LAKATOS, P.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

MODULAR GROUP RINGS AND WREATH PRODUCTS OF GROUPS P. LAKA.Tos Mathematical Institut of L. Kossuth University H-4010, Debrecen Received August 10, 1988 Now we give some definitions and notations. Let A and B be two groups. Define AB to be the set of all functions f: B -+ A such that f(b) " 1 for all but a finite number of b E B, and with componentwise multiplication fg(b) = f(b)g(b) for all bE B. Then AB is a restricted direct product of copies of A. The standard restricted wreath product of A by B, A ( B is defined as the extension of AB by B given by p(b') = f(b'b- 1) for all f E AB; b, b' E B. As we only use the standard restricted wreath product, Vie will simply call it the wreath product. In the following let Zpn be the ring of the integers mod pn, where p is a prime number, and let G = A ( B be a wreath product of a cyclic group A by a finite p-group B. We denote by K = AB the base group of G. If we consider K as an additive group, then an arbitrary b E B generates an automorphism (Pb of group K defined in the follovving way kerb = b-1kb; bE B; k E K . Denoting k CPb = kb, the additive group K can be identified ,\ith aB-module, just as G = A ( B may be realized as the group ring Zp" [B] extended by B in its right regular representation. Using automorphism CPb element [k, b] can he written in the form k(l b), where k E K, b E B. As it is known, the augmentation ideal J(Zpn, B) of the group ring Zpn[B] as an additive group can be generated hy elements of form 1 b, bE Band Jk(Zpn, B) can be generated hy elements (1 - bi,l) ... (1 - bi,l:) where bi,j are arbitrary elements of group B. It can be seen from these ([2]) that Yi(G) = [KJi-l(Zp'" B)]ytCB) (1) 58 'where fi(G) and y/B) denote the i-th member of lower central series of cor-responding groups. Denoting by J!(n, B) the nilpotency class of I(Zp'" B) and using (1) we obtain cl (Cpn ( B). Therefore ,\-e get the cl (A ( B) by determining J!(n, B), where A is an ahelian group with finite exponent pn and B is an arbitrary finite p-group. '.:'40w we show some simple connections between nilpotent wreath prod-ucts and modular group rings. Let B = fl(B):::l )'2(B)~ ... ~ YH I(B) = 1 he the lower central series of B. We denote by r(j) the rank of Yi(B)/Yi+1(B) and Bi = {b(i,j), 1 j <;;;'r(j)} ~ ~ Yi(B) Yi+l(B) denotes an independent set of generators for Yi(B) modulo l'i+l(B), such that ib(i,j)I, = p(i,j) where p(i, 1), .. . ,p(i, rei»~ are the orders, indescending powers of p, of cyclic factors of y/B)/Yi+l(B). Theorem 1. [5]. If A is an abelian p-group of exponent pn and W = = A ( B, then L r(j) cl(W) =:E :Ei(p(i,j) -1) -!- (n -l)mp(m, I)(p i=l j=l mp(m, 1) = max {i p(i, In. l:5:i:5:L l)p-l + 1 We define the J ennings-series of a nilpotent p-group G by the equation KlG) = If Yn(G)pi, n·pj~i where )In(G) is the n-ts member of the lower central series of G(i 2: 1). Let B be a finite p-group and d a maximal number such that Ka(B) -;-<- 1. d Let peU) = /Kj(B)/Kj+1(B)/ and a = 1 + (p - 1) :E je(j), b = (p l)d. j=l Theorem 2. [6]. If A is a nilpotent p-group of class c, B is a finite p-group and the exponent of YwCA) equals pS(W) (1 < w c), then cl (A ( B) = max {aw -T (s(w) - I)bII w :s:: c}. This shows that to kno'w the lower central series is important for the investi-gation of nilpotent wreath products. Corollary 3. If A is an abelian p-group, and exponent of A equals pn, then l'(n, B) = F(I, B) + (n - l)(p l)d. Proof. Applying theorem 2. with c = 1 the statement follows from a = vel, B) ([I)). _,raVULAR GROUP RISGS 59 Remark: The class of the radical of Zp,,[B] is equal to pen, B) if )!