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2-Adic Properties for the Numbers of Involutions in the Alternating Groups

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2-Adic Properties for the Numbers of Involutions in the Alternating Groups 2-adic properties for the numbers of involutions in the alternating groups 著者 KODA Tatsuhiko, SATO Masaki, TAKEGAHARA Yugen journal or Journal of Algebra and Its Applications publication title volume 14 number 4 page range 1550052 year 2015-05 URL http://hdl.handle.net/10258/00008883 doi: info:doi/10.1142/S0219498815500528 2-adic properties for the numbers of involutions in the alternating groups 著者 KODA Tatsuhiko, SATO Masaki, TAKEGAHARA Yugen journal or Journal of Algebra and Its Applications publication title volume 14 number 4 page range 1550052 year 2015-05 URL http://hdl.handle.net/10258/00008883 doi: info:doi/10.1142/S0219498815500528 2-adic properties for the numbers of involutions in the alternating groups. Tatsuhiko Koda, Masaki Sato, and Yugen Takegahara∗ Muroran Institute of Technology, 27-1 Mizumoto, Muroran 050-8585, Japan September 23, 2014 Abstract We study the 2-adic properties for the numbers of involutions in the alter- native groups, and give an affirmative answer to a conjecture of D. Kim and J. S. Kim [14]. Some analogous and general results are also presented. 1 Introduction Let Sn be the symmetric group of degree n, and let An be the alternating group of degree n. Let ϵ be the identity of a group. Given a positive integer m, we denote m by an(m) the number of permutations σ 2 Sn such that σ = ϵ. Let p be a prime. By definition and Wilson's theorem, ap(p) = 1 + (p − 1)! ≡ 0 (mod p). Moreover, an(m) ≡ 0 (mod gcd(m; n!)) by a theorem of Frobenius (see, e.g., [10]). u Let u be a positive integer. There exist remarkable p-adic properties of an(p ) (cf. Theorems 4.2{4.4). The beginning of them is due to H. Ochiai [16] and K. Conrad [4]. For each integer a, ordp(a) denotes the exponent of p in the decomposition of a into prime factors. As a pioneer work, the formula [ ] [ ] n n ord (a (p)) ≥ − p n p p2 (cf. Corollary 4.5) was given in [6, 7, 9], which was also shown by various methods (cf. [4, 11, 13, 14]); moreover, the equality holds for all n such that n − [n=p2]p2 ≤ p − 1 (see, e.g., [6, 11, 13]). When p = 2, this formula was found by S. Chowla, I. N. ∗This work was supported by JSPS KAKENHI Grant number 22540004 ∗E-mail: [email protected] 2000 Mathematics Subject Classification. Primary 05A15; Secondary 11S80, 20B30, 20E22. Keyword. symmetric group, alternating group, Artin-Hasse exponential, p-adic analytic function, wreath product. 1 2-adic properties for the numbers of involutions 2 Herstein, and W. K. Moore [2]. The precise formula for ord2(an(2)) is known as 8 [ ] [ ] > n n <> − + 1 if n ≡ 3 (mod 4); [ 2 ] [ 4 ] ord2(an(2)) = > n n : − otherwise 2 4 (cf. Example 4.6). The value of ord2(an(4)) is also determined (cf. Proposition 4.7). m We denote by tn(m) the number of even permutations σ 2 An such that σ = ϵ. Recently, D. Kim and J. S. Kim [14] proved that for any nonnegative integer y, ord2(t4y(2)) = y + χo(y); ord2(t4y+2(2)) = ord2(t4y+3(2)) = y; where χo(y) = 1 if y is odd, and χo(y) = 0 if y is even. They also conjectured that for any nonnegative integer y, there exists a 2-adic integer α satisfying ord2(t4y+1(2)) = y + χo(y) · (ord2(y + α) + 1) (see [14, Conjecture 5.6]). According to [14], α = 1 + 2 + 23 + 28 + 210 + ··· satisfies the condition for all y ≤ 1000. In this paper, we solve affirmatively their conjecture (cf. Theorem 5.1), and present some analogous and general results, including the result for ord2(tn(4)) (cf. Theorems 5.4). We adapt K. Conrad's methods presented u in [4] to the case of tn(2 ). u u Sections 2{5 are devoted to the study of ordp(an(p )) and ord2(tn(2 )). In addition to the above results, we also show that, if r = 0 or r = 1, then there exists a 2-adic integer