Motose, Kaoru

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Mathematical Journal of OkayamaUniversityVolume 44, Issue 1 2002 Article 7JANUARY 2002On the Nilpotency Index of the Radical of aGroup Algebra. XIKaoru Motose∗∗Hirosaki UniversityCopyright c©2002 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. 44(2002), 51{56ON THE NILPOTENCY INDEX OFTHE RADICAL OF A GROUP ALGEBRA. XIKaoru MOTOSELet t(G) be the nilpotency index of the radical J(KG) of a group algebraKG of a ¯nite p-solvable group G over a ¯eld K of characteristic p > 0.Then it is well known by D. A. R. Wallace [7] thatpe ¸ t(G) ¸ e(p¡ 1) + 1;where pe is the order of a Sylow p-subgroup of G.H. Fukushima [1] characterized a group G of p-length 2 satisfying t(G) =e(p¡ 1) + 1, see also [4]. Unfortunately, his characterization holds under acondition such that the p0-part V = Op0;p(G)=Op(G) of G is abelian.In this paper, using Dickson near ¯elds, we shall give an explicit example(see Example 1) such that a group G of p-length 2 has the non abelian p0-part V and satis¯es t(G) = e(p¡ 1)+1. This example will be new and havea contributions in our research. Example 2 is also very interesting becausequite di®erent objects (see [3] and [5]) are uni¯ed on the ground of Dicksonnear ¯elds.Let H be a sharply 2-fold transitive group on ¢ = f0; 1; ®; ¯; : : : ; °g(see [8, p. 22]). Let V = H0 be a stabilizer of 0, and let U be the setconsisting of the identity " and ¯xed point-free permutations in H. Then Uis an elementary abelian p-subgroup of H with the order ps (see Lemma 1).Let ¾ be a permutation of order p on ¢ satisfying conditions¾H¾¡1 µ H; ¾p = 1; ¾(0) = 0 and ¾(1) = 1:Then it is easy to see ¾U¾¡1 µ U and ¾V ¾¡1 µ V . We set W = h¾i andCV (¾) = fv 2 V j ¾v = v¾g. Assume that there exists a normal subgroup Tof WV contained in V such that V is a semi-direct product of T by CV (¾).We set G = hW;T;Ui.Now, we present the well known results Lemmas 1 and 2 for completenessof this paper.Lemma 1. U is a normal and elementary abelian p-subgroup of H.Proof. First we shall prove, for k 2 ¢¤ = ¢ n f0g, there exists only oneuk 2 U with uk(0) = k, equivalently, the following map º from U to ¢ isbijective:º : u! u(0):This paper was ¯nancially supported by the Grant-in-Aid for Scienti¯c Research fromJapan Society for the Promotion of Science (Subject No. 1164003).511Motose: On the Nilpotency Index of the Radical of a Group Algebra. XIProduced by The Berkeley Electronic Press, 200252 K. MOTOSEFor ¿ 2 U n f"g, there exists ½ 2 H0 with ½(¿(0)) = k since ¿(0) 6= 0 and H0is transitive on ¢¤. We set uk = ½¿½¡1. Then uk 2 U and uk(0) = k. Thusº is surjective. It follows from de¯nition of H and U thatU = H n[a2¢(Ha n f"g); (Ha n f"g) \ (Hb n f"g) = ; for a 6= b:Using jHj = jHajjaH j = jHajj¢j, where aH is an orbit of a, we can seejU j = j¢j. Hence º is injective.Assume ´¿ has a ¯xed point ` for ´; ¿ 2 U . Then we may assume ` = 0since H is transitive on ¢ and ½U½¡1 = U for ½ 2 H. Thus ¿ = ´¡1follows from ´¡1 2 U , ¿(0) = ´¡1(0) and the above observation. Thismeans ´¿ 2 U . Hence U is a normal subgroup of H because ½U½¡1 = U forall ½ 2 H.Now, we shall show U is elementary abelian. Let p be a prime factor ofjU j