| Table algebras were initially defined by Arad and Blau in 1991,in order to study in a uniform way the decomposition of products of conjugacy classes and irreducible characters of finite groups.Table algebras are generalization of the center of group algebras of finite groups,Bose-Mesner algebras of commutative association schemes and the algebras of all complex valued class functions on finite groups,etc.The classification of table algebras generated by a faithful non-real element of small degree has been an interesting topic,but all papers assume that either the identity element is the only linear element or the identity element is the only basis element of degree one.For an integral table algebra(A,B),the set of basis elements of degree one L1(B)is contained in the set of linear elements L(B),i.e.,L1(B)(?)L(B).If(A,B)is a normalized integral table algebra,then L1(B)=L(B)and L1(B)is an abelian group under the multiplication of A.This thesis is mainly focus on the classification of normalized integral table algebras generated by a faithful non-real element of degree 2 and with prime linear elements,and normalized integral table algebras generated by a faithful real element of degree 2 and with 2 and 4 linear elements.This thesis is divided into five chapters.In chapter 1,the research background,main results and the innovation of this thesis are stated.In chapter 2,firstly,we give some known results and terminologies on table algebras.Secondly,we give several examples of integral table algebra.In chapter 3,we mainly study the normalized integral table algebras(A,B)generated by a faithful element b2 of degree 2 and having two linear elements.If b2 is non-real,then(A,B)is exactly isomorphic to(Ch(GL(2,3)),Irr(GL(2,3)))or(Ch(SL(2,5)(?)C2)),Irr(SL(2,5)(?)C2)).If b2 is real,we proved that(A,B)is exactly isomorphic to(Ch(G),Irr(G))or(Ch(D2(2n|1)),Irr(D2(2n|1))),where G=<a,b,a|a4=b3=c2=abc>is the binary octahedral group and D2(2n+1))is the dihedral group of order 2(2n+1).In chapter 4,we mainly investigate the normalized integral table algebras(A,B)generated by a faithful non-real element b2 of degree 2 and |L1(B)|=p,where p≥3 is a prime.Firstly,since |L1(B)|=p,let L1(B)=<q1>(?)Cp.We prove that b22=d2+d2g1 or b22=g1i+b3g1i,where d2∈B/{b2g1l|l=1,2,…,p},b3∈B and 1≤i≤p-1.Secondly,we completely determine the table algebras satisfy b22=d2+d2q1.The basis and the multiplications on basis elements are given.Moreover,we also discuss the exactly isomorphic problem for such table algebras.Finally,we classify the table algebras satisfying b22=g1i+b3g1i.It is proved that(A,B)is exactly a NITA of dimension 9p or(A,B)(?)x(Ch(SL(2,3),Irr(SL(2,3))).In Chapter 5,we completely classify the normalized integral table algebras(A,B)generated by a faithful real element b2 of degree 2 and |L1(B)|=4.First of all,it is prove that there is no b3∈B such that b22=1+b3.Secondly,we classify the NITA satisfying L1(B)=<g1>(?)C4.It is proved that b22=1+g12+g1+g1 or there exists c2,2∈B such that b22=1+g12+c2,2.If b22=1+g12+g1+g1,then B={1,g1,g12,g1,b2} with g14=1 and b22=1+g12+g1+g1.Moreover,in this case,(A,B)is not exactly isomorphic to any table algebra of generalized characters.If b22=1+g12+c2,2 and for any α∈B,α≠αg1.We show that(A,B)(?)x(Ch(T4(2n|1)),Irr(T4(2n|1))),where T4(2n|1)=<a,b|α2(2n+1)=1,α2n|1=b2,b-1αb=α1>is a group of order 4(2n+1).If b22=1+g12+c2,2 and there exists some α∈B such that α=αg1,it is proved that(A,B)is exactly a NITA of dimension 2n+3(n≥2).Finally,if L1(B)(?)C2×C2,we show that(A,B)(?)x(Ch(D4n),Irr(D4n)),where D4n is the dihedral group of order 4n. |
