On the number of cyclic subgroups of a finite Abelian group

Transcription

1 Bull. Math. Soc. Sci. Math. Roumanie Tome 55103) No. 4, 2012, On the number of cyclic subgroups of a finite Abelian group by László Tóth Abstract We prove by using simple number-theoretic arguments formulae concerning the number of elements of a fixed order and the number of cyclic subgroups of a direct product of several finite cyclic groups. We point out that certain multiplicative properties of related counting functions for finite Abelian groups are immediate consequences of these formulae. Key Words: Finite Abelian group, cyclic subgroup, order of elements, multiplicative arithmetic function Mathematics Subject Classification: Primary 20K01, 20K27, Secondary 11A25. 1 Introduction Consider an arbitrary finite Abelian group G of order #G > 1. Let c G) and cg) denote the number of cyclic subgroups of order #G) and the number of all cyclic subgroups of G, respectively. Let #G = p a1 1 par r be the prime power factorization of #G and let G G 1 G r be the primary decomposition of the group G, where #G i = p ai i 1 i r). Furthermore, let LG) and LG i ) denote the subgroup lattices of G and G i 1 i r), respectively. It is known the lattice isomorphism LG) LG 1 ) LG r ). See R. Schmidt [7], M. Suzuki [10]. It follows that c G) = c 1 G 1 ) c r G r ), 1) where = 1 r and i = p bi i with some exponents b i a i 1 i r). Also, cg) = cg 1 ) cg r ). 2) Consider now a p-group G p) of type λ 1,..., λ k ) with 1 λ 1... λ k. It is known that the number of cyclic subgroups of order p ν of G p) is c p ν G p) ) = pk jν 1 p λ0+λ1+...+λjν +k jν 1)ν 1), 3) p 1 The author gratefully acknowledges support from the Austrian Science Fund FWF) under the project Nr. M1376-N18.

2 424 László Tóth where λ jν < ν λ jν+1 and λ 0 = 0. See G. A. Miller [6], Y. Yeh [13]. For every finite Abelian group G the value of c G) is therefore completely determined by formulae 1) and 3). Obviously, cg) = #G c G), leading to a formula for cg). Recently, formula 3) was recovered by M. Tărnăuceanu [11, Th. 4.3], in a slightly different form, by a method based on properties of certain matrices attached to the invariant factor decomposition of an Abelian group G, i.e., G C i1 C is, where i 1 i s = #G, i 1 i 2... i s, C k denoting the cyclic group of order k. For arbitrary positive integers r, n 1,..., n r consider in what follows the direct product C n1 C nr and denote by cn 1,..., n r ) the number of its cyclic subgroups. Let φ denote Euler s function. Theorem 1. For any n 1,..., n r 1, cn 1,..., n r ) is given by the formula φd 1 ) φd r ) cn 1,..., n r ) = φlcmd 1,..., d r )). 4) Formula 4) was derived in [12] using the orbit counting lemma Burnside s lemma). Consequently, the number of all cyclic subgroups of a finite Abelian group can be given by formula 4) in a more compact form. Namely, consider the invariant factor decomposition of G given above and apply 4) by selecting r = s and n 1 = i 1,..., n s = i s. Note that 4) can be used also starting with the primary decomposition of G. In the case r = 2, by using the identity φd 1 )φd 2 ) = φgcdd 1, d 2 ))φlcmd 1, d 2 )) d 1, d 2 1) we deduce from 4) the next result: Corollary 1. For every n 1, n 2 1 the number of cyclic subgroups of C n1 C n2 is cn 1, n 2 ) = φgcdd 1, d 2 )). 5) d 1 n 1,d 2 n 2 It is the purpose of the present paper to give another direct proof of Theorem 1, using simple number-theoretic arguments. In fact, we prove the following formulae concerning the number o n 1,..., n r ) of elements of order in C n1 C nr. Let n := lcmn 1,..., n r ). Obviously, the order of every element of the direct product is a divisor of n. Let µ be the Möbius function. Theorem 2. For every n 1,..., n r 1 and every n, o n 1,..., n r ) = e gcde, n 1 ) gcde, n r )µ/e) 6) = φd 1 ) φd r ). 7) lcmd 1,...,d r)= Let c n 1,..., n r ) denote the number of cyclic subgroups of order n) of the group C n1 C nr. Since a cyclic subgroup of order has φ) generators, c n 1,..., n r ) = o n 1,..., n r ). 8) φ) Now 4) follows immediately from 7) and 8) by cn 1,..., n r ) = n c n 1,..., n r ).

