ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI

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1 Mth. J. Okym Univ. 44(2002), ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI Koru MOTOSE Let t(g) be the nilpotency index of the rdicl J(KG) of group lgebr KG of finite p-solvble group G over field K of chrcteristic p > 0. Then it is well known by D. A. R. Wllce [7] tht p e t(g) e(p 1) + 1, where p e is the order of Sylow p-subgroup of G. H. Fukushim [1] chrcterized group G of p-length 2 stisfying t(g) = e(p 1) + 1, see lso [4]. Unfortuntely, his chrcteriztion holds under condition such tht the p -prt V = O p,p(g)/o p (G) of G is belin. In this pper, using Dickson ner fields, we shll give n explicit exmple (see Exmple 1) such tht group G of p-length 2 hs the non belin p - prt V nd stisfies t(g) = e(p 1) + 1. This exmple will be new nd hve contributions in our reserch. Exmple 2 is lso very interesting becuse quite different objects (see [3] nd [5]) re unified on the ground of Dickson ner fields. Let H be shrply 2-fold trnsitive group on = {0, 1, α, β,..., γ} (see [8, p. 22]). Let V = H 0 be stbilizer of 0, nd let U be the set consisting of the identity ε nd fixed point-free permuttions in H. Then U is n elementry belin p-subgroup of H with the order p s (see Lemm 1). Let σ be permuttion of order p on stisfying conditions σhσ 1 H, σ p = 1, σ(0) = 0 nd σ(1) = 1. Then it is esy to see σuσ 1 U nd σv σ 1 V. We set W = σ nd C V (σ) = {v V σv = vσ}. Assume tht there exists norml subgroup T of W V contined in V such tht V is semi-direct product of T by C V (σ). We set G = W, T, U. Now, we present the well known results Lemms 1 nd 2 for completeness of this pper. Lemm 1. U is norml nd elementry belin p-subgroup of H. Proof. First we shll prove, for k = \ {0}, there exists only one u k U with u k (0) = k, equivlently, the following mp ν from U to is bijective: ν : u u(0). This pper ws finncilly supported by the Grnt-in-Aid for Scientific Reserch from Jpn Society for the Promotion of Science (Subject No ). 51

2 52 K. MOTOSE For τ U \ {ε}, there exists ρ H 0 with ρ(τ(0)) = k since τ(0) 0 nd H 0 is trnsitive on. We set u k = ρτρ 1. Then u k U nd u k (0) = k. Thus ν is surjective. It follows from definition of H nd U tht U = H \ (H \ {ε}), (H \ {ε}) (H b \ {ε}) = for b. Using H = H H = H, where H is n orbit of, we cn see U =. Hence ν is injective. Assume ητ hs fixed point l for η, τ U. Then we my ssume l = 0 since H is trnsitive on nd ρuρ 1 = U for ρ H. Thus τ = η 1 follows from η 1 U, τ(0) = η 1 (0) nd the bove observtion. This mens ητ U. Hence U is norml subgroup of H becuse ρuρ 1 = U for ll ρ H. Now, we shll show U is elementry belin. Let p be prime fctor of U nd let τ be n element of order p in the center of Sylow p-subgroup of U. We set η U \ {ε}. Then there exists ρ H 0 with ρ(τ(0)) = η(0). Thus ρτρ 1 = η follows from ρτρ 1 U nd ρτρ 1 (0) = η(0). Thus the order of every element in U is p or 1 nd so η is in the center of p-group U. Thus U is elementry belin. The next shows is ner field of chrcteristic p. Lemm 2. is ner field of chrcteristic p nd σ is n utomorphism of. Proof. First, we shll prove tht is ner field. We cn set structure of ner field in set by the following method. It follows from Lemm 1 tht there exists only one u U with u (0) = for. It is esy to see tht for = \ {0}, there exists only one v V = H 0 with v (1) =. It is cler from definition tht u 0 = v 1 = ε. We define the sum nd the product of elements, b in by using the bove v nd u b : + b := u b (), b := v (b) for 0 nd 0b := 0. First we shll prove the next equtions: These follow from u u b = u b+, v v b = v b nd v u b v 1 = u b. u u b (0) = u (b) = b + = u b+ (0), v v b (1) = v (b) = b = v b (1), v u b v 1 (0) = v u b (0) = v (b) = b = u b (0).

