Unit Groups of Semisimple Group Algebras of Abelian p-groups over a Field*

Transcription

1 JOURNAL OF ALGEBRA 88, ARTICLE NO. JA96686 Unit Grous of Semisimle Grou Algebras of Abelian -Grous over a Field* Nako A. Nachev and Todor Zh. Mollov Deartment of Algebra, Plodi Uniersity, 4000 Plodi, Bulgaria Communicated by Susan Montgomery Received December 8, 995 Let G be an abelian -grou, let K be a field of characteristic different from, and let KG be the grou algebra of G over K. In this aer we give a descrition of the unit grou UKG of KG when Ž i. K is a field of the first kind with resect to and the first Ulm factor GG is a direct sum of cyclic grous and when Ž ii. G is an arbitrary abelian -grou and K is a field of the second kind with resect to. The indicated descrition of UKG extends results of Chatzidakis and Paas. 997 Academic Press J. London Math. Soc , 93. INTRODUCTION Let G be a torsion abelian grou and let K be a field such that the characteristic of K does not divide the orders of the elements of G. Let UKG be the unit grou and let VŽ KG. be the grou of augmentation in the grou algebra KG. Berman and Rossa, 3 have described the -torsion subgrou SŽ KG. of VŽ KG. when G is a countable abelian -grou. Nachev, 3 has given a descrition of the torsion subgrou tvž LG. of VŽ LG. when G is an abelian -grou and L is a commutative ring with identity and of characteristic not equal to such that L contains n the th roots of unity, n. Karilovsky 6,.5 Theorem,. 6 has determined the isomorhism class of UŽ G., i.e., he has given a descrition of UŽ G. when G is a finitely generated abelian grou. Mollov 7, 9 has described the torsion subgrou tvž KG. of VŽ KG. when G is an abelian -grou. He has also described VŽ KG. 8 when Ž. a G is an infinite *Suorted by the National Scientific Fund of the Ministry of Science and Education of Bulgaria under Contract MM $5.00 Coyright 997 by Academic Press All rights of reroduction in any form reserved. 580

2 UNIT GROUPS 58 direct sum of cyclic -grous and K and Ž b. G is an abelian -grou and K. Chatzidakis and Paas 4 have described UKG when the torsion abelian grou G is a direct sum of countable grous. Let G be an abelian -grou and G G n n. We give a descri- tion of UKG in the case when at least one of the following conditions Ž i. or Ž ii. is fulfilled: Ž. i the first Ulm factor GG of G is a direct sum of cyclic grous and K is a field of the first kind with resect to. Ž ii. G is an arbitrary abelian -grou and K is a field of the second kind with resect to. These results extend the mentioned investigations of Berman and Rossa, Nachev, Mollov, and Chatzidakis and Paas. The descrition of UKG, given by us, has been announced in 4. This aer is organized as follows. In Section we set u notation. In Section we give a descrition of UKG under conditions Ž i. or Ž ii... NOTATION AND CONVENTIONS Throughout this aer G denotes an abelian -grou n n n 4 n G g gg, n,g G and K a field of characteristic not equal to, i.e., char K. Let be the grou of the n th roots of unity with n ranging over. The field K of characteristic not equal to is called of the first kind with resect to if ŽKŽ.: K. is infinite; otherwise it is of the second kind with resect to. All direct roducts of grous are assumed to be restricted direct roducts, i.e., they are direct sums and the concet direct roduct will mean a restricted direct roduct. Let G denote the restricted direct roduct of coies of G, where is a cardinal number. Let K* be the multilicative grou of the field K and let K be the -comonent of the torsion subgrou tk* of K*. If is a cardinal number then we shall use the notation K* of Chatzidakis and Paas 4, which in our case will be K* K*. Let K be a field of the first kind with resect to and let i, i, be i a rimitive th root of unity. Then the grou KŽ. i, where i if and i if, is cyclic. Hence for this i there exists a ositive integer such that K i. We call the number the constant of the field K with resect to. Let be the cardinality of. We set G n gg 0

