FACTORIZATION NUMBERS OF FINITE ABELIAN GROUPS. Communicated by Bernhard Amberg. 1. Introduction

Transcription

1 Iteratoal Joural of Grou Theory ISSN (rt): , ISSN (o-le): Vol 2 No 2 (2013), 1-8 c 2013 Uversty of Isfaha wwwtheoryofgrousr wwwuacr FACTORIZATION NUMBERS OF FINITE ABELIAN GROUPS M FARROKHI D G Commucated by Berhard Amberg Abstract The umber of factorzatos of a fte abela grou as the roduct of two subgrous s comuted two dfferet ways ad a combatoral detty volvg Gaussa bomal coeffcets s reseted 1 Itroducto A grou G s factorzed f G = AB for some subgrous A ad B of G ad such a exresso s called a factorzato of G The factorzato of grous has a very log hstory the theory of fte ad fte grous such a way that how the structure of subgrous the factorzato flueces the structure of the whole grou Also, t s mortat to kow what grous have a otrval factorzato by roer subgrous ad to determe all factorzatos of a gve fte or fte grou (see [13] for detals o factorzatos of fte smle grous) Coutg the umber of factorzatos of grous wth a fte umber of factorzatos s of some terestg combatoral flavor, for f we are able to comute the factorzato umber of a grou two dfferet ways, the we may obta combatoral dettes, whch s of deedet terest The umber of factorzatos of a grou, the factorzato umber, also ca be aled to comute the subgrou ermutablty degree of fte grous recetly defed by Tǎrǎuceau [22] If F 2 (G) deotes the factorzato umber of a grou G, the the subgrou ermutablty degree of G s sd(g) = 1 L(G) 2 F 2 (H), H G MSC(2010): Prmary: 20D40; Secodary: 20K01, 20K27, 20K30 Keywords: Factorzato umber, Abela grou, subgrou, Gaussa bomal coeffcet Receved: 25 February 2012, Acceted: 11 August

2 2 It J Grou Theory 2 o 2 (2013) 1-8 M Farrokh D G where L(G) s the lattce of all subgrous of G Recetly, the author ad Saeed [17, 18] comuted the factorzato umbers of subgrous of P SL(2, ) ad used them to obta the subgrou ermutablty degree of P SL(2, ) The am of ths aer s to obta the factorzato umbers of fte abela grous two dfferet ways, from whch we get also a detty volvg Gaussa bomal coeffcets Sce F 2 (H K) = F 2 (H)F 2 (K) for fte grous H ad K of corme orders, the roblem of comutg the factorzato umber of fte abela grous reduces to just fte abela -grous Hece, throughout ths aer we choose a fxed rme ad all grous wll be fte abela -grous I what follows, we set the followg otatos for a gve o-creasg seuece of atural umbers A = (a 1,, a ): () A = (a 1,, a ), for each = 1,, ; () G (A) = Z a 1 Z a s a abela -grou of tye A; () S(A) = {(b 1,, b m ) : m, b m b 1 ad b a, for = 1,, m}, ad (v) f B = (b 1,, b m ) S(A), the T (A : B) s the set of all m-tules (x 1,, x m ) of elemets of G (A) such that x 1,, x m = G (B) ad x = b, for each = 1,, m For a -grou G, the subgrou geerated by all elemets of order at most s deoted by Ω (G) ad the subgrou geerated by all th ower of elemets of G s deoted by G I other words, Ω (G) = x G : x = 1 ad G = x : x G Also, f G s a grou, the d(g) stad for the mmum umber of geerators of G 2 Ma Results To beg comutg the factorzato umber of a fte abela -grou, we frst eed to obta some rcal lemmas about secal subsets of the grou Lemma 21 If A = (a 1,, a ) s a o-creasg seuece of atural umbers ad B = (b 1,, b m ) S(A), the T (A : B) = ( ) µ b (A) µ b 1(A)+µ b (B 1 ) µ, b 1(B 1 ) =1 where µ (C) = max{j : c j } + c max{j:cj } c k for every C = (c 1,, c k ) S(A) Proof Let G = G (A) ad H = G (B) If x 1,, x m G such that x = b ad x 1,, x m = H, the we may choose x 1 to be ay elemet of Ω b1 (G) \ Ω b1 1(G) ad ductvely x to be ay elemet of Ω b (G) \ Ω b 1(G) x 1,, x 1, for = 2,, m Hece the umber of such m-tules (x 1,, x m ) s O the other had, Ω b (G) \ Ω b 1(G) x 1,, x 1 =1 Ω b (G) \ Ω b 1(G) x 1,, x 1 = Ω b (G) \ Ω b 1(G)Ω b ( x 1,, x 1 )