3 ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI 53 Next we shll prove the next equtions from the first eqution nd the commuttivity of U: + (b + c) = u b+c () = u c u b () = u c ( + b) = ( + b) + c, + b = u +b (0) = u b u (0) = u u b (0) = u (b) = b +, + 0 = 0 + = u (0) =, + u 1 (0) = u 1 (0) + = u (u 1 (0)) = ε(0) = 0. We shll prove the next equtions from the second eqution for, b. For = 0 or b = 0, it is esy to prove our equtions: (bc) = v (bc) = v (v b (c)) = v v b (c) = v b (c) = (b)c, 1 = v (1) = = ε() = v 1 () = 1, v 1 (1) = v (v 1 (1)) = ε(1) = 1. For, v 1 (1) 0 follows from v (0) = 0 1 nd we cn see v v 1 (1) = v 1(1). Thus we hve v 1 by v v 1 (1) v 1 (1) = v v 1 (1) () = v 1 () = v 1 (v (1)) = 1. The next eqution follows from the third eqution: (1) = (b + c) = v (b + c) = v (u c (b)) = v u c v 1 (v (b)) = u c (b) = b + c. Thus is ner field by our definition of the sum nd the product. Moreover is of chrcteristic p becuse u p 1 = u p 1 = ε = u 0. Next we shll show σ is n utomorphism of. It is esy to see from the definitions of U nd V tht σuσ 1 U nd σv σ 1 V. It follows from the definitions of u nd v tht by equtions nd σu σ 1 = u σ() nd σv b σ 1 = v σ(b) σu σ 1 (0) = σu (0) = σ() = u σ() (0) σv b σ 1 (1) = σv b (1) = σ(b) = v σ(b) (1). Since σ is permuttion on, it follows from the next equtions tht σ is n utomorphism of : nd u σ(+b) = σu +b σ 1 = σu σ 1 σu b σ 1 = u σ() u σ(b) = u σ()+σ(b) v σ(b) = σv b σ 1 = σv σ 1 σv b σ 1 = v σ() v σ(b) = v σ()σ(b).

4 54 K. MOTOSE We cn see from Lemm 2 nd the clssifiction of finite ner fields (see [9]) tht is Dickson ner field becuse hs n utomorphism of order p where p is the chrcteristic of. Lemm 3. W T is Frobenius group with kernel T nd complement W. Proof. We note W V = {ε} since σ(1) = 1. Let x = σ k v be n element of W T \ W, where v T, nd let x 1 σ s x = σ t ε be n element of x 1 W x W. Then we my ssume s = 1 becuse the order of σ is p. Thus x 1 W x W contins v 1 σv = σ t. The element σ t 1 = v 1 σvσ 1 is contined in W V = {ε}. Hence σv = vσ. Thus v C V (σ) T = {ε} nd x = σ k v = σ k is contined in W. Therefore we hve Lemm 4. G = T C G (σ)t. x 1 W x W = {ε} for x W T \ W. Proof. Clerly T C G (σ)t contins T nd W. Let u δ be n rbitrry element of U \ {ε}, where δ is n rbitrry element in = \ {0}. Then v δ = v γ v λ = v γλ where v γ T nd v λ C V (σ), nmely, σ(λ) = λ. Thus δ = γλ nd so u δ = v γ u λ vγ 1 T C G (σ)t. It follows from U T C G (σ)t tht G = T C G (σ)t. Lemm 5. (J(KW ) ˆT KG) n J(KW ) n ˆT KG, where ˆT = t T t. Proof. Since T is norml in W V nd G = T C G (σ)t by Lemm 4, we cn see sσ = σs for every s ˆT KG ˆT = ˆT KC G (σ) ˆT. Clerly the result holds for n = 1. Assume tht the result holds for n. Then using the lst ssertion, we conclude tht (J(KW ) ˆT KG) n+1 J(KW ) n ˆT KGJ(KW ) ˆT KG = J(KW ) n ˆT KG ˆT J(KW )KG J(KW ) n+1 ˆT KG. Theorem. Let S be subgroup of V contining T nd let p s+1 be the order of Sylow p-subgroup W U of M = S, W, U. Then t(m) = (s+1)(p 1)+1. Proof. Let v be n rbitrry element of S. Then v = tc where t T nd c C V (σ). Hence we hve vσv 1 = tcσc 1 t 1 = tσt 1 G = T, W, U. Noting T is norml in V, we hve tht G is norml in M nd the index M : G is reltively prime to p. Therefore we obtin t(m) = t(g) nd it is enough to prove in cse M = G. Since the rdicl J(KG) contins the kernel J(KU)KG of the nturl homomorphism ν of the group lgebr KG onto K(G/U), it follows tht ν(j(kg)) = ν(j(kw ) ˆT ) by Lemm 3 nd