3 58 NACHEV AND MOLLOV g n, 4 n. Let KŽ. i * be the multilicative grou of the field KŽ. i, i. The set s Ž K. i 04 KŽ. KŽ. 4 i i is said to be the sectrum of the field K with resect to 9.. DESCRIPTION OF UKG In this section we shall make use of the descrition of UKG given by Chatzidakis and Paas 4, Theorem 3.6, when G is a torsion abelian grou which is a direct roduct of countable grous. Let be the grou of all n th roots of unity and let n be a rimitive nth root of unity. THEOREM. 4, Theorem 3.6. Let G be a torsion grou, such that its -comonents G are direct sums of countable grous. Let P be the set of rimes such that G Ž.,and let D be the maximal diisible subgrou of G and D D. For eery finite S P and eery integer n relatiely P rime to the elements of S, we define a cardinal number mž S, n. as follows: Ž. a m, ; Ž b. if Ž G. Ž.,then mž 4,. G, otherwise mž 4,. 0; Ž. c ms, mž 4,. S ; Ž d. ms,n ms, i, where i KŽ.: K a GD n n n a has or- der n 4. Then mž S, n. ž Ž n. S/ S, n UK G K,. In the following theorem conditions Ž. and Ž. 3 give a more recise form of conditions Ž. 4 and Ž. 5 of Theorem 8 of. THEOREM., Theorem 8. Let G be an infinite abelian -grou such that GG is a direct roduct of cyclic grous and let K be a field of the first kind with resect to. If H is another abelian -grou then KG KH as K-algebras if and only if the following hold. Ž. KGG Ž. KŽHH.. Ž. If either G or H is infinite or triial, then G H. Ž. 3 GG and G ZŽ. if and only if HH 0 0 and H ZŽ..

4 UNIT GROUPS 583 In fact this theorem is equivalent to Theorem 8 of since KGG Ž. KŽHH. and, Lemma imly condition. of Theorem 8, i.e., HH is a direct roduct of cyclic grous. LEMMA.3. Let H be a subgrou of the infinite abelian grou G and let GH be a direct roduct of cyclic grous. Then G G0 G, where G is a direct roduct of cyclic grous, H G 0, and Ž. i if H is infinite, then G H 0 ; Ž ii. if H is finite, then G. 0 0 Proof. By the theorem of Szele 5, Proosition 6., there exists a ure subgrou C of G such that H C and C H,if His infinite and C 0,if His finite. Since GH is a direct roduct of cyclic grous, there exist subgrous G0 and B of the grou G such that GH G0H BH, G C, 0 and C G 0. Hence G G0B, G0BHC G 0. It is not difficult to see that G0 is ure in G. Since GG0 BH is a direct roduct of cyclic grous, it follows from the theorem of Kulikov 5, Theorem 8. that G G0 G, where G is a subgrou of G. The fields KŽ,. in formula n of Theorem. can be isomorhic by different integers n but their reetitions are not taken into account. In the next result we eliminate this deficiency using the sectrum s Ž K. of the field K for the case when G is an abelian -grou. Actually, in the following theorem we give a descrition of UKG when G is an abelian -grou such that GG is a direct roduct of cyclic grous and K is a field of the first kind with resect to. This result exends the results of Chatzidakis and Paas 4, Theorem 3.6, when G is an abelian -grou. THEOREM.4. Let G be an abelian -grou such that its first Ulm factor GG is a direct roduct of cyclic grous and K is a field of the first kind with resect to with char K and with constant with resect to. If GG is Ž. of finite exonent ex GG, then we set ex ŽGG. and max,. If GG is of infinite exonent then we set. Then where j UŽ KG. K* KŽ.* KŽ.* j j, if, ½ Ž GG., if, 0, if G, Ž 3 ½. G, if G, and the numbers j,,... are defined in the following manner. j