3 It J Grou Theory 2 o 2 (2013) 1-8 M Farrokh D G 3 ad Ω b 1(G)Ω b ( x 1,, x 1 ) = Ω b 1(G) Ω b ( x 1,, x 1 ) Ω b 1( x 1,, x 1 ) so that the umber of m-tules (x 1,, x m ) s =1 ( Ω b (G) Ω b 1(G) Ω b ( x 1,, x 1 ) Ω b 1( x 1,, x 1 ) Hece the roblem reduces to comutg the order of Ω t (K) for a gve abela -grou K of tye C = (c 1,, c k ) ad a ostve teger t If α k (C) = max{ : c k}, the Ω t (K) s of tye ( t,, t, c α t (C)+1,, c k) ad coseuetly Ω t (K) = tαt(c)+c α t (C)+1+ +c k = µt(c) Now, by assumto x 1,, x 1 = b 1+ +b 1 ad the umber of m-tules (x 1,, x m ) s ( ) µ b (A) µ b 1(A)+µ b (B 1 ) µ, b 1(B 1 ) as reured =1 The above lemma has a terestg alcato the case where A = B Corollary 22 Let G be a fte abela -grou of tye A The () Aut(G) = T (A : A), ad () f A = (a 1,, a 1,, a m,, a m ) = (b 1,, b ), where the umber of a s k ad a > a +1, the Aut(G) = N =1 j=n 1 +1 ) ( b jn +b N +1+ +b m (b j 1)N +b N +1+ +b m+j 1 ) ad artcular, the sze of the Sylow -subgrou of Aut(G) s Aut(G) = N =1 j=n 1 +1 (b j 1)N +b N +1+ +b m+j 1, where = + 1 Sg(a +1 a + 1) ad N = k k, for each = 1,, m Proof () Let G = x 1,, x ad x = a, for each = 1,, The, the result follows from the fact that the ma Aut(G) T (A : A), whch seds a automorhsm ϕ Aut(G) to (ϕ(x 1 ),, ϕ(x )) s a bjecto () The result follows by comutg the values of µ bj (A), µ bj 1(A), µ bj (A j 1 ) ad µ bj 1(A j 1 ), for each 1 j To ed ths, let N 1 < j N Now, a easy observato shows that µ bj (A) = b j N + b N b, µ bj 1(A) = (b j 1)N + b N b, µ bj (A j 1 ) = (j 1)b j, µ bj 1(A j 1 ) = (j 1)(b j 1), where = + 1 Sg(a +1 a + 1), as reured

