arxiv:1502.04776v1 [math.gr] 17 Feb 2015 The ormal subgroup structure of ZM-groups Marius Tărăuceau February 17, 2015 Abstract The mai goal of this ote is to determie ad to cout the ormal subgroups of a ZM-group. We also idicate some ecessary ad sufficiet coditios such that the ormal subgroups of a ZM-group form a chai. MSC (2010): Primary 20D30; Secodary 20D60, 20E99. Key words: ZM-groups, ormal subgroups, chais. 1 Itroductio The startig poit for our discussio i give by the paper [2], where the class G of fiite groups that ca be see as cyclic extesios of cyclic groups has bee cosidered. The mai theorem of [2] furishes a explicit formula for the umber of subgroups of a group cotaied i G. I particular, this umber is computed for several remarkable subclasses of G: abelia groups of the form Z m Z, dihedral groups D 2m, ad Zassehaus metacyclic groups (ZM-groups, i short). I group theory the study of the ormal subgroups of(fiite) groups plays a very importat role. So, the followig questio cocerig the class G is atural: Which is the umber of ormal subgroups of a group i G? The purpose of the curret ote is to aswer partially this questio, by fidig this umber for the above three subclasses of G. Sice all subgroups of a abeliagroupareormal, forthefirstsubclasstheaswer isgiveby[2]. The 1
umber of ormal subgroups of the dihedral group D 2m is also well-kow, amely τ(m) + 1 if m is odd, ad τ(m) + 3 if m is eve (as usually, τ(m) deotes the umber of distict divisors of m N ). Therefore we will focus oly o describig ad coutig the ormal subgroups of ZM-groups. Most of our otatio is stadard ad will ot be repeated here. Basic defiitios adresults ogrouptheory cabefoudi[5, 6,8]. Forsubgroup lattice theory we refer the reader to [7, 9]. First of all, we recall that a ZM-group is a fiite group with all Sylow subgroups cyclic. By [5], such a group is of type ZM(m,,r) = a,b a m = b = 1, b 1 ab = a r, where the triple (m,, r) satisfies the coditios gcd(m,) = gcd(m,r 1) = 1 ad r 1(modm). It is clear that ZM(m,,r) = m, ZM(m,,r) = a (cosequetly, we have ZM(m,,r) = m) ad ZM(m,,r)/ZM(m,,r) is cyclic of order. Oe of the most importat (lattice theoretical) property of the ZM-groups is that these groups are exactly the fiite groups whose poset of cojugacy classes of subgroups forms a distributive lattice (see Theorem A of [1]). We ifer that they are DLN-groups, that is groups with distributive lattice of ormal subgroups. The subgroups of ZM(m,,r) have bee completely described i [2]. Set { } L = (m 1,,s) N 3 m 1 m,, s < m 1, m 1 s r 1. r 1 The there is a bijectio betwee L ad the subgroup lattice L(ZM(m,,r)) of ZM(m,,r), amely the fuctio that maps a triple (m 1,,s) L ito the subgroup H (m1,,s) defied by H (m1,,s) = α(,s) k a m 1 = a m 1,α(,s), where α(x,y) = b x a y, for all 0 x < ad 0 y < m. Remark also that H (m1,,s) = m m 1, for ay s satisfyig (m 1,,s) L. By usig this result, we are able to describe the ormal subgroup structure of ZM(m,,r). 2
Theorem 1. The ormal subgroup lattice N(ZM(m,, r)) of ZM(m,, r) cosists of all subgroups where H (m1,,s) L(ZM(m,,r)) with (m 1,,s) L, L = { (m 1,,s) N 3 m 1 gcd(m,r 1),, s = 0 } L. We ifer that, for every m 1 m ad, ZM(m,,r) possesses at most m oe ormal subgroup of order m 1. I this way, all ormal subgroups of ZM(m,, r) are characteristic. I particular, the above theorem allows us to cout them. Corollary 2. The followig equality holds N(ZM(m,,r)) = τ(gcd(m,r 1)). (1) I the followig we will deote by d the multiplicative order of r modulo m, that is d = mi{k N r k 1(modm)}. Clearly, the sum i the right side of (1) depeds o d. For m or primes, this sum ca be easily computed. Corollary 3. If m is a prime, the while if is a prime, the N(ZM(m,,r)) = τ()+τ( ), (2) d N(ZM(m,,r)) = τ(m)+1. (3) Metio that the umber of ormal subgroups of the dihedral group D 2m with m odd ca be obtaied from (3), by takig = 2. Nextwewillfocusofidigthetriples(m,,r)forwhichN(ZM(m,,r)) becomes a chai. Theorem 4. The ormal subgroup lattice N(ZM(m,,r)) of ZM(m,,r) is a chai if ad oly if either m = 1 ad is a prime power, or both m ad are prime powers ad gcd(m,r k 1) = 1 for all 1 k <. 3
