Finite P-groups With Only Two Different Character Degrees

Let G denote a finite p-group of class 2 and cd(G)the set of irreducible charac-ter degrees of G respectively.The main objective of this thesis is to study G when cd(G)contains only two elements.In general case,we obtain the structure of G'and the number of generators of G.For a special case when cd(G)= {1,p2} with an assumption that G/Z(G):<p5 and |G'|?p2,we get the structure of G.Our main results are:Theorem 3.1 Let G be a finite p-group of class 2 and cd(G)= {1,pk}.Then the derived group G' is elementary abelian.Corollary 3.1 Let G be a finite p-group and cd(G)= {1,pk}.Then d(G)? 2k,where d(G)is the number of generators of G.Theorem 4.1 Let G be a finite p-group.G/Z(G)is an elementary abelian group and p(?)cd(G).Suppose M is a maximal subgroup of G which contains Z(G).then M' = G'.In particular,we have |G'| ? pCn-1 2 if |G/Z(G)| = pn and n ? 4.Theorem 4.2 Let G be a finite p-group of class 2 and cd(G)= {1,p2}.Then the following holds:1.|G/Z(G)| =p.(1.1)If |G'| = p,then G =(G1*G2)· Z(G)where G1,G2 are both minimal non-abelian p-groups.(1.2)If |G'| = p2,then G =<x1,x2,x3,x4,Z(G)|[xi,x3]=[x2,x4]=1,[x2,xx3]= a,[x1,x4]= at1,[x1,x2]= b,[x3,x4]= al1bk2>,where G' =(a)×(b)and(t1,p)(k2,p)= 1.2.|G/Z(G)|=p5.(2.1)If |G'| = p,there is no such p-group.(2.2)If |G'| = p2,then G =<x1,x2><x3,x4,x5,Z(G)>,where |x1,x2|=[x1,x5]=[x2,x4]=[x3,x4]=[x3,x5]=[x4,x5]= 1,[x1,x3]= l1,[x1,x4]=l2,[x2,x4]= l2,[x2,x3]=l1r1,[x2,x5]=l1t1l2t2,(r1,p)=(r2,p)= 1,and G' =<l1>×<l2>.