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A CENTRAL SERIES ASSOCIATED with V (G) a Dissertation Submitted A CENTRAL SERIES ASSOCIATED WITH V (G) A dissertation submitted to Kent State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy by Nabil Mlaiki August 2011 Dissertation written by Nabil Mlaiki B.S., Kent State University, 2004 M.A., Kent State University, 2006 Ph.D., Kent State University, 2011 Approved by Mark L. Lewis, Chair, Doctoral Dissertation Committee Stephen M. Gagola, Jr., Member, Doctoral Dissertation Committee Donald L. White, Member, Doctoral Dissertation Committee David Allender, Member, Doctoral Dissertation Committee Mike Mikusa, Member, Doctoral Dissertation Committee Accepted by Andrew Tonge, Chair, Department of Mathematical Sciences Timothy S. Moerland, Dean, College of Arts and Sciences ii TABLE OF CONTENTS ACKNOWLEDGEMENTS . iv INTRODUCTION . 1 1 BACKGROUND ABOUT FINITE GROUPS . 8 2 BACKGROUND ABOUT V (G) ........................... 13 3 GENERAL LEMMAS FOR THE CENTRAL SERIES ASSOCIATED WITH V (G) 16 4 PROOF OF THE FIRST TWO THEOREMS AND THEOREM 4 . 21 5 CAMINA PAIRS AND CAMINA TRIPLES . 23 6 FUTURE RESEARCH QUESTIONS . 31 BIBLIOGRAPHY . 32 iii ACKNOWLEDGEMENTS I would like to express my sincere gratude to Dr. Lewis for being a good advisor and mentor. I enjoyed our sessions and thank him for guidance, support and encouragement he has given me throughout this process. THANK YOU! I also, like to thank Dr. White and Dr. Gagola for all their help. I thank my mother, and my brother Neji Mlaiki for all their support. iv INTRODUCTION Throughout this dissertation, G is a finite group. Denote by C the set of complex numbers. Define the general linear group, GL(n; C); to be all invertible n×n matrices over C: A C-representation of G is a homomorphism < : G ! GL(n; C): For many purposes, the set of representations has too much information about the group. So, it will be enough to consider the traces of the representations. Recall that the trace of a matrix is the sum of the diagonal entries of the matrix. The character χ afforded by < is the function given by χ(g) = tr<(g) for g 2 G; where tr is the trace of <(g): Thus, χ is a function from G to C: However, except when n = 1; this function χ is not a homomorphism. We will be interested in the set of all characters that can occur for a group G: One can show that the sum of characters is a character. A character is said to be irreducible if it cannot be written as a sum of other characters of G: Every character is a sum of irreducible characters, and if one knows the irreducible characters of G; then one can determine all characters of G: Hence, we focus on the irreducible characters of G: We write Irr(G) for the set of irreducible characters of G: Let g be an element of G: The conjugacy class of g is the set denoted by cl(g) and defined by cl(g) = fxgx−1 j x 2 Gg: The conjugacy classes of G partition G: That is, if x; y 2 G; then either cl(x) \ cl(y) is empty or cl(x) = cl(y): One can show that characters are constant on the conjugacy classes of a group. Hence, characters are what we call class functions. A class function of G is any function from G to C that is constant on conjugacy classes. A class function is a character if and only if it is afforded by some representation. One can define addition ' + γ of class functions by (' + γ)(x) = '(x) + γ(x); and scalar multiplication c' by (c')(x) = c'(x): Note that with this addition and scalar multiplication the set of class functions of a group G forms a vector space, and one can show that the set 1 2 of irreducible characters forms a basis for this vector space. There is another basis for this vector space. Let C be a conjugacy class for G: Define the class function fc by fc(g) = 1 if g 2 C and fc(g) = 0 if g 62 C: The set ffc j C is a conjugacy class of Gg is also a basis whose size is the number of conjugacy classes of G: This implies that the number of irreducible characters equals the number of conjugacy classes. As proved in Corollary 2.7 P 2 in [4], χ2Irr(G) χ(1) = jGj: Given