(n - 1, B) < pen, B) for all n ([2]). By corollary 3. the last property is true for all finite p-groups. It means that determination of nilpotency classes of nilpotent wreath products has an important application in the theory of modular group rings. Theorem 4. If B is the group defined in the theorem 1, then the nil-potency class of the radical of Z p[ B] equals L r(j) p(1, B) = ~ ~ (i p(i,j) - 1) + 1, i=1 j=1 and the length of the J ennillgs-series of B equals -1 . (. 1) x = p lllax L p L, • l:S;i:S;L Proof: Combining theorem 1. and corollary 3. 'we obtain the statement (omitting the term depending on the n). Example. Similarly to the example 4.3 in [5] determine the v(n, Cp'" ~ Cp) (where cpm denotes a cyclic group of order pm). Let B = Cpm ( Cp, A = Cpn and K be the base group of B. Then [B[ = pmp +1 and by the arguments in [7] B/K abelian, j'2(B) C K, Yl(B)!?'2(B) = CpmX Cp, rei) = 1, p(i, 1) = P for i = 2,3, ... , m(p - 1) + 1. Thus by the theorem 1. the class of G = A ( B is also ( ) m(p - 1) + 2 ( v( n, B) = pm 1 + n(p - 1) + (p - 1) - m(p 2 1) 1). Theorem 5. Let G = ( ... (Ak 2 A k- I ) ( h A A X 4 '14 I - ~(j,i) were _":1.j = -":1. j ,lX • •• - j,r(j)'.- j,i - P , p<:.(j,l) > p,,-(j,2) > ... >p~(j,r(i» (j = 1,2, ... , k) then k-I r(j) k-2 r(j) cl (G) 2 II (~ (p,,-(j,i) 1) + 1) + 11 (~(p~(j,i) 1) + l)(a(k, 1) - 1) . j=1 i=l j=1 i=1 .(p~(k-I,I)-I(p _ 1) + 1). (2) If rU) = 1 for all j = 1,2, ... , k or x(k, 1) = x(k, 2) = ... = x(k, r(k)) then the equality holds in (2). Proof. We apply the induction by the number of wreath factors. If j = k 1, then r(k-I) cl (A ) 4 ) - '" (p,,(k-I,i) k (- le-I - ~ i=l 1) + (a(k, 1) - 1) + (aCk, 1) 1)· 60 P. LAKATOS If A = ( ... (Ak ( A k_1) ( ••• ) (Aj_1 then cl(Ap ( A) ): cl (Gp ( A) . cl (A) hecause in the case of exp (Z(A»)=p by theorem 2. cl (A ( Aj)=v(l, Aj)cl(A) (see also [4] theorem 2.5) and exp (Z(A») > p, sew) > 1 for all 1 < w < c. r(j) As cl (Cp ( A) = ~ (p~(j,i) - 1) + 1 i=l the induction hypothesis implies the statement of theorem 5. If exp (Aj) = p for all j = 1,2, ... , k then k-1 rei) cl (G) = ~ II (p~(i,j) - 1) + 1 holds. i=l j=l If every A/j = L2, ... , k) is cyclic, then reducing the determination of nil-potency class of G to the determination of nilpotency class of two cyclic groups ([8]), we obtain k-1 k-1 k-1 ::E ~(i,1) ::E ~(i,l)-l :E ~(i,1)-1 cl (G) = rx(k, l)(pi=l - pi=l ) + pi=l . This completes the proof. References 1. JEN:t-."'INGS, S. A.: The strncture of the group ring of a p-group over a modular field -Trans. Amer. Math. Soc. 50 (1941), 175-185. 2. BUCKLEY, J.: Polinominal functions and the wreath products - Ill. J. of )fath. 14 (1970) 274-282. 3. SANDLING, R.: Modular augmentation ideals - Proc. Camb. Phil. Soc. 71, (1972) 25-32. 4. SA:t-.l)LING, R.: Subgroups dual to dimension subgroups - Proc. Camb. Phi!. Soc. 71 (1972) 33-38. 5. liloRLEY, L. J. and PERKEL, M.: The nilpotency class of extensions of certain p-groups -Communications in algebra (8 (ll). 1053-1069 (1980). 6. :M:ARCONI RICCARDO: Sulla classe di nilpotenza dei prodotti intrccciati - Bollettino U.M.I. Algebra e Geometria Serie VI. Vo!. II-D.N.l. 9-19. (1983). 7. J1Al-{ATOlll, n.: 0 CKpYKType CIJJleTeHH5I 1\ByX l.\HKJIH'IeCKJlX rpynn nOp5I1\KOB cTeneHeH npocToro tIHCJla. PubI. Math., 1974, tom. 22. pp. 293-305. 8. flAl-{ATOlll, n.: l{JIacc H!fJlbnOTeHTHOCTH KpaTHblx clmeTeHHH l.\HlOIlltIetIKHX rpynn nOp5ItJ;KoB CTeneI-IH npoCToro tIIlCJIa. Pub!. l.Iath. 1985. tom. 31. pp. 153-156. Piroska LAK.ATOS H-4010 Debrecen, Pf. 12. 

MODULAR GROUP RINGS AND WREATH PRODUCTS OF GROUPS

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MODULAR GROUP RINGS AND WREATH PRODUCTS OF GROUPS

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