Fuma, Michitaku

Ninomiya, Yasushi

Okayama University Scientific Achievement Repository

Mathematical Journal of OkayamaUniversityVolume 46, Issue 1 2004 Article 28JANUARY 2004Hasse Principle” for Finite p-Groups withCyclic Subgroups of Index p2Michitaku Fuma∗ Yasushi Ninomiya†∗Shinshu University†Shinshu UniversityCopyright c©2004 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouMath. J. Okayama Univ. 46 (2004), 31{38\HASSE PRINCIPLE" FOR FINITE p-GROUPS WITHCYCLIC SUBGROUPS OF INDEX p2Michitaku FUMA and Yasushi NINOMIYA1. IntroductionLet G be a group. A map f : G ¡! G satisfying f(xy) = f(x)f(y)x forevery x; y 2 G, where f(y)x = xf(y)x¡1, is called a cocycle of G. Let f be acocycle of G. If, for every x 2 G, there exists a 2 G such that f(x) = a¡1axthen f is called a local coboundary, and if there exists a 2 G such thatf(x) = a¡1ax for every x 2 G then f is called a (global) coboundary.G is said to enjoy \Hasse principle" if every local coboundary of G is acoboundary. Abelian groups trivially enjoy \Hasse principle". It is knownthat a ¯nite group G enjoys \Hasse principle" if and only if every conjugacypreserving automorphism of G is an inner automorphism ([6], Theorem 3.1).Some types of groups enjoying \Hasse principle" are known ([1], [2], [3],[5], [6], [7], [8], [9]). For ¯nite p-groups, it is known that the following groupsenjoy \Hasse principle".(1) ¯nite p-groups with cyclic subgroups of index p ([1]);(2) extraspecial p-groups ([1]);(3) ¯nite p-groups of order p4 ([2]).Among the known results, the following are useful for our study:Theorem 1 ([2]). Metacyclic groups enjoy \Hasse principle".Theorem 2 ([3]). Let H be a central subgroup of G. If G=H is generated byxH and yH (x; y 2 G) and every element of G=H can be written as xrysH,then G enjoys \Hasse principle".Recently, M. Kumar and L. R. Vermani [3] proved that for an odd primep, every non-abelian ¯nite p-group of order pm having a normal cyclic sub-group of order pm¡2 but having no element of order pm¡1 enjoys \Hasseprinciple". Further they have described that there are fourteen 2-groups(up to isomorphism) of order 2m of the above type and they showed thattwelve of them enjoy \Hasse principle" but remaining two do not enjoy\Hasse principle". In [4], for any prime p, all ¯nite non-abelian p-groupsof order pm having cyclic subgroups of order pm¡2 but having no elementof order pm¡1 are classi¯ed. From the result we see that there is a missinggroup in a description in [3], which is given byha; b j a2m¡2 = 1; b4 = a2m¡3 ; b¡1ab = a¡1i311Fuma and Ninomiya: Hasse Principle" for Finite p-Groups with Cyclic Subgroups of IndProduced by The Berkeley Electronic Press, 200432 M. FUMA AND Y. NINOMIYA(see [4], Remark 3 (1)). This group is metacyclic, and so enjoys \Hasseprinciple". Further, two groups given in [3], Theorem 3.4 are isomorphic(see [4], Remark 3 (2)).In this note we report that every non-abelian p-group of order pm havinga cyclic subgroup of order pm¡2 but having no normal cyclic subgroup oforder pm¡2 and no element of order pm¡1 enjoys \Hasse principle". Fromnow on suppose that G is a non-abelian p-group of this type.