4 4 It J Grou Theory 2 o 2 (2013) 1-8 M Farrokh D G Note that, the order of automorhsm grou of a fte abela -grou s also obtaed alteratvely by several authors ad we may refer the reader to [3, 7, 8, 9, 19, 20] for more detals There are varous aers, whch volved wth the comutato of certa subgrous of a gve fte abela -grou Let G be a fte abela -grou Mller [14, 15, 16] gves some artal results o the the umber of certa subgrous of G, say cyclc subgrous etc, Stehlg [21] gves a recursve formula for the umber of subgrou of a gve order, ad Delsarte [6], Kosta [11] ad Yeh [23] gve dfferet formulas for the umber of subgrous of a gve tye Also, Daves [5] comutes the umber of subgrous of secal tye wth a gve secal factor grou, where the grou G has also a secal tye Utlzg Lemma 21, we obta a alteratve formula for the umber of subgrous of a gve tye a arbtrary fte abela -grou We ote that, our roof of the formula s both shorter ad smler tha Delsarte s, Kosta s ad Yeh s methods Lemma 23 The umber of subgrous of tye B of a fte abela -grou of tye A s A = T (A : B) B T (B : B) Proof The result s obvous by the deftos Corollary 24 The umber of subgrous of a fte abela -grou G of tye A s L(G) = A B where L(G) s the set of all subgrous of G B S(A) Now, we are able to obta our frst formula for the factorzato umber of a fte abela -grou Theorem 25 If G s a fte abela -grou of tye A, the A F 2 (G) = L(G) 2 F 2 (G (B)) B B S(A)\{A}, Proof Sce AB G for all subgrous A ad B of G, we have L(G) 2 = 1 = 1 = F 2 (H) H G A,B G H=AB A,B H G Now sce S(A) s the set of all tyes of subgrous of G, we get L(G) 2 = A F 2 (G (B)) = F 2 (G) + B B S(A) from whch the result follows B S(A)\{A} A F 2 (G (B)), B It s worth otg that, we may obta a recursve formula for the umber of factorzatos of a fte abela -grou to k subgrous the same way as the roof of Theorem 25 To get the ext formula for the factorzato umber of a gve fte abela -grou G, we use the fact that f G = CD, the G = C D I fact, we take a ordered ar of subgrous (A, B) of G such

5 It J Grou Theory 2 o 2 (2013) 1-8 M Farrokh D G 5 that G = AB ad we shall cout the umber of ordered ars of subgrous (C, D) of G wth G = CD ad (C, D ) = (A, B) I what follows, we set the followg otatos For ay real umber > 0 ad teger, the umbers [] ad []! deote the -umber ad -factoral defed by [] = 1 1 ad []! = [] [ 1] [2] [1], resectvely Moreover, the Gaussa bomal coeffcets are defed terms of -factorals by []! = []![ ]! = ( 1) ( 1) ( 1) ( 1)( 1) ( 1) ad as usual the Gaussa olyomal coeffcets are defed by []! = 1,, k [ 1 ]! [ k ]![ 1 k ]!, where k Gve a rme ower, the Gaussa bomal coeffcet s the umber of subsaces of dmeso a vector sace of dmeso over the feld of order We refer the terested reader to [1, 2, 4, 10, 12] for more detals o -umbers ad related tocs As we have see before, Lemma 23 geeralzes the Gaussa bomal coeffcets as the umber of subgrous of a gve tye of a fte abela -grou We beg wth two rcal lemmas Lemma 26 Let G be a elemetary abela -grou ad X G The, the umber of subgrous Y of G of order ( d(g) d(x)) such that X Y = 1 s d(g) d(x) d(x) Proof To cout the umber of -tules (y 1,, y ) of elemets of G such that y 1,, y s a subgrou of order tersectg trvally wth X, we may choose y 1 G \ X ad ductvely y G \ X, y 1,, y 1, for every = 2,, O the other had, to cout the umber of -tules (z 1,, z ) geeratg a gve subgrou Y = y 1,, y of order, we may choose z 1 Y \ {1} ad ductvely z Y \ z 1,, z 1, for every = 2,, Hece, the umber of subgrous Y s as reured 1 =0 d(g) d(x)+ 1 d(x)+ = =0 d(g) d(x) 1 1 ] = d(x) [ d(g) d(x), Lemma 27 Let G be a elemetary abela -grou ad X Y G The, the umber of subgrous Z of G of order d(g) d(y )+ ( d(y ) d(x)) such that X Z = 1 ad Y Z = G s d(y ) d(x) d(x)+(d(y ) )(d(g) d(y ))