5 584 NACHEV AND MOLLOV If j, then and j j Ž GG. Ž GG., if GG 0, Ž K Ž.: K j j. Ž 4. GG, if GG 0 Ž GG., if and GG 0, Ž K Ž. : K. Ž GG. Ž GG. KŽ. : K if and GG 0, 0, if and ex Ž GG., GG, if GG and ex GG. 0 Ž 5. Proof. Ž. i We consider the case when G is finite or a direct roduct of countable grous. In the notation of Theorem. we have P 4 and DD. Then S or S 4. Let D be the maximal divisible sub- grou of G. Then n n Ž GD. Ž GD. i n, n. KŽ.:K By Theorem. we have n Ž n. S S, n ms,n UŽ KG. K,. Ž 6. If GD is of finite exonent then we set ex Ž GD. and maxž,..ifgdis of infinite exonent then we set. In order to obtain Ž. we consider the only ossible cases Ž. I S and Ž II. S 4. Ž. I Let S. Then n 04 and taking into account the definition of the sectrum of K we get in Ž. 6 a direct factor n mž,. Ž n. Ž j. K * K* K *. n0 j j

6 UNIT GROUPS 585 Ž 0. Ž r. r Since m, and m, i we obtain, if, ½ i, if, j i j, if j, 7 Ý r Ý r i r, if and j, i r, if and j. Ž II. Let S 4. Since Ž n,. then by roerty Ž d. of Theorem. it holds that n. Consequently and in Ž. n 6 we have a direct factor KŽ.*, mž 4,.. The selection of the direct factors of UKG examined in the only ossible cases I. and II. shows that j UŽ KG. K* KŽ.* KŽ.* Ž 8. j j holds. Proerty Ž. b of Theorem. shows that for Ž. 8, i.e., for Ž. formula Ž. 3 is fulfilled for. We shall obtain reliminary formulas for and. From Ž. j 7, for we obtain Ž GD., i.e.,, if, ½ Ž 9. Ž GD., if holds. Besides Ž. 7 in view of the formula for i j easily imlies j j Ž GD. Ž GD. j, j,,... KŽ.: K Ž j. Ž GD. 0, Ž KŽ. : K. if, Ž GD. Ž GD., KŽ.: K if.

7 586 NACHEV AND MOLLOV Therefore if G is finite or a direct roduct of countable grous then Ž. 8, Ž.Ž. 3, 9, and Ž 0. hold. If we relace GD by GG in the formulas Ž. 9 and Ž 0. we obtain Ž., Ž. 4, and Ž. 5 and therefore Ž.. This comletes the roof of the theorem in this case. Ž ii. Let G be an infinite grou which is not a direct roduct of countable grous. We shall rove that G is infinite. If we suose that G is finite. then, by Lemma.3, G G G, where G G, G , and G is a direct roduct of cyclic grous. Hence G is a direct roduct of countable grous which is a contradiction. Therefore G is an infinite grou. We define a grou H ŽGG. D, where D Ž.. G Then it follows from Theorem. that KG KH. Therefore UKG UKH. Since H is a direct roduct of countable grous, we may aly to the grou UKH the case Ž i.. Then H D, H G, H G, HH GG Ž. Ž. and ex HH ex GG comletes the roof of the theorem. Remark. When the grou G is infinite GG is a direct roduct of cyclic grous and G is a finite grou; we can relace the cardinalities ŽGG. n in Theorem.4 by G n. Really, ŽG G. n G n for all n, which imlies ŽGG. n G n. Chatzidakis and Paas 4, Theorem 3.4 have given a descrition of UKG in terms of the grou G and the field K when G is a countable abelian -grou and K is a field of the second kind with resect to. In the following theorem we generalize this result for an arbitrary abelian -grou. THEOREM.5. Let G be an abelian -grou and let K be a field of the second kind with resect to. Then where. If, then UŽ KG. K* KŽ.* G, if G 0, Ž K Ž.: K. G, if G. 0