(I) For an odd prime p, there are seven possibilities about G. Usingnotation given in [4], we here list these groups:G1 = hx; y; z j xpm¡2 = 1; yp = zp = 1; xy = yx; z¡1xz = xy;yz = zyi (m ¸ 3);G5 = hx; y; z j xpm¡2 = 1; yp = zp = 1; xy = yx; z¡1xz = xy;z¡1yz = xpm¡3yi (m ¸ 4);G6 = hx; y; z j xpm¡2 = 1; yp = zp = 1; xy = yx; z¡1xz = xy;z¡1yz = xrpm¡3yi (m ¸ 4);where r is a quadratic nonresidue mod p.G7 = hx; y; z j xpm¡2 = 1; yp = zp = 1; y¡1xy = x1+pm¡3 ;z¡1xz = xy; yz = zyi (m ¸ 4);G9 = hx; y j xpm¡2 = 1; yp2 = 1; y¡1xy = x1+pi (m ¸ 5);G10 = hx; y j xp2 = 1; xpp¡3 = yp2 ; y¡1xy = x1¡pi (m ¸ 6);G11 = hx; y; z j x9 = 1; y3 = 1; z3 = x3; xy = yx; z¡1xz = xy;z¡1yz = x6yiBy Theorem 2, G1 enjoys \Hasse principle", and because G9 and G10 aremetacyclic by Theorem 1, they also enjoy \Hasse principle".(II) For p = 2, there are twelve possibilities about G. Again, using nota-tion in [4], we list these groups:G5 = hx; y; z j x2m¡2 = 1; y2 = z2 = 1; xy = yx; z¡1xz = xy;yz = zyi (m ¸ 4);G9 = hx; y j x2m¡2 = 1; y4 = 1; x¡1yx = y¡1i (m ¸ 5);G13 = hx; y; z j x2m¡2 = 1; y2 = z2 = 1; xy = yx; z¡1xz = x¡1y;yz = zyi (m ¸ 5);G14 = hx; y; z j x2m¡2 = 1; y2 = 1; z2 = x2m¡3 ; xy = yx;z¡1xz = x¡1y; yz = zyi (m ¸ 5);2Mathematical Journal of Okayama University, Vol. 46 [2004], Iss. 1, Art. 28http://escholarship.lib.okayama-u.ac.jp/mjou/vol46/iss1/28\HASSE PRINCIPLE" FOR FINITE p-GROUPS 33G17 = hx; y; z j x2m¡2 = 1; y2 = z2 = 1; y¡1xy = x1+2m¡3 ;z¡1xz = xy; yz = zyi (m ¸ 5);G18 = hx; y; z j x2m¡2 = 1; y2 = 1; z2 = y; y¡1xy = x1+2m¡3 ;z¡1xz = x¡1yi (m ¸ 5);G21 = hx; y j x2m¡2 = 1; x2m¡3 = y4; x¡1yx = y¡1i (m ¸ 6);G22 = hx; y; z j x2m¡2 = 1; y2 = z2 = 1; xy = yx;z¡1xz = x1+2m¡4y; z¡1yz = x2m¡3yi (m ¸ 6);G23 = hx; y; z j x2m¡2 = 1; y2 = z2 = 1; xy = yx;z¡1xz = x¡1+2m¡4y; z¡1yz = x2m¡3yi (m ¸ 6);G24 = hx; y; z j x2m¡2 = 1; y2 = z2 = 1; y¡1xy = x1+2m¡3 ;z¡1xz = x¡1+2m¡4y; yz = zyi (m ¸ 6);G25 = hx; y; z j x2m¡2 = 1; y2 = 1; z2 = x2m¡3 ; y¡1xy = x1+2m¡3 ;z¡1xz = x¡1+2m¡4y; yz = zyi (m ¸ 6);G26 = hx; y; z j x8 = 1; y2 = 1; z2 = x4; y¡1xy = x5;z¡1xz = xy; yz = zyiBy Theorem 2, G5, G13 and G14 enjoy \Hasse principle". Because G9and G21 are metacyclic, they also enjoy \Hasse principle".In what follows, we denote by AutcG and InnG the set of automorphismswhich preserves each conjugacy class ofG and the inner automorphism groupof G, respectively.2. The case p oddIn [2], it has been shown that if every f 2 AutcG that ¯xes one of thegenerating elements of G is in InnG, then G enjoys \Hasse principle". Let pbe an odd prime. We here show that the groups G5, G6, G7 and G11 givenin (I) enjoy \Hasse principle".G5 and G6 enjoy \Hasse principle".Proof. Let f 2 AutcG5 such that f(z) = z. Then there