6 6 It J Grou Theory 2 o 2 (2013) 1-8 M Farrokh D G Proof Let N X be a subgrou of order If Z G such that Y Z = G, the Y Z = ad the umber of such Z euals to the roduct of the umber of subgrous Z 1 of Y such that Z 1 = ad X Z 1 = 1 by the umber of subgrous Z 2 /N of G/N such that X/N Z 2 /N = G/N ad X/N Z 2 /N = 1 G/N By Lemma 26, the frst ad secod umbers are d(y ) d(x) d(x) ad d(g) d(y ) (d(g) d(y ))(d(y ) ) = (d(y ) )(d(g) d(y )), d(g) d(y ) resectvely Therefore, the umber of subgrous Z s d(y ) d(x) d(x)+(d(y ) )(d(g) d(y )) ad the roof s comlete Utlzg the above lemmas, we ca obta our secod formula for the factorzato umber of a fte abela -grou Theorem 28 Let G be a fte abela -grou The F 2 (G) = Ω 1(G ) 2+d(A)+d(B) Ω 1 (A) d(a) Ω 1 (B) d(b) G =AB where = d(ω 1 (G)) d(ω 1 (G )) 0 +j d(a)+jd(b) Ω 1 (G j, ) +j, j Proof Frst we ote that f X G, the X = Y for some subgrou Y of G I artcular, f X = x 1 x m, the Y = y 1 y m U, where U s a elemetary abela -subgrou such that U G = 1 ad y = x, for = 1,, m Now, f z s ay elemet wth z = x, the (z y 1 ) = 1 that s z y 1 Ω 1 (G) Let Z = z 1 z m V be aother subgrou of G wth Z = X The, we may assume that z = w y for some w Ω 1 (G) Suose that Y = Z The U = V ad Ω 1 (X)U = Ω 1 (X)V Sce y 1 Z, there exst tegers α 1,, α m ad elemet v V such that y 1 = (w 1 y 1 ) α 1 (w m y m ) αm v Thus x 1 = x α 1 1 xαm m, whch mles that α 1 1 (mod x 1 ) that s α 1 = 1+ x 1 t 1 for some teger t 1 Hece w1 1 = y x 1 t 1 1 (w 2 y 2 ) α 2 (w m y m ) αm v ad coseuetly x α, for each = 2,, m, say α = x t Therefore w 1 = y x 1 t 1 1 y m xm tm v Ω 1 (Y ) ad smlarly w Ω 1 (Y ), for each = 2,, m Coversely, t ca be easly see that f Ω 1 (X)U = Ω 1 (X)V ad w Ω 1 (Y ), the the subgrous Y ad Z are eual Now let G = AB be a factorzato of G ad = α(g) β(g), where α(g) = d(ω 1 (G)) ad β(g) = d(ω 1 (G )) The, by Lemmas 26 ad 27, the umber of factorzatos G = CD wth (C, D ) = (A, B) such that [Ω 1 (C) : Ω 1 (A)] = ad [Ω 1 (D) : Ω 1 (B)] = j (j = + k) s Ω 1 (G) d(a) Ω 1 (C) d(a) Ω 1(G) d(b) β(g) kβ(g)+( )(β(g)+ k), Ω 1 (D) d(b) k

7 It J Grou Theory 2 o 2 (2013) 1-8 M Farrokh D G 7 whch smlfes to (21) Ω 1 (G) d(a)+d(b) Ω 1 (G ) +j Ω 1 (A) d(a) Ω 1 (B) d(b) ( )( j) d(a)+jd(b) j Now, by uttg = ad j = j (21) ad much more smlfcato, t yelds Ω 1 (G ) 2+d(A)+d(B) d(a)+j d(b) Ω 1 (A) d(a) Ω 1 (B) d(b) Ω 1 (G j ) +j, j factorzatos G = CD wth the gve codtos, from whch the result follows Corollary 29 If (V, F ) s a vector sace of dmeto over a fte feld F of order, the the umber of factorzatos of V as the sum of two subsaces s F 2 (V ) = j, j 0 +j Proof If = s a rme, the V ca be detfed wth the elemetary abela -grou of order, ad the result follows by Theorem 28 Now f s ay rme ower, the by substtutg grous by vector saces ad subgrous by subsaces Lemmas 26 ad 27, oe ca rerove Theorem 28 for vector saces, from whch the result follows Oce we combe Theorems 25 ad 28 the secal case of elemetary abela -grous, we wll obta the followg combatoral detty for Gaussa bomal coeffcets, whch s of deedet terest Note that, the Gaussa bomal coeffcets occur the theory of arttos ad coutg of symmetrc olyomals Corollary 210 For each atural umber ad rme ower, we have ( ) 2 = j, j, k =0 0 +j+k Proof Sce the both sdes of the gve detty are olyomals, t s eough to show that the result holds whe s a rme We kow from the roof of Theorem 25 that for a elemetary abela -grou of order (22) ( =0 ) 2 = =0 0 j+k m F 2 (Z ) Also, by Corollary 29, for a elemetary abela -grou of order m, we have F 2 (Z m ) = m jk j, k Hece =0 F 2 (Z ) = jk =0 0 j+k j, k = jk, j, k 0 +j+k,