5 ON THE NILPOTENCY INDEX OF THE RADICAL OF A GROUP ALGEBRA. XI 55 [2, Theorem 4] nd so J(KG) = J(KW ) ˆT KG + J(KU)KG. Since U is norml nd elementry belin subgroup of order p s, it is cler tht the nilpotency index of J(KU)KG is s(p 1) + 1. On the other hnd, Lemm 5 shows tht (J(KW ) ˆT KG) p = 0. Since J(KW ) ˆT KG nd J(KU)KG re right idels of KG, it follows tht J(KG) (s+1)(p 1)+1 = (J(KW ) ˆT KG + J(KU)KG) p+s(p 1) = 0, nd so t(g) (s + 1)(p 1) + 1. On the other hnd (s + 1)(p 1) + 1 t(g) by [7, Theorem 3.3]. This completes the proof. Exmple 1. Let (q, n) be Dickson pir where p is prime nd q = p r for positive integer r. Then (q p, n) is lso Dickson pir becuse q p 1 mod 4 if nd only if q 1 mod 4. Let F = F q pn be finite field of order q pn nd let D = D q pn be finite Dickson ner field defined by the utomorphism τ : x x qp of F. Then n utomorphism σ : x x qn of F is lso of D by [9, Stz 18] or [6, Theorem 5] becuse p rn = q n 1 mod n (see lso [6, Theorem 1]). Let ω be genertor of the multiplictive group F nd we set = ω n, b = ω in F. Then the multiplictive group D of D hs the structure D =, b m = 1, b n = t, bb 1 = qp, where m = qpn 1 n, t = m q p 1. Here we use the usul symbol s the product in D for simplicity. Do not confuse with the product in F. We consider some permuttions on D: u c : x x + c for c D, v c : x cx for c D. Then we hve some reltions u c u d = u d+c, v c v d = v cd, v c u d v 1 c on u c, v c, σ. We set nd = u cd, σu c σ 1 = u σ(c), σv c σ 1 = v σ(c) U = {u c c D}, V = {v c c D }, W = σ T = {v c V c qn 1 n }. It is esy to see tht UV is shrply 2-fold trnsitive on D, T is norml in W V nd the order of T is qpn 1 q n 1 becuse products of nd x in D re the sme in F. On the other hnd, the set C V (σ) is equl to F q n s set nd the order of C V (σ) is q n 1. Since qpn 1 q n 1 nd qn 1 re reltively prime, we hve V = C V (σ)t, C V (σ) T = {ε}. Let S be subgroup of V contining T nd M = S, W, U. Then t(m) = (rpn+1)(p 1)+1 by Theorem, where p rpn+1 is the order of Sylow p-subgroup W U of M.

6 56 K. MOTOSE If we put D = F for the extreme cse n = 1, we hve the sme exmple s in [3]. Exmple 2. If (q, n) (3, 2) nd p is not divisor of r, then D q n hs no utomorphisms of order p, nd so we consider D q pn. But D 3 2 hs n utomorphism σ of order 3 nd we cn consider the ffine group G = σ, V, U over D 3 2 where D 3 2 is Dickson ner fields defined by n utomorphism x x 3 of F 3 2 = F 3 [x]/(x 2 + 1) = {s + ti i 2 = 1, s, t F 3 }, σ is defined by σ(s + ti) = s + t + ti, nd the permuttion group U, V re defined s in Exmple 1. This group G is isomorphic to Qd(3), nmely, group defined by semi-direct product of F (2) 3 by SL(2, 3) using the nturl ction, where F (2) 3 is 2-dimensionl vector spce over F 3 nd SL(2, 3) is the specil liner group over F (2) 3. In this cse 33 is the order of Sylow 3-subgroup of G nd it is known form [5] tht t(g) = 9 > 7 = 3(3 1) + 1. This observtion is very interesting becuse quite different objects (see [3] nd [5]) re unified on the ground of Dickson ner fields. References [1] H. Fukushim, On groups G of p-length 2 whose nilpotency indices of J(KG) re (p 1) + 1, Hokkido Mth. J. 20(1991), [2] K. Motose, On rdicls of group rings of Frobenius groups, Hokkido Mth. J. 3(1974), [3] K. Motose, On the nilpotency index of the rdicl of group lgebr. III, J. London Mth. Soc. (2) 25(1982), [4] K. Motose, On the nilpotency index of the rdicl of group lgebr. IV, Mth. J. Okym Univ. 25(1983), [5] K. Motose, On the nilpotency index of the rdicl of group lgebr. V, J. Algebr 90(1984), [6] K. Motose, On finite Dickson ner fields, Bull. Fc. Sci. Technol. Hiroski Univ. 3(2001), [7] D. A. R. Wllce, Lower bounds for the rdicl of the group lgebr of finite p-soluble group, Proc. Edinburgh Mth. Soc. (2) 16(1968/69), [8] H. Wielndt, Finite permuttion groups, Acdemic Press, [9] H. Zssenhus, Über endliche Fstkörper, Abh. Mth. Semin. Univ. Hmburg 11(1935/36), Koru Motose Deprtment of Mthemticl System Science Fculty of Science nd Technology Hiroski University Hiroski , Jpn e-mil ddress: skm@cc.hiroski-u.c.jp (Received April 25, 2002 )