exist a = xiyjzk,b = xryszt 2 G5 with 0 · i; r < pm¡2, 0 · j; k; s; t < p such that f(x) =a¡1xa; f(y) = b¡1yb, and sof(x) = z¡ky¡jx¡i ¢ x ¢ xiyjzk = z¡kxzk;f(y) = z¡ty¡sx¡r ¢ y ¢ xryszt = z¡tyzt:3Fuma and Ninomiya: Hasse Principle" for Finite p-Groups with Cyclic Subgroups of IndProduced by The Berkeley Electronic Press, 200434 M. FUMA AND Y. NINOMIYAAs z¡1xz = xy and z¡1yz = xpm¡3y we havez¡kxzk = x1+(1+2+¢¢¢+(k¡1))pm¡3yk = x1+k(k¡1)2pm¡3yk:We also have z¡tyzt = xtpm¡3y. Therefore f(x) = x1+k(k¡1)2pm¡3yk, f(y) =xtpm¡3y. Since f is an automorphism,f(z)¡1f(x)f(z) = f(z¡1xz) = f(xy) = f(x)f(y):We havef(z)¡1f(x)f(z) = z¡1(x1+k(k¡1)2pm¡3yk)z= (z¡1xz)1+k(k¡1)2pm¡3(z¡1yz)k= x1+(k+k(k¡1)2)pm¡3y1+k;f(x)f(y) = x1+k(k¡1)2pm¡3ykxtpm¡3y= x1+(t+k(k¡1)2)pm¡3y1+k:Therefore the following congruence holds:1 +³k +k(k ¡ 1)2´pm¡3 ´ 1 +³t+k(k ¡ 1)2´pm¡3 (mod pm¡2):From this it follows that k ´ t (mod p). Then because 0 · k; t < p, wehave k = t. Thus we have f(x) = z¡kxzk, f(y) = z¡kxzk, f(z) = z¡kzzk.This shows that f 2 InnG5, and so G5 enjoys \Hasse principle". By ananalogous argument we can show that G6 enjoys \Hasse principle". ¤In the rest of the paper, we proceed with a similar argument as above.Given f 2 AutcG, the image f(g) of g 2 G will be denoted by g.G7 enjoys \Hasse principle".Proof. Let f 2 AutcG7 such that z = z. Then there exist a = xiyjzk,b = xryszt 2 G7 with 0 · i; r < pm¡2, 0 · j; k; s; t < p such that x = a¡1xa,y = b¡1yb. We then have x = x1+jpm¡3yk, y = x¡rpm¡3y, z = z. Sincef is an automorphism, z¡1x z = x y. Because x y = x1+(j¡r)pm¡3yk+1,z¡1x z = x1+jpm¡3yk+1, we havez¡1x z = x y () rpm¡3 ´ 0 (mod pm¡2):Thus we have x = x1+jpm¡3yk; y = y; z = z. Therefore setting u = yjzk, wehavef(x) = u¡1xu; f(y) = u¡1yu; f(z) = u¡1zu;and so f 2 InnG7. ¤G11 enjoys \Hasse principle".4Mathematical Journal of Okayama University, Vol. 46 [2004], Iss. 1, Art. 28http://escholarship.lib.okayama-u.ac.jp/mjou/vol46/iss1/28\HASSE PRINCIPLE" FOR FINITE p-GROUPS 35Proof. Let f 2 AutcG11 such that z = z. Then there exist a = xiyjzk,b = xryszt 2 G11 with 0 · i; r < 9, 0 · j; k; s; t < 3 such that x = a¡1xa,y = b¡1yb. We then have x = x1+3k(k¡1)yk, y = x6ty, z = z. Sincef is an automorphism, z¡1x z = x y. Because x y = x1+6t+3k(k¡1)yk+1,z¡1x z = x1+6k+3k(k¡1)yk+1, we have z¡1x z = x y () k = t. Thus we havex = x1+3k(k¡1)yk, y = x6ky, z = z. Therefore setting u = zk, we havef(x) = u¡1xu; f(y) = u¡1yu; f(z) = u¡1zu;and so f 2 InnG11. ¤3. The case p = 2We here show that the groups G17, G18, G22, G24, G25 and G26 given in(II) enjoy \Hasse principle".G17 enjoys \Hasse principle".Proof. Let f 2 AutcG17 such that z = z. Then there exist a = xiyjzk, b =xryszt 2 G17 with 0 · i; r < 2m¡2, 0 · j; k; s; t < 2 such that x = a¡1xa,y = b¡1yb. We then have x = x1+j2m¡3yk, y = xr2m¡3y, z = z. Since f isan automorphism, we have z¡1x z = x y. Because x y = x1+(j+r)2m¡3yk+1,z¡1x z = x1+j2m¡3yk+1, z¡1x z = x y () r ´ 0 (mod 2). Thus we havex = x1+j2m¡3yk, y = y, z = z. Therefore setting u = yjzk, we havex = u¡1xu, y = u¡1yu, z = u¡1zu; and so f 2 InnG17. ¤G18 enjoys \Hasse principle".Proof. Let f 2 AutcG18 such that z = z. Then there exist a = xiyjzk,b = xryszt 2 G18 with 0 · i; r < 2m¡2, 0 · j; k; s; t < 2 such thatx = a¡1xa, y = b¡1yb. We then have x = xj2m¡3+(¡1)k(1+k2m¡4)+k22m¡4yk,y = xr2m¡3y, z = z. Since f is an automorphism, we have z2 = y. Be-cause y = z2 = xr2m¡3y, z2 = y () r ´ 0 (mod 2). Thus we have x =xj2m¡3+(¡1)k(1+k2m¡4)+k22m¡4yk, y = y, z = z. Therefore setting u = yjzk,we have x = u¡1xu, y = u¡1yu, z = u¡1zu, and so f 2 InnG18. ¤G22 enjoys \Hasse principle".Proof. Let f 2 AutcG22 such that z = z. Then there exist a = xiyjzk, b =xryszt 2 G22 with 0 · i; r < 2m¡2, 0 · j; k; s; t < 2 such that x = a¡1xa,y = b¡1yb. We then have x = x1+k2m¡4yk, y = xt2m¡3y, z = z. Since f isan automorphism, we have z¡1x z = x1+2m¡4 y. Becausex1+2m¡4y = x1+t2m¡3+(1+k)2m¡4y1+k; z¡1x z = x1+k2m¡3+(1+k)2m¡4y1+k;z¡1x z = x1+2m¡4y () k = t:5Fuma and Ninomiya: Hasse Principle" for Finite p-Groups with Cyclic Subgroups of IndProduced by The Berkeley Electronic Press, 200436 M. FUMA AND Y. NINOMIYAThus we have x = x1+k2m¡4yk, y = xk2m¡3y, z = z. Therefore settingu = zk, we have x = u¡1xu, y = u¡1yu, z = u¡1zu; and so f 2 InnG22. ¤G23 enjoys \Hasse principle".Proof. Let f 2 AutcG23 such that z = z. Then there exist a = xiyjzk, b =xryszt 2 G23 with 0 · i; r < 2m¡2, 0 · j; k; s; t < 2 such that x = a¡1xa,y = b¡1yb. We then havex =(x (k = 0)x¡1+2m¡4y (k = 1); y = xt2m¡3y; z = z:Since f is an automorphism, we have z¡1x z = x¡1+2m¡4 y. If k = 0,x¡1+2m¡4y = x¡1+2m¡4+t2m¡3y; z¡1x z = x¡1+2m¡4y:Therefore z¡1x z = x¡1+2m¡4 y () t = 0. If k = 1,x¡1+2m¡4y = x1+(t¡1)2m¡3; z¡1x z = x:Therefore z¡1x z = x¡1+2m¡4 y () t = 1. Thus we havex =(x (k = 0)x¡1+2m¡4y (k = 1); y = xk2m¡3y; z = z:Therefore setting u = zk, we have x = u¡1xu, y = u¡1yu, z = u¡1zu; andso f 2 InnG23. ¤G24 enjoys \Hasse principle".Proof. Let f 2 AutcG24 such that z = z. Then there exist a = xiyjzk, b =xryszt 2 G24 with 0 · i; r < 2m¡2, 0 · j; k; s; t < 2 such that x = a¡1xa,y = b¡1yb. We then havex =(x1+j2m¡3(k = 0)x¡1+2m¡4+j2m¡3y (k = 1); y = xr2m¡3y; z = z:Since f is an automorphism, we have z¡1x z = x¡1+2m¡4 y. If k = 0,x¡1+2m¡4y = x¡1+2m¡4+(r¡j)2m¡3y; z¡1x z = x¡1+2m¡4+j2m¡3y:Therefore z¡1x z = x¡1+2m¡4 y () r ´ 0 (mod 2). If k = 1,x¡1+2m¡4y = x1+(r¡j)2m¡3= x1+(r+j)2m¡3; z¡1x z = xj2m¡3x = x1+j2m¡3:Therefore z¡1x z = x¡1+2m¡4 y () r ´ 0 (mod 2). Thus we havex =(x1+j2m¡3(k = 0)x¡1+2m¡4+j2m¡3y (k = 1); y = y; z = z:6Mathematical Journal of Okayama University, Vol. 46 [2004], Iss. 1, Art. 28http://escholarship.lib.okayama-u.ac.jp/mjou/vol46/iss1/28\HASSE PRINCIPLE" FOR FINITE p-GROUPS 37Therefore setting u = yj , we have x = u¡1xu, y = u¡1yu, z = u¡1zu; andso f 2 InnG24. ¤G25 enjoys \Hasse principle".Proof. Let f 2 AutcG25 such that z = z. Then there exist a = xiyjzk, b =xryszt 2 G25 with 0 · i; r < 2m¡2, 0 · j; k; s; t < 2 such that x = a¡1xa,y = b¡1yb. We then havex =(x1+j2m¡3(k = 0)x¡1+2m¡4+j2m¡3y (k = 1); y = xr2m¡3y; z = z:Since f is an automorphism, we have z¡1x z = x¡1+2m¡4 y. If k = 0,x¡1+2m¡4y = x¡1+2m¡4+(j+r)2m¡3y; z¡1x z = x¡1+2m¡4+j2m¡3y:Therefore z¡1x z = x¡1+2m¡4 y () r ´ 0 (mod 2). If k = 1,x¡1+2m¡4y = x1+(r¡j)2m¡3= x1+(r+j)2m¡3; z¡1x z = x1+j2m¡3:Therefore z¡1x z = x¡1+2m¡4 y () r ´ 0 (mod 2). Thus we havex =(x1+j2m¡3(k = 0)x¡1+2m¡4+j2m¡3y (k = 1); y = y; z = z:Therefore setting u = yj , we have x = u¡1xu, y = u¡1yu, z = u¡1zu; andso f 2 InnG25. ¤G26 enjoys \Hasse principle".Proof. Let f 2 AutcG26 such that z = z. Then there exist a = xiyjzk,b = xryszt 2 G26 with 0 · i; r < 8, 0 · j; k; s; t < 2 such that x = a¡1xa,y = b¡1yb. We then have x = x1+4jyk, y = x4ry, z = z. Since f isan automorphism, we have z¡1x z = x y. Because x y = x1+4j+4ry1+k,z¡1x z = x1+4jy1+k, z¡1x z = x y () r ´ 0 (mod 2). Thus we havex = x1+4jyk, y = y, z = z. Therefore setting u = yjzk, we have x = u¡1xu,y = u¡1yu, z = u¡1zu; and so f 2 InnG26. ¤Acknowledgement. The authors thank the referee for his or her com-ments.References[1] M. Kumar and L. R. Vermani, \Hasse principle" for extraspecial p-groups, Proc.Japan Acad. 76A (2000), 123{125.[2] M. Kumar and L. R. Vermani, \Hasse principle" for groups of order p4, Proc. JapanAcad. 77A (2001), 95{98.[3] M. Kumar and L. R. Vermani, On automorphisms of some p-groups, Proc. JapanAcad. 78A (2002), 46{50.7Fuma and Ninomiya: Hasse Principle" for Finite p-Groups with Cyclic Subgroups of IndProduced by The Berkeley Electronic Press, 200438 M. FUMA AND Y. NINOMIYA[4] Y. Ninomiya, Finite p-groups with cyclic subgroups of index p2, Math. J. OkayamaUniv. 36 (1994), 1{21.[5] T. Ono, \Hasse principle" for PSL2(Z) and PSL2(Fp), Proc. Japan Acad. 74A(1998), 130{131.[6] T. Ono, Shafarevich-Tate sets for pro¯nite groups, Proc. Japan Acad. 75A (1999),96{97.[7] T. Ono and H. Wada, \Hasse principle" for free groups, Proc. Japan Acad. 75A(1999), 1{2.[8] T. Ono and H. Wada, \Hasse principle" for symmetric and alternating groups, Proc.Japan Acad. 75A (1999), 61{62.[9] H. Wada, \Hasse principle" for SLn(D), Proc. Japan Acad. 75A (1999), 67{69.Michitaku FumaDepartment of Mathematical SciencesFaculty of ScienceShinshu UniversityMatsumoto 390-8621, JapanYasushi NinomiyaDepartment of Mathematical SciencesFaculty of ScienceShinshu UniversityMatsumoto 390-8621, Japane-mail address: ysninom@gipac.shinshu-u.ac.jp(Received December 2, 2003 )8Mathematical Journal of Okayama University, Vol. 46 [2004], Iss. 1, Art. 28http://escholarship.lib.okayama-u.ac.jp/mjou/vol46/iss1/28

Hasse Principle" for Finite p-Groups with Cyclic Subgroups of Index p2

Abstract

Similar works

Full text

Available Versions

Okayama University Scientific Achievement Repository

Okayama University Digital Information Repository