8 8 It J Grou Theory 2 o 2 (2013) 1-8 M Farrokh D G from whch the result follows Ackowledgmets The author would lke to thak the referee for some helful correctos Refereces [1] G E Adrews ad K Erksso, Iteger Parttos, Cambrdge Uversty Press, Cambrdge, 2004 [2] W N Baley, A ote o certa -dettes, Quart J Math, 12 (1941) [3] J N S Bdwell ad M J Curra, Automorhsms of fte Abela grous, Math Proc R Ir Acad, 110A o 1 (2010) [4] H Coh, Projectve geometry over F 1 ad the Gaussa bomal coeffcets, Amer Math Mothly, 111 (2004) [5] I J Daves, Eumerato of certa subgrous of abela -grous, Proc Edb Math Soc (2), 13 (1962) 1 4 [6] S Delsarte, Foctos de Mobus sur les groues Abeles fs, A of Math (2), 49 (1948) [7] M Golasńsk ad D L Goçalves, O automorhsms of fte Abela -grous, Math Slovaca, 58 o 4 (2008) [8] C J Hllar ad D L Rhea, Automorhsms of fte abela grous, Amer Math Mothly, 11 (2007) [9] N J S Hughes, The structure ad order of the grou of cetral automorhsms of a fte grou, Proc Lodo Math Soc, 52 o 2 (1951) [10] F H Jackso, Certa -dettes, Quart J Math, 12 (1941) [11] Y Kosta, O a eumerato of certa subgrous of a -grou, J Osaka Ist Sc Tech, Part I, 1 (1949) [12] J Kovala, Geeralzed bomal coeffcets ad the subset-subsace roblem, Adv Al Math, 21 (1998) [13] M W Lebeck, C E Praeger ad J Saxl, The maxmal factorzatos of the fte smle grous ad ther automorhsm grous, Mem Amer Math Soc, 86 o 432 (1990) v+151 [14] G A Mller, Form of the umber of the rme ower subgrous of a abela grou, Proc Natl Acad Sc USA, 12 (1926) [15] G A Mller, Number of the subgrous of ay gve abela grou, Proc Natl Acad Sc USA, 25 (1939) [16] G A Mller, Ideedet geerators of the subgrous of a abela grou, Proc Natl Acad Sc USA, 25 (1939) [17] F Saeed ad M Farrokh D G, Factorzato umbers of some fte grous, Glasgow J Math, 54 (2012) [18] F Saeed ad M Farrokh D G, Subgrou ermutablty degree of P SL(2, ), to aear Glasgow J Math [19] P R Saders, The cetral automorhsms of a fte grou, J Lodo Math Soc, 44 (1969) [20] K Shoda, Über de Automorhsme eer edlsche Abelsche Grue, Math A, 100 (1928) [21] T Stehlg, O comutg the umber of subgrous of a fte Abela grou, Combatorca, 12 o 4 (1992) [22] M Tǎrǎuceau, Subgrou commutatvty degrees of fte grous, J Algebra, 321 o 9 (2009) [23] Y Yeh, O rme ower abela grous, Bull Amer Math Soc, 54 (1948) Mohammad Farrokh Derakhshadeh Ghoucha Deartmet of Pure Mathematcs, Ferdows Uversty of Mashhad, POBox 1159, 91775, Mashhad, Ira Emal: mfarrokhdg@gmalcom