3 On the number of cyclic subgroups 425 Corollary 2. For every prime p and every a 1,..., a r 1, 1 ν maxa 1,..., a r ), 1 c p ν p a1,..., p ar ) = p ν 1 p minν,a1)+...+minν,ar) p minν 1,a1)+...+minν 1,ar)). 9) p 1) Here 9) follows at once by 6). In case of a p-group of type λ 1,..., λ k ) and choosing r = k, a 1 = λ 1,..., a k = λ k, 9) reduces to 3). The proof of Theorem 2 is contained in Section 2. Additional remarks are given in Section 3. We point out, among others, that the multiplicative properties 1) and 2) are direct consequences of the formulae included in Theorems 1 and 2. 2 Proof of Theorem 2 Let ox) denote the order of a group element x. Let G = C n1 C nr in short. Let x i C ni such that ox i ) = n i 1 i r). Then G = {x = x i1 1,..., xir r ) : 1 i 1 n 1,..., 1 i r n r }. Using elementary properties of the order of elements in a group and of the gcd and lcm of integers, respectively we deduce that for every x = x i1 1,..., xir r ) G, ) ox ox) = lcmox i1 1 ) 1 ),..., oxir r )) = lcm gcdox 1 ), i 1 ),..., ox r ) gcdox r ), i r ) ) ) n 1 = lcm gcdn 1, i 1 ),..., n r n = lcm gcdn r, i r ) gcdn, i 1 n/n 1 ),..., n gcdn, i r n/n r ) n = gcdi 1 n/n 1,..., i r n/n r, n). Assume that ox) =, where n is fixed. Then gcdi 1 n/n 1,..., i r n/n r, n) = n/. Write i 1 n/n 1 = j 1 n/,..., i r n/n r = j r n/. Then gcdj 1,..., j r, ) = 1 and j 1 = i 1 /n 1,..., j r = i r /n r are integers, that is i 1 0 mod n 1 ),..., i r 0 mod n r ). We obtain, as solutions of these linear congruences, that i 1 = k 1 n 1 / gcd, n 1 ) with 1 k 1 gcd, n 1 ),..., i r = k r n r / gcd, n r ) with 1 k r gcd, n r ). Therefore, the number of elements of order in G is the number of ordered r-tuples k 1,..., k r ) satisfying the conditions i) 1 k 1 gcd, n 1 ),..., 1 k r gcd, ) n r ) and ii) gcd k 1 gcd,n,..., k 1) r gcd,n, r) = 1. Using the familiar formula d n µd) = 1 for n = 1 and 0 otherwise, this can be written as o n 1,..., n r ) = µe) 1. e k 1 gcd,n 1) gcd,n 1) Here the linear congruence k 1 e), and mod gcd, n 1 )) it has exactly gcd e, gcd, n 1 ) e gcd, n 1 ) e k 1 gcd,n 1 ) 0 mod e) has gcde, ) k r gcd,n r) e k r gcd,nr ) gcd,n 1) ) solutions in k 1 mod = gcd gcd, n 1 ), ) ) = gcd e e, n 1

4 426 László Tóth solutions similar for the indexes 2,..., r). Hence, o n 1,..., n r ) = e µe) gcd/e, n 1 ) gcd/e, n r ) = e µ/e) gcde, n 1 ) gcde, n r ), which is formula 6). Now using the identity d n φd) = n, o n 1,..., n r ) = e = µ/e) d 1 gcde,n 1) φd 1 ) φd r ) φd 1 ) ab lcmd 1,...,d r)= d r gcde,n r) µa), φd r ) where the inner sum is 0 unless lcmd 1,..., d r ) =, and in this case it is 1. This gives 7). The proof of Theorem 2 is complete. 3 Remarks The function cn 1,..., n r ) is representing a multiplicative function of r variables, i.e., cn 1 m 1,..., n r m r ) = cn 1,..., n r )cm 1,..., m r ) 10) holds for any n 1,..., n r, m 1,..., m r 1 such that gcdn 1 n r, m 1 m r ) = 1. Therefore, cn 1,..., n r ) = p cp epn1),..., p epnr) ), where n i = p pepni) is the prime power factorization of n i, the product being over the primes p and all but a finite number of the exponents e p n i ) are zero 1 i r). The property 10) follows by the next simple number-theoretic argument: According to 4), φn1) φnr) φlcmn 1,...,n r)) with cn 1,..., n r ) is the r variables convolution of the function n 1,..., n r ) the constant 1 function, both multiplicative. Since convolution preserves the multiplicativity we deduce that cn 1,..., n r ) is multiplicative, too. See [12, Section 2] for details. Also, let n 1,..., n r, m 1,..., m r 1 such that gcdn 1 n r, m 1 m r ) = 1. Let n := lcmn 1,..., n r ), m := lcmm 1,..., m r ). For nm write = ab with a n, b m which are unique). Then, c n 1 m 1,..., n r m r ) = c a n 1,..., n r )c b m 1,..., m r ), 11) which follows from 6) by a short computation. Now, 10) and 11) show the validity of 2) and 1) quoted in the Introduction. Let An 1,..., n r ) denote the arithmetic mean of the orders of elements in C n1 C nr. From 7) we deduce

