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Vo.Lt the EULERIAN FUNCTIONS of CYCLIC Nl /vo.lt THE EULERIAN FUNCTIONS OF CYCLIC GROUPS, DIHEDRAL GROUPS, AND P- GROUPS THESIS Presented to the Graduate Council of the University of North Texas in Partial Fulfillment of the Requirements For the Degree of MASTER OF ARTS by Cynthia M. Sewell, B.S. Denton, Texas August, 1992 Z3~:A Sewell, Cynthia M., The Eulerian Functions of Cyclic Groups, Dihedral Groups, and P-Groups. Master of Arts (Mathematics), August, 1992, 50 pp., 13 illustrations, bibliography, 5 titles. In 1935, Philip Hall developed a formula for finding the number of ways of generating the group of symmetries of the icosahedron from a given number of its elements. In doing so, he defined a generalized Eulerian function. This thesis uses Hall's generalized Eulerian function to calculate generalized Eulerian functions for specific groups, namely: cyclic groups, dihedral groups, and p- groups. TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS ... ............. iv Chapter I. INTRODUCTION .. ............. 1 Historical significance Definition of the Eulerian Function II. THE MOBIUS FUNCTION..... .. ...... 4 Definition and calculation of the MObius function The Eulerian function of cyclic groups Frattini subgroup III. THE EULERIAN FUNCTION OF DIHEDRAL GROUPS... ... 14 Classification of subgroups Calculation of Mt*bius function IV. THE EULERIAN FUNCTION OF P-GROUPS.... ... 35 Weisners Theorem BIBLIOGRAPHY. ..... ............... 50 iii LIST OF ILLUSTRATIONS Figure Page 1. Mbuis function example....... ..... 2. MObius function example.. ......... 3. Subgroup structure of the integers modulo 12. 4. Subgroup structure of the integers modulo p . .18 5. Subgroup structure of the integers modulo pq. .20 6. Subgroup structure of the Quaternion group. .. 4..21 7. Subgroup structure of D2p . .2518 8. Subgroup structure of D2p n . 0 20 . .39 9. Subgroup structure of D2pq... .......... 0 21 10. Subgroup structure of D2pqs. ......... 25 11. Subgroup structure of the elementary abelian group of order 4.............. .......... .. .39 12. Subgroup structure of the elementary abelian group of order 9............ ........... .. .40 13. Subgroup structure of the elementary abelian group of order 8............. ........... .. .44 iv CHAPTER I INTRODUCTION In 1935, Philip Hall developed a formula for finding the number of ways of generating the group of symmetries of the icosahedron from a given number of its elements. In doing so, he showed that this problem can be solved for any finite group whose subgroups are sufficiently known. Given any finite group G, Hall defined the function $n (G) to be the number of ways of generating G with n of its elements. This function is known as the generalized Eulerian function and is defined as follows: #n(G) = p,(H,G) JH| H: H < G [In In order to find the Eulerian function of a finite group G, we must first calculate, for each subgroup H of G, the integer valued function p(H,G), known as the M6bius function. Only those subgroups whose Mtbius functional values differ from zero need to be considered. Chapter II provides a theorem for determining these subgroups. The MObius function is. defined and discussed in Chapter II. The second chapter also provides examples of the calculation of both the M*bius and Eulerian functions and is concluded with the generalized Eulerian function for cyclic groups. 1 2 The third chapter is devoted to dihedral groups. This chapter addresses the problem of finding all the subgroups of a given finite dihedral group and calculating the Mbius functional value for each subgroup. Again a Eulerian function is calculated for finite dihedral groups. In the final chapter, the Eulerian function of p-groups is determined. The calculation of the Mobius function on the subgroups of a given p-group is simplified, through the use of several theorems and lemmas, to the calculation of the Mbius function on an elementary abelian group. It is here that Weisner's Theorem is employed. The chapter concludes with the generalized Eulerian function for p-groups. CHAPTER BIBLIOGRAPHY P. Hall, The Eulerian Functions of a Croup, Quart. J. Math., 7 (1936), 134-151. 3 CHAPTER II THE MOBIUS FUNCTION The Mbius function , p , is the integer-valued function defined inductively on the set S of all subgroups of a finite group G by: (*) tp(G,G) = 1 and for each subgroup H of G (**) p(K,G) = 0. K:H < K Consider any group G such that all the subgroups of G are known. Since the elements of S are subgroups, we can arrange them according to their order; that is, the elements of S are arranged in the following way: G = Si, S 2 , S3 ..., Sn = {e}, where IGI 1S2 1 > --- >> Sn.