Remark that Theorem 4 gives a method to costruct fiite (both abelia ad oabelia) groups whose lattices of ormal subgroups are chais of prescribed legths. Fially, we idicate a ope problem with respect to the above results. Ope problem. Describe ad cout the ormal subgroups of a arbitrary fiite group cotaied i G. Also, exted these problems to arbitrary fiite metacyclic groups, whose structure is well-kow (see, for example, [4]). 2 Proofs of the mai results Proof of Theorem 1. First of all, we observe that uder the otatio i Sectio we have This implies that Sice α(x 1,y 1 )α(x 2,y 2 ) = α(x 1 +x 2,r x 2 y 1 +y 2 ). α(x,y) k = b kx a y rkx 1 r x 1, for all k Z, ad α(x,y) 1 = α( x, r x y). α(x,y) 1 α(,s)α(x,y) = α(,t x,y ), where t x,y = r y +r x s+y, oe obtais H α(x,y) (m 1,,s) = α(x,y) 1 H (m1,,s)α(x,y)= = = = α(x,y) 1 α(,s) k α(x,y) 1 a m 1 = ( α(x,y) 1 α(,s)α(x,y) ) k a m 1 = α(,t x,y ) k a m 1 = H (m1,,t x,y) 4
with the covetio that t x,y is possibly replaced by t x,y modm 1. The H (m1,,s) is ormal i ZM(m,,r) if ad oly if we have t x,y s(modm 1 ), or equivaletly m 1 s(r x 1) y(r 1), (4) for all 0 x < ad 0 y < m. Take x = 0 i (4). It follows that m 1 y(r 1), for all 0 y < m, ad so m 1 r 1. We get m 1 s(r x 1), for all 0 x <. By puttig x = 1 ad usig the equality gcd(m,r 1)=1, it results m 1 s. But s < m 1, therefore s = 0. Hece we have proved that the subgroup H (m1,,s) is ormal if ad oly if m 1 gcd(m,r 1) ad s = 0, as desired. Proof of Theorem 4. Suppose first that N(ZM(m,,r)) is a chai. The ZM(m,,r) is a moolithic group, that is it possesses a uique miimal ormal subgroup. By Theorem 5.9 of [3] it follows that either m = 1 ad is a prime power, or m is a prime power ad r k 1(modm) for all 1 k <. O the other had, we observe that N(ZM(m,, r)) cotais the sublattice L 1 = { H (1,1,0) }, which is isomorphic to the lattice of all divisors of. Thus is a prime power, too. I order to prove the last assertio, let us assume that gcd(m,r k 1) = m 1 1 for some 1 k < ad cosider k to be miimal with this property. It follows that k. The the subgroup H (m1,k,0) belogs to N(ZM(m,,r)), but it is ot comparable to H (1,,0) = ZM(m,,r), a cotradictio. Coversely, ifthetriple(m,,r)satisfiesoeofthecoditiositheorem 4, the N(ZM(m,,r)) is either a chai of legth v for m = 1 ad = q v (q prime), amely H (1,q v,0) H (1,q v 1,0) H (1,1,0), or a chai of legth u+v, for m = p u ad = q v (p,q primes), amely H (p u,q v,0) H (p u 1,q v,0) H (1,q v,0) H (1,q v 1,0) H (1,1,0). This completes the proof. Ackowledgemets. The author is grateful to the reviewer for its remarks which improve the previous versio of the paper. 5
Refereces [1] Bradl, R., Cutolo, G., Riauro, S., Posets of subgroups of groups ad distributivity, Boll. U.M.I. 9-A (1995), 217-223. [2] Calhou, W.C., Coutig subgroups of some fiite groups, Amer. Math. Mothly 94 (1987), 54-59. [3] De Medts, T., Tărăuceau, M., Pseudocomplemetatio i (ormal) subgroup lattices, Comm. Algebra 39 (2011), 247-262, doi: 10.1080/00927870903527493. [4] Hempel, C.E., Metacyclic groups, Comm. Algebra 28 (2000), 3865-3897. [5] Huppert, B., Edliche Gruppe, I, Spriger Verlag, Berli, 1967. [6] Isaacs, I.M., Fiite group theory, Amer. Math. Soc., Providece, R.I., 2008. [7] Schmidt, R., Subgroup lattices of groups, de Gruyter Expositios i Mathematics 14, de Gruyter, Berli, 1994. [8] Suzuki, M., Group theory, I, II, Spriger Verlag, Berli, 1982, 1986. [9] Tărăuceau, M., Groups determied by posets of subgroups, Ed. Matrix Rom, Bucureşti, 2006. Marius Tărăuceau Faculty of Mathematics Al.I. Cuza Uiversity Iaşi, Romaia e-mail: tarauc@uaic.ro 6