and let ¿ be an element of order p in the center of a Sylow p-subgroupof U . We set ´ 2 U n f"g. Then there exists ½ 2 H0 with ½(¿(0)) = ´(0).Thus ½¿½¡1 = ´ follows from ½¿½¡1 2 U and ½¿½¡1(0) = ´(0). Thus theorder of every element in U is p or 1 and so ´ is in the center of a p-groupU . Thus U is elementary abelian. ¤The next shows ¢ is a near ¯eld of characteristic p.Lemma 2. ¢ is a near ¯eld of characteristic p and ¾ is an automorphismof ¢.Proof. First, we shall prove that ¢ is a near ¯eld. We can set a structureof a near ¯eld in a set ¢ by the following method. It follows from Lemma 1that there exists only one ua 2 U with ua(0) = a for a 2 ¢. It is easy tosee that for a 2 ¢¤ = ¢ n f0g, there exists only one va 2 V = H0 withva(1) = a. It is clear from de¯nition that u0 = v1 = ".We de¯ne the sum and the product of elements a; b in ¢ by using theabove va and ub:a+ b := ub(a); ab := va(b) for a 6= 0 and 0b := 0:First we shall prove the next equations:uaub = ub+a; vavb = vab and vaubv¡1a = uab:These follow fromuaub(0) = ua(b) = b+ a = ub+a(0);vavb(1) = va(b) = ab = vab(1);vaubv¡1a (0) = vaub(0) = va(b) = ab = uab(0):2Mathematical Journal of Okayama University, Vol. 44 [2002], Iss. 1, Art. 7http://escholarship.lib.okayama-u.ac.jp/mjou/vol44/iss1/7ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI 53Next we shall prove the next equations from the ¯rst equation and thecommutativity of U :a+ (b+ c) = ub+c(a) = ucub(a) = uc(a+ b) = (a+ b) + c;a+ b = ua+b(0) = ubua(0) = uaub(0) = ua(b) = b+ a;a+ 0 = 0 + a = ua(0) = a;a+ u¡1a (0) = u¡1a (0) + a = ua(u¡1a (0)) = "(0) = 0:We shall prove the next equations from the second equation for a; b 2 ¢¤.For a = 0 or b = 0, it is easy to prove our equations:a(bc) = va(bc) = va(vb(c)) = vavb(c) = vab(c) = (ab)c;a1 = va(1) = a = "(a) = v1(a) = 1a;av¡1a (1) = va(v¡1a (1)) = "(1) = 1:For a 2 ¢¤, v¡1a (1) 6= 0 follows from va(0) = 0 6= 1 and we can see vv¡1a (1) =v¡1a by vv¡1a (1)(1) = v¡1a (1). Thus we havev¡1a (1)a = vv¡1a (1)(a) = v¡1a (a) = v¡1a (va(1)) = 1:The next equation follows from the third equation:a(b+ c) = va(b+ c) = va(uc(b)) = vaucv¡1a (va(b)) = uac(ab) = ab+ ac:Thus ¢ is a near ¯eld by our de¯nition of the sum and the product. Moreover¢ is of characteristic p because up¢1 = up1 = " = u0.Next we shall show ¾ is an automorphism of ¢. It is easy to see from thede¯nitions of U and V that¾U¾¡1 µ U and ¾V ¾¡1 µ V:It follows from the de¯nitions of ua and va that¾ua¾¡1 = u¾(a) and ¾vb¾¡1 = v¾(b)by equations¾ua¾¡1(0) = ¾ua(0) = ¾(a) = u¾(a)(0)and¾vb¾¡1(1) = ¾vb(1) = ¾(b) = v¾(b)(1):Since ¾ is a permutation on ¢, it follows from the next equations that ¾ isan automorphism of ¢:u¾(a+b) = ¾ua+b¾¡1 = ¾ua¾¡1¾ub¾¡1 = u¾(a)u¾(b) = u¾(a)+¾(b)andv¾(ab) = ¾vab¾¡1 = ¾va¾¡1¾vb¾¡1 = v¾(a)v¾(b) = v¾(a)¾(b): ¤3Motose: On the Nilpotency Index of the Radical of a Group Algebra. XIProduced by The Berkeley Electronic Press, 200254 K. MOTOSEWe can see from Lemma 2 and the classi¯cation of ¯nite near ¯elds(see [9]) that ¢ is a Dickson near ¯eld because ¢ has an automorphismof order p where p is the characteristic of ¢.Lemma 3. WT is a Frobenius group with kernel T and complement W .Proof. We note W \ V = f"g since ¾(1) = 1. Let x = ¾kv be an elementof WT n W , where v 2 T , and let x¡1¾sx = ¾t 6= " be an element ofx¡1Wx \ W . Then we may assume s = 1 because the order of ¾ is p.Thus x¡1Wx \W contains v¡1¾v = ¾t. The element ¾t¡1 = v¡1 ¢ ¾v¾¡1 iscontained in W \ V = f"g. Hence ¾v = v¾. Thus v 2 CV (¾)\ T = f"g andx = ¾kv = ¾k is contained in W . Therefore we havex¡1Wx \W = f"g for x 2WT nW: ¤Lemma 4. G = TCG(¾)T .Proof. Clearly TCG(¾)T contains T and W . Let u± be an arbitrary elementof U n f"g, where ± is an arbitrary element in ¢¤ = ¢ n f0g. Then v± =v°v¸ = v°¸ where v° 2 T and v¸ 2 CV (¾), namely, ¾(¸) = ¸. Thus ± = °¸and so u± = v°u¸v¡1° 2 TCG(¾)T . It follows from U ½ TCG(¾)T thatG = TCG(¾)T . ¤Lemma 5. (J(KW )T^KG)n µ J(KW )nT^KG, where T^ =Pt2T t.Proof. Since T is normal in WV and G = TCG(¾)T by Lemma 4, we cansee s¾ = ¾s for every s 2 T^KGT^ = T^KCG(¾)T^ . Clearly the result holdsfor n = 1. Assume that the result holds for n. Then using the last assertion,we conclude that(J(KW )T^KG)n+1 µ J(KW )nT^KGJ(KW )T^KG= J(KW )nT^KGT^J(KW )KGµ J(KW )n+1T^KG: ¤Theorem. Let S be a subgroup of V containing T and let ps+1 be the orderof a Sylow p-subgroup WU of M = hS;W;Ui. Then t(M) = (s+1)(p¡1)+1.Proof. Let v be an arbitrary element of S. Then v = tc where t 2 T andc 2 CV (¾). Hence we havev¾v¡1 = tc¾c¡1t¡1 = t¾t¡1 2 G = hT;W;Ui:Noting T is normal in V , we have that G is a normal in M and the indexjM : Gj is relatively prime to p. Therefore we obtain t(M) = t(G) and itis enough to prove in case M = G. Since the radical J(KG) contains thekernel J(KU)KG of the natural homomorphism º of the group algebra KGonto K(G=U), it follows that º(J(KG)) = º(J(KW )T^ ) by Lemma 3 and4Mathematical Journal of Okayama University, Vol. 44 [2002], Iss. 1, Art. 7http://escholarship.lib.okayama-u.ac.jp/mjou/vol44/iss1/7ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI 55[2, Theorem 4] and so J(KG) = J(KW )T^KG + J(KU)KG. Since U isa normal and elementary abelian subgroup of order ps, it is clear that thenilpotency index of J(KU)KG is s(p¡1)+1. On the other hand, Lemma 5shows that (J(KW )T^KG)p = 0. Since J(KW )T^KG and J(KU)KG areright ideals of KG, it follows thatJ(KG)(s+1)(p¡1)+1 = (J(KW )T^KG+ J(KU)KG)p+s(p¡1) = 0;and so t(G) · (s+1)(p¡1)+1. On the other hand (s+1)(p¡1)+1 · t(G)by [7, Theorem 3.3]. This completes the proof. ¤Example 1. Let (q; n) be a Dickson pair where p is a prime and q = pr for apositive integer r. Then (qp; n) is also a Dickson pair because qp ´ ¡1 mod 4if and only if q ´ ¡1 mod 4. Let F = F qpn be a ¯nite ¯eld of order qpn andlet D = Dqpn be a ¯nite Dickson near ¯eld de¯ned by the automorphism¿ : x ! xqp of F . Then an automorphism ¾ : x ! xqn of F is also of Dby [9, Satz 18] or [6, Theorem 5] because prn = qn ´ 1 mod n (see also [6,Theorem 1]).Let ! be a generator of the multiplicative group F ¤ and we set a = !n,b = ! in F ¤. Then the multiplicative group D¤ of D has the structureD¤ = ha; b j am = 1; bn = at; bab¡1 = aqpi;where m = qpn¡1n , t =mqp¡1 . Here we use the usual symbol as the product inD for simplicity. Do not confuse with the product in F . We consider somepermutations on D:uc : x! x+ c for c 2D; vc : x! cx for c 2D¤:Then we have some relationsucud = ud+c; vcvd = vcd; vcudv¡1c = ucd; ¾uc¾¡1 = u¾(c); ¾vc¾¡1 = v¾(c)on uc; vc; ¾. We setU = fuc j c 2Dg; V = fvc j c 2D¤g; W = h¾iandT = fvc 2 V j c 2 haqn¡1n ig:It is easy to see that UV is sharply 2-fold transitive on D, T is normal inWV and the order of T is qpn¡1qn¡1 because products of a and x in D are thesame in F . On the other hand, the set CV (¾) is equal to F ¤qn as a set andthe order of CV (¾) is qn¡1. Since qpn¡1qn¡1 and qn¡1 are relatively prime, wehave V = CV (¾)T , CV (¾) \ T = f"g. Let S be a subgroup of V containingT andM = hS;W;Ui. Then t(M) = (rpn+1)(p¡1)+1 by Theorem, whereprpn+1 is the order of a Sylow p-subgroup WU of M .5Motose: On the Nilpotency Index of the Radical of a Group Algebra. XIProduced by The Berkeley Electronic Press, 200256 K. MOTOSEIf we put D = F for the extreme case n = 1, we have the same exampleas in [3]. ¤Example 2. If (q; n) 6= (3; 2) and p is not a divisor of r, then Dqn has noautomorphisms of order p, and so we consider Dqpn . But D32 has an auto-morphism ¾ of order 3 and we can consider the a±ne group G = h¾; V; Uiover D32 where D32 is a Dickson near ¯elds de¯ned by an automorphismx! x3 of F 32 = F 3[x]=(x2+1) = fs+ ti j i2 = ¡1; s; t 2 F 3g, ¾ is de¯nedby ¾(s+ ti) = s+ t+ ti, and the permutation group U , V are de¯ned as inExample 1. This group G is isomorphic to Qd(3), namely, a group de¯nedby semi-direct product of F (2)3 by SL(2; 3) using the natural action, whereF(2)3 is 2-dimensional vector space over F 3 and SL(2; 3) is the special lineargroup over F (2)3 . In this case 33 is the order of a Sylow 3-subgroup of G andit is known form [5] that t(G) = 9 > 7 = 3(3¡ 1) + 1.This observation is very interesting because quite di®erent objects (see [3]and [5]) are uni¯ed on the ground of Dickson near ¯elds. ¤References[1] H. Fukushima, On groups G of p-length 2 whose nilpotency indices of J(KG) area(p¡ 1) + 1, Hokkaido Math. J. 20(1991), 523{530.[2] K. Motose, On radicals of group rings of Frobenius groups, Hokkaido Math. J.3(1974), 23{34.[3] K. Motose, On the nilpotency index of the radical of a group algebra. III, J. LondonMath. Soc. (2) 25(1982), 39{42.[4] K. Motose, On the nilpotency index of the radical of a group algebra. IV, Math. J.Okayama Univ. 25(1983), 35{42.[5] K. Motose, On the nilpotency index of the radical of a group algebra. V, J. Algebra90(1984), 251{258.[6] K. Motose, On ¯nite Dickson near ¯elds, Bull. Fac. Sci. Technol. Hirosaki Univ.3(2001), 69{78.[7] D. A. R. Wallace, Lower bounds for the radical of the group algebra of a ¯nitep-soluble group, Proc. Edinburgh Math. Soc. (2) 16(1968/69), 127{134.[8] H. Wielandt, Finite permutation groups, Academic Press, 1964.[9] H. Zassenhaus, ÄUber endliche FastkÄorper, Abh. Math. Semin. Univ. Hamburg11(1935/36), 187{220.Kaoru MotoseDepartment of Mathematical System ScienceFaculty of Science and TechnologyHirosaki UniversityHirosaki 036-8561, Japane-mail address: skm@cc.hirosaki-u.ac.jp(Received April 25, 2002 )6Mathematical Journal of Okayama University, Vol. 44 [2002], Iss. 1, Art. 7http://escholarship.lib.okayama-u.ac.jp/mjou/vol44/iss1/7

On the Nilpotency Index of the Radical of a Group Algebra. XI

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