8 UNIT GROUPS 587 where. If, then UŽ KG. K* KŽ.* Ž 3. GG K Ž.: K GG, if G, Ž 4. G, if G 0 and G, 0, if G. Proof. Ž. I If G is a finite grou or a direct roduct of countable grous, then we rove the theorem similarly to case Ž. i in the roof of Theorem.4 bearing in mind that KŽ. KŽ. for and KŽ. KŽ. for. Ž II. Let G be an infinite grou which is not a direct roduct of countable grous. Then in view of, Proosition 7 the algebra KG has a grou basis H which is a direct roduct of countable grous. Therefore KG KH and UKGUKH. Consequently and Ž 3. hold for UKH with the requirements and Ž 4., resectively. We shall consider two subcases: Ž. and Ž.. Ž. Let. Then in view of H G we have UŽ KG. UŽ KH. K* KŽ.*K* KŽ.*, 0 H G where the second isomorhism is obtained from and which are alied to the grou H. Therefore is valid for UKG with requirement Ž.. Let. Then Ž 3. and Ž 4. hold for UKH, i.e., H, if H, UŽ KH. K*KŽ.*, ½ Ž 5. 0, if H. HH We consider two subcases:. K KŽ. and. K KŽ... Let K KŽ.. Then it follows from Ž 5. U KH K* K*. H G Therefore formula Ž 3. is fulfilled for UKG with requirement Ž 4... Let K KŽ..

9 588 NACHEV AND MOLLOV.. Let G. Then by, Corollary H and HH GG hold. Then Ž 5. imlies UŽ KG. UŽ KH. K* KŽ.*. GG Since H G then Ž 3. is fulfilled for UKG with requirement Ž Let G. Then in view of, Corollary we have H. Then Ž 5. imlies U KG U KH K* K*, H G i.e., Ž 3. is fulfilled for UKG with requirement Ž 4.. The theorem is roved. REFERENCES. S. D. Berman and T. Zh. Mollov, Isomorhism of semisimle grou algebras of abelian grous, Pliska Stud. Math. Bulgar. 8 Ž 986., 07. Russian. S. D. Berman and A. R. Rossa, The Sylow -subgrou of a grou algebra of a countable abelian -grou, Dooidi Akad. Nauk. Ukrain. RSR Ser. A Ž 968., Ukrainian 3. S. D. Berman and A. R. Rossa, The Sylow -subgrous of the grou algebras of countable abelian -grous, in Proc. of the XXIX Scientific Conference of Professors and Instructors of the Staff of the Uzhgorod University, Det. of Mathematical Sciences, Uzhgorod Ž 975., Russian 4. Z. Chatzidakis and P. Paas, Units in abelian grou rings, J. London Math. Soc. 44 Ž 99., L. Fuchs, Infinite Abelian Grous, Vol. I, Academic Press, New York, G. Karilovsky, Commutative Grou Algebras, Dekker, New York, T. Zh. Mollov, On multilicative grous of semisimle grou algebras of abelian -grous, C. R. Acad. Bulgare Sci. 35 Ž 98., 696. Russian 8. T. Zh. Mollov, On multilicative grous of real and rational grou algebras of abelian -grous, C. R. Acad. Bulgare Sci. 37 Ž 984., 553. Russian 9. T. Zh. Mollov, Sylow -subgrous of the grou of the normalized units of semisimle grou algebras of uncountable abelian -grous, Pliska Stud. Math. Bulgar. 8 Ž 986., Russian 0. T. Zh. Mollov, Multilicative grous of semisimle grou algebras, Pliska Stud. Math. Bulgar. 8 Ž 986., Russian. T. Zh. Mollov, UlmKalansky invariants of Sylow -subgrous of normalized units of semisimle grou algebras of infinite searable abelian -grous, Pliska Stud. Math. Bulgar. 8 Ž 986., 006. Russian. N. A. Nachev, UlmKalansky invariants of the grou of the normalized units of grou algebras of abelian -grous over a commutative ring in which is non unit, C. R. Acad. Bulgare Sci. 33 Ž 980., Russian

10 UNIT GROUPS N. A. Nachev, UlmKalansky invariants of the grou of the normalized units of grou algebras of abelian -grous over a commutative ring in which is non unit, Pliska Stud. Math. Bulgar. 8 Ž 986., 33. Russian 4. N. A. Nachev and T. Zh. Mollov, Unit grous of semisimle grou algebras of abelian -grous over field, C. R. Acad. Bulgare Sci. 46 Ž 993., 79.