . 106 6.2 Group Representations and Group Characters . 110 6.3 Cyclotomic Classes of G and the Primitive Central Orthogonal Idempotents of FG ........................112 6.4 The Isomorphism Problem . 131 6.5 Table of U(FG) for selected group algebras (G abelian) . 140 7 Perlis Walker Theorem - Finding U(FG) 142 7.1 A General Approach . 145 7.2 U(FG) where Maschke's Theorem does not apply . 149 7.3 The Unit Groups . 150 7.4 General Table of U(FG)(G abelian) . 167 8 Conclusion 176 References 179 Bibliography 182 ii Abstract Group Algebras can be used to construct Low Density Parity Check Codes (LDPC) and Convolution Codes. These codes have applications within digi- tal 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 commu- tative 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. iii 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, re- cent techniques by Broche and Del Rio (2007) are used to find the decompo- sition 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 × 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 Lm exponent of the group, then FG ' FH. In this case, FG ' FH ' i=1 Fq L[n−(m)]=2 3 ⊕ i=1 Fq2 . An example of this is F7(C2 × C4 × C8) ' F7(C4 ). iv Acknowledgements First of all, thanks to my wonderful wife Shona Heffernan for supporting me while I studied. She must often have wondered when or if I would ever finish. Also to my parents, Joan and Lionel, and to Margaret and Padraic Heffernan I am very grateful for helping out in so many ways. Second of all thanks to Grace Corcoran, who met me for coffee many moons ago and put me in touch with her colleague Leo Creedon who became my supervisor for this project. Thanks to Leo for introducing me to linear and abstract algebra, a subject I knew nothing about and a subject that has fascinated me ever since. Thanks for listening to my questions and answering usually with another question. It has stood me in good stead. Apologies for scribbling in the margins of your T opics in Algebra book Leo! Thanks to Ian McLoughlin and Faye Monaghan for their support and encouragement and for adding a new vibrancy to IT Sligo's mathematical research. To my colleages in the research lab (some of whom have escaped already), thanks for helping me to do battle with printers, binders, computers and sandwichmakers and for providing lots of laughs along the way. Thanks also to Aoife, Carmel, Catriona and Taraneh in the Engineering Office who are always happy to assist. Finally thanks to John Bartlett and Mary McLoughlin in the research office for all of their work and to Sligo County Council for contributing to my college fees. v Frequently Used Notation C a code c a codeword G a generator matrix H a parity check matrix w(c) the minimum weight of c dmin the minimum distance of a code n (n; k) code a k dimensional subspace of Fq (n; k; d) code an (n; k) code with minimum distance d p a prime integer n a positive integer Z the set of integers N the set of positive integers a(mod n) the remainder when a is divided by n (a; b) the greatest common divisor of a; b φ(n) Euler's totient function - the number of a ≤ n such that (a; n) = 1 G a group jGj the order of the group G hxi the group generated by x xy yxy−1, the conjugate of x by y jxj o(x), the order of the group element x H < G H is a subgroup of G H/GH is a normal subgroup of G G=H the group of cosets of the normal subgroup H G × H the direct product of groups G and H N o H the semidirect product of groups N and H with N a normal subgroup Cn the cyclic group of order n k Cn the direct product of k copies of the cyclic group of order n vi 1 ≤ i ≤ n Dn the dihedral group of order n Sn the symmetric group on n objects Z(G) the centre of G G0 the commutator subgroup of G Inn(G) the group of inner automorphisms of G Aut(G) the group of automorphisms of G Aut(G : H) the group of automorphisms of G which fix the subgroup H Aut0GG AutnG Aut(Autn−1G) χ an irreducible character of a group G∗ the group of irreducible characters of G R a ring F a field Fn the finite field with n elements Z=nZ the ring of integers mod n (Z=nZ)× the multiplicative group of units of the ring Z=nZ F [x] the algebra of polynomials in x with coefficients in F GLn(F ) the general linear group of degree n over the field F GLn(Z=mZ) the general linear group of degree n over Z=mZ