5 On the number of cyclic subgroups 427 Corollary 3. For every n 1,..., n r 1, 1 An 1,..., n r ) = φd 1 ) φd r ) lcmd 1,..., d r ). n 1 n r The function n 1,..., n r ) An 1,..., n r ) is also multiplicative. properties of the average of orders of elements in finite Abelian groups. Consider the special case n 1 =... = n r = n. According to 6), See [1, 2] for further o n,..., n) = e e r µ/r) = φ r ) 12) is the Jordan function of order r. Note that φ 1 ) = φ) is Euler s function. This means that the number of elements of order n) in the direct product C n C n, with r factors, is φ r ) = r p 1 1/pr ). This is a less known group-theoretic interpretation of the Jordan function. See G. A. Miller [4, 5], J. Schulte [8, Th. 7]. Hence the number of cyclic subgroups of order n) of the group C n C n, with r factors, is φ r )/φ). Note that the function k φ r k)/k r 2 φk)) = k p k 1+1/p+1/p /p r 1 ) r 2) was considered quite recently in [9], in connection with Robin s inequality and the Riemann hypothesis. As a byproduct we obtain from 7) and 12) that for every positive integer, φd 1 ) φd r ) = φ r ), 13) lcmd 1,...,d r)= where the sum is over all ordered r-tuples d 1,..., d r ) such that lcmd 1,..., d r ) =. The function in the left hand side of 13) is the von Sterneck function, cf. [3, p. 14]. A final remark: The number sn 1, n 2 ) of all subgroups of the group C n1 C n2 is given by sn 1, n 2 ) = gcdd 1, d 2 ) n 1, n 2 1). 14) d 1 n 1,d 2 n 2 Compare 14) to 5). A proof of 14) will be presented elsewhere. References [1] J. von zur Gathen, A. Knopfmacher, F. Luca, L. G. Lucht, I. E. Shparlinski, Average order in cyclic groups, J. Théor. Nombres Bordeaux, ), [2] F. Luca, I. E. Shparlinski, Average multiplicative orders of elements modulo n, Acta Arith., ), [3] P. J. McCarthy, Introduction to Arithmetical Functions, Springer, [4] G. A. Miller, On the totitives of different orders, Amer. Math. Monthly, ),

6 428 László Tóth [5] G. A. Miller, Some relations between number theory and group theory, American J. Math., ), [6] G. A. Miller, Number of the subgroups of any given Abelian group, Proc. Nat. Acad. Sci. U.S.A., ), [7] R. Schmidt, Subgroup Lattices of Groups, de Gruyter Expositions in Mathematics 14, de Gruyter, Berlin, [8] J. Schulte, Über die Jordansche Verallgemeinerung der Eulerschen Funktion, Results Math., ), [9] P. Solé, M. Planat, The Robin inequality for 7-free integers, Integers, ), Article A65, 8 pp. [10] M. Suzuki, On the lattice of subgroups of finite groups, Trans. Amer. Math. Soc., ), [11] M. Tărnăuceanu, An arithmetic method of counting the subgroups of a finite abelian group, Bull. Math. Soc. Sci. Math. Roumanie N.S.), 53101) 2010), [12] L. Tóth, Menon s identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, ), [13] Y. Yeh, On prime power Abelian groups, Bull. Amer. Math. Soc., ), Received: , Revised: , Accepted: Institute of Mathematics, Department of Integrative Biology, Universität für Bodenkultur, Gregor Mendel-Straße 33, A-1180 Wien, Austria and Department of Mathematics, University of Pécs, Ifjúság u. 6, H-7624 Pécs, Hungary ltoth@gamma.ttk.pte.hu