αr such that u u+1 ord2(t2u+1y+r(2 )) = (2 − u − 2)y + χo(y) · (ord2(y + αr) + u) for any nonnegative integer y (cf. Theorem 5.6). Let Cp o Sn be the wreath product of Cp by Sn, where Cp is a cyclic group of order p, and let C2 o An be the wreath product of C2 by An. We are also interested m u in the number of elements x of these wreath products such that x = ϵ. Let bn(p ) pu u be the number of elements x of Cp o Sn such that x = ϵ, and let qn(2 ) be the 2u number of elements x of C2 o An such that x = ϵ. In Sections 6{8, we focus on the u u p-adic properties of bn(p ) and the 2-adic properties of qn(2 ). When u = 1, we are successful in finding the fact that [ ] [ ] ([ ]) n n + 1 n ord (b (p)) = n − and ord (q (2)) = + χ p n p 2 n 2 o 2 (cf. Examples 7.4 and 8.2). The former fact with p = 2 is due to T. Yoshida [20]. u u The results for ordp(bn(p )) and ord2(qn(2 )) with u ≥ 2 are similar to those for u−1 u−1 ordp(an(p )) and ord2(tn(2 )), while there are slight differences between the proofs (cf. Example 7.5, Proposition 7.6, Theorems 8.3, 8.5, and 8.7). 2-adic properties for the numbers of involutions 3 2 Generating functions pu For each σ 2 Sn, σ = ϵ if and only if the cycle type of σ is of the form (1j0 ; pj1 ;:::; (pu)ju ); P k where j0; j1; : : : ; ju are nonnegativeQ integers satisfying k jkp = n. Since the u kjk number of such a permutations is n!= k=0 p jk! (see, e.g., [12, Lemma 1.2.15] or [18, Chap. 4 x2]), it follows that X u n! an(p ) = Q : (1) u kjk u k=0 p jk! j0+j1p+···+jup =n 0 u u Set an(p ) = an(p ), and define X ··· (−1)j0+j1+ +ju n! a1 (pu) = Q : (2) n u kjk u k=0 p jk! j0+j1p+···+jup =n Then we have a0 (pu) + (−1)na1 (pu) t (pu) = n n : (3) n 2 u u (Obviously, an(p ) = tn(p ) if p =6 2.) Let \ denotes both 0 and 1. We always \ u assume that a0(p ) = 1. By Eqs. (1){(3), we have ! X1 \ u Xu a (p ) 1 k n Xn = exp (−1)\ Xp (4) n! pk n=0 k=0 and ! ! X1 u Xu Xu t (2 ) 1 1 k 1 1 k n Xn = exp X2 + exp X − X2 n! 2 2k 2 2k n=0 k=0 k=1 f \ g1 (see also [3] and [18, Chap. 4, Problem 22]). Let cn n=0 be a sequence given by ! X1 X1 1 k c\ Xn = exp (−1)\ Xp : (5) n pk n=0 k=0 \ Then by [5, Proposition 1] (see also [15, p. 97, Exercise 18]), cn 2 Zp \ Q, where Zp is the ring of p-adic integers. When \ = 0, this formal power series is called the x x 0 Artin-Hasse exponential (cf. [5], [15, Chap. IV 2], [19, 48]). We write cn = cn for u the sake of simplicity. By definition, cr = ar(p )=r! for any nonnegative integer r less than pu+1. According to Mathematica, we have the following lemma. 2-adic properties for the numbers of involutions 4 \ Lemma 2.1 If p = 2, then the values of cr for integers r with 0 ≤ r ≤ 17 are as follows : r 0 1 2 3 4 5 6 7 8 9 10 11 0 2 2 7 16 67 88 617 2626 18176 cr 1 1 1 3 3 15 45 315 315 2835 14175 155925 1 − 1 − 1 1 1 − 5 − 8 43 − 74 − 559 cr 1 1 0 3 3 5 45 63 105 405 14175 17325 r 12 13 14 15 16 17 0 6949 423271 2172172 19151162 58438907 899510224 cr 66825 6081075 42567525 638512875 638512875 10854718875 1 697 − 13232 − 30727 450991 − 5519014 8250311 cr 18711 552825 14189175 49116375 91216125 144729585 For any nonnegative integer r less than pu+1, we set X1 \ u a u+1 (p ) H\ (X) = p y+r (−(−1)\pu+1)yXy; u;r (pu+1y + r)! y=0 f \ g1 and define a sequence dn;r n=0 by 0 1 ! X1 X1 X1 \ pi(u+1) " p i d\ Xn = @ c\ (−(−1)\pu+1)jXjA exp Xp ; n;r pu+1j+r pu+i+1 n=0 j=0 i=1 where "\ = −1 if p = 2 and \ = 0, and "\ = +1 otherwise. Lemma 2.2 Let r be a nonnegative integer less than pu+1. Then X1 \ \ n Hu;r(X) = exp(X) dn;rX : n=0 Proof. Using Eqs. (4) and (5), we have ! ! X1 \ u X1 X1 a (p ) 1 k n Xn = c\ Xn exp −(−1)\ Xp : n! n pk n=0 n=0 k=u+1 This formula yields 0 1 X1 \ u X1 a u+1 (p ) u+1 u+1 p y+r Xp y+r = @ c\ Xp j+rA (pu+1y + r)! pu+1j+r y=0 j=0 ! X1 1 u+i+1 × exp −(−1)\ Xp : pu+i+1 i=0 2-adic properties for the numbers of involutions 5 Omit Xr and substitute (−(−1)\pu+1)X for Xpu+1 .
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