- The M5bius function assigns one and only one numerical value to each element of S. The initial condition (*) stipulates that p(G,G) = 1. To assign a value to S2 one must first find all the members of S that contain S2 . The sum of the functional values of these members of S containing S2 along with the functional value of S2 must be zero. S2 is 4 5 S2S3 -S2i SE Figure 1 Figure 2 of a subgroup of G and only G; therefore, by (**): p(S 2 ,G) + p(G,G) = 0. Solving for p(S2 ,G) we find that the functional value of S2 is -1. Now let us consider S3 . Suppose that S3 is not a subgroup of S2 . Then by (**): g(S 3 ,G) + p(G,G) = 0 and hence, A(S53 ,G) = -1. (See Figure 1) However, if S3 is a subgroup of S2 (then again by (**)): p(Sa,G) + p(S 2 ,G) + p(G,G) = 0 and hence, 1(4S 3 ,G) = - ( p(52 ,G) + IL(G,G) ) = - ( -1 + 1 ) = 0 (See Figure 2) One can continue in this manner to assign a functional value to each subgroup. Example 2.1. Let us now demonstrate this procedure on the abelian group (112 = { ,iHTHH,7,H,9,ThK} of the 6 3iZ1-2 < 22> <3 "< I<I< FT < 0 > Figure 3 integers modulo 12 under addition. The subgroups of E12 are as follows (Figure 3): E1 2 < > ={,,1,i, ,~} <H > = {5,,,9}, < > = <K > = {tH}. < > = {}. The M6bius functional values can now be calculated for each of the above subgroups. (See figure 3) p1(112 ,112) = 1 S >2112) = -1 p( 3>,iE12) 7 p Dz pq = < X > [II -1< XP> < - {e} E-] {e} LIII| Figure 4 Figure 5 The subgroup <~4 > is contained in 112 and < > ; and, therefore, by (**): Pu(112 ,F1 2) + A(<~ >,1!>2) + pt(<24 >,12) = 0. Hence A(< 4 >,112) = 0. The subgroup < ~B> is contained in 912, < ~ >, and < ~3 >; and, again by (**): (1 2 ,Z112) + p(< ~2 >,12) + (< ~3 >,L12) + A(<6 >,12) = 0. Hence p( 6>,2712) = - The subgroup < ~ > is contained in every subgroup of E12. Once again using (**) we find that: A(< b~ >, 112) = 0. Once the M5bius function values are determined for every subgroup of 112, the Eulerian Function for 112 can be calculated as follows: I On (E2) = p(HA12) Hn _ 12 [ 1_ ]+ 2 n IJ J [J + [ We now have a function for determining the number of n-bases that generate 112. For example: 8 01 (11 2 ) = 4. This reveals that E12 can be generated by four distinct elements of the group. Example 2.2. The Eulerian Function for HP, for any prime p, is as follows (See Figure 4): #n (21P)=n n Example 2.3. Consider the group Epq such that p and q are distinct primes. (See Figure 5) The Mbbius Function is easily computed and the Eulerian Function for "pq is given by the following formula: On(11pq) = [pq [] [n]+[] Example 2.4. Consider the Quaternion Group, Q = { 1, + i, + j,2=k }where i2 = j2 = k2 = -1; ij = k; jk = i; ki = j; ji = -k; kj = -i; ik = -j; and the usual rules apply for multiplying by -- 1. The subgroups of G are: G, < i > = { i, -i, -1, 1 }, < j > = { j, -j, -1, 1 }, < k > = { k, -k, -1, 1 }, < -1 > = { -1, 1 }, < 1 > = { I }. See figure 6 on the following page. The Eulerian Function of the Quaternion Group is #n (G) = []- 3 [ ] 2 [J. The following theorem is useful in calculating Eulerian functions on larger groups. 9 G i> < <j> < k> < -1> {e} Figure 6 THEOREM 2.1. Let G be a group and H be any subgroup of G. The Mdbius function p(H,G) can differ from zero if and only if H can be written as the intersection of a certain number of maximal subgroups of G. Proof. The proof of the theorem is by induction. Let us suppose that Theorem 2.1 holds for all Ki > H, where H < G and Ki < G for all i. Suppose further that H is not the meet of any set of maximal subgroups of G. In other words, there does not exist a set {M1 ,M2 ,...,M} of maximal subgroups of G such that H = M1 f M2 f ... fl M. Let M represent the intersection of all the maximal subgroups of G that contain H. Hence, M = Mi where H < Mi for i=1 1 to n. Using the definition of the Mbbius function we know: or K:HK< K p(KG) = 0 10 [ pK(K,G) ] + p(H,G) = 0. K:H < K Therefore, ( K:HK (K,G) = - pa(H,G). For all K such that H < K but K does not contain M, we know that K is not the meet of any set of maximal subgroups of G; hence, for all such K, p(K,G) = 0 (by hypothesis). Thus, (*) can be rewritten as follows: I(KG) = - u(H,G). K:K M But M is, itself, a subgroup of G, and therefore pL(KG) = 0.
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