a character χ of G; note that χ(1) = n; where n is the integer, so that < : G ! GL(n; C) is a representation affording χ. Let χ be a character of a group G: We say that χ(1) is the degree of χ. Characters of degree 1 are called linear characters. Linear characters are exactly the characters that are homomorphisms from G to C∗ =∼ GL(1; C): If λ is a linear character, then λ(g) is invertible for all g 2 G: Hence, λ(g) 6= 0 for all g 2 G: An irreducible character χ of G is nonlinear if χ(1) 6= 1: For nonlinear characters the situation is different. The following result is Theorem 3.15 in [4], and is due to Burnside: if χ is an irreducible character of G with χ(1) > 1; then χ(g) = 0 for some g 2 G: We write nl(G) for the set of nonlinear irreducible characters of G: Let X be a subset of G: Define hXi to be the subgroup generated by X (i.e., the smallest subgroup of G containing X): Now, define the vanishing off subgroup V (χ) of a character χ to be the subgroup generated by all the elements g 2 G where χ(g) is not 0: Note that the set of elements where χ(g) is not 0 tends not to be a subgroup. That is why we consider the subgroup V (χ) = hg 2 G j χ(g) 6= 0i: If λ is a linear character, then observe that V (λ) = G: We mention that V (χ) is the smallest subgroup of G such that χ vanishes on G n V (χ): Also, if χ is nonlinear, then we know χ must vanish on some elements of G; but it is possible to have the set of elements where χ does not vanish generate all of G: Hence, we can have V (χ) = G for χ 2 nl(G): In the groups we are interested in, we will have V (χ) < G for χ 2 nl(G): In this dissertation, we are going to study a related subgroup. We define the vanishing 3 off subgroup of G, denoted by V (G); by V (G) = hg 2 G j there exists χ 2 nl(G) such that χ(g) 6= 0i: It was first introduced by Lewis in [8], which appeared in 2008. We exclude linear characters so that it is possible to have V (G) < G: Also, note that V (G) is the smallest subgroup of G such that all nonlinear irreducible characters vanish on G n V (G): In this dissertation we are interested in the case where V (G) is proper in G: Note that V (G) may have some elements g such that for every χ 2 nl(G); χ(g) = 0: A series of normal subgroups Ni C G such that N0 ≤ N1 ≤ · · · ≤ Nn; is called a central series in G if Ni+1=Ni ≤ Z(G=Ni) for all i = 0; ··· ; n − 1: We need to consider the commutator of two elements in a group. Let x; y 2 G: The commutator of x and y is denoted by [x; y] and is defined by [x; y] = x−1y−1xy: Also, if H; K are two subgroups of G; the commutator subgroup of H and K is [H; K] = hh−1k−1hk j h 2 H; k 2 Ki: Now, consider the term Gi as the i-th term in the lower central series, which is defined by 0 G1 = G; G2 = G = [G; G]; and Gi = [Gi−1;G] for i ≥ 2: We are going to study a central series associated with the vanishing off subgroup, defined by V1 = V (G), and Vi = [Vi−1;G] for i ≥ 2: Lewis, in [8], proved that Gi+1 ≤ Vi ≤ Gi: Our main focus is on the case when Vi < Gi for some i > 3: In [8], Lewis showed that in the case when Vi < Gi; we have Vj < Gj for all j such that 1 ≤ j ≤ i: An abelian group is a group with the property that for every x 2 G we have jcl(x)j = 1: If, in such group, every nonidentity element has the same prime order p; then the group is called an elementary abelian p-group. In [8], Lewis proved that if V2 < G2, then there exists a prime p such that Gi=Vi is an elementary abelian p-group for all i ≥ 1: Also, he proved that if V3 < G3 there exists a positive integer n such that 0 2 2n jG : V1j = jG : V2j = p . The main questions that we are trying to answer in Chapters 3 and 4 are can we generalize the results in [8] to the case where Vi < Gi for i > 3? What can we say about the index of Vi−1 in Gi−1? 0 To answer these questions, we define some subgroups. First, set D3=V3 = CG=V3 (G =V3): p Lewis, in [8], proved that if V3 < G3, then either jG : D3j = jG : V1j or D3 = V1. To 4 study the case when i > 3; we will add an additional hypothesis and a new definition.
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