Group algebras of finite abelian groups and their applications to ...

Download PDF
3 downloads 0 Views 308KB Size Report
Comment
Oct 15, 2005 - concerning the Davenport constant, we refer to [16, Chapter 5]. Note however, that the problem of determining d(G) for all finite abelian groups ...
    

        

         

            

                 !      

                                                                    ½          ! "                " #           $                       %&       ' $( )*+  , (  "# $# %     -     %&     #  . /   0     1            0  (    &( 2 3 4  (            

    (  '# ' 1     (                    5 #    !               ½          ! "                "   %&(      (           " 

!  ½      











!









   (&      !"#    $%&'( )*+& '&$* %!"#(  ,&!("

-"&*.& 1( //+0& 2!3 1( +* '&&*(-"&( ("##!+&/ * #+ '   2+1 %*+ *"'& 45 */ &.&0&/ ' +1& "(+* .&*.&  6!7&.+! ' +1& &/+!( !* "*/ .+!'&  */ 8559:; * &0(&/
6"3

        

6"7

  ! "8 ( !  "(      



 



9         +              #   #     ' $( )*+  ,             &( # 333  33: 

(  / 

        - !  ;           -   

  ; 





  / .  





     !   !   /       !   )) 

1    (   Æ                     ( )  ! )     ! (  $( 2  ?37  # :/@ #(      (     (  +       )  ,                    %&      (   1           (          ( *# +( &( 2 3     A 3   (          ½  

           













#              -     A ;  ? 

 =    

6"B

(              

  /                      







    /    !   

/













/







½   ½

;     /

    / 



!  ½      









! "  

 

       

 !       +               + / + /   ;          

     -    ! "

!!, +

         ;                   

     5   ½        ½    ! "  

              !    !    !  



2  /;   # //      (                   '-  @;                  &          ½  /

   

·½



/ ! "   

      # # //          2     # //     & (      C      -(  @       ! ))  ))   ! ))( # //     D (  +*# +(           (   !            ;  ;(

        

6"6

              ½  /       /   %  %  A ;  -           A ?       #              # //         #   # /(/   2  /;     A @ 1 (  A 3    # //                         (  # .# -# ++# +'

+              !

"  1   )       !  )      (   !      1     (          





















        (           1   (                        "(   &             (           )(



.    .   ! "    



!    !   ! "

  (                  2 (         &( 2 3 E         (                 (                     ( /        1  

!   

 



     

     (   !     + !

  



 



  (  .    . 

/  



 =   

6":

                 !            #                  (                 % E   &( 2 3          +       (      .              %&      + / 1    (    &

(

' / !01  (          (          / ! "       #              &               (        



  



   !

























  



E             !               %&               %&      #    

5 



      

 



!



    &  1  

 5 



 









    

                

  &   2  (   *      

  (    *          %&     2 (      (          



    %&             (     

   &         

        

6/"

     !     (    &   (    +  !    !        ( - !      !    ! /                       ! /     /   / E   =    ! =               '     =       -  

&   5            





  ! 



  











1  -( A /B(            !           (    (                ! (     !    !  (       (                !             (  ! =    (        (           ! "       ! "      =    (  &( D 33;  (   ! "     =    (     -  (                 

 ! /       

 

 





E             

/  '            !         /                    ! / ;              /  /  













 ½   /



   





/ !  "





  

 =   

6//

"

 E     ;         (   5           &    !    # )  !   /   /    (  /   !   



!





 







 







!    







  

A !              (   ! "   ! /(      ! "(         %&       ! /(    / ! "     /    %&     =          %&     ; E      ½  /

   

  /   A   (  -        !  

    # ! / ! "  ! / ! "    / (

    







/















/  '+

        /                                 ;   )     )         )  "

   ! -  / A   ! /     (   !      (                     !   ;  5 )   ))      #   !  ! "     ))   /      %&   )  #  -    )  

        

6/;

  ! "   ! "(        ! "    !   ! "(  

 )



 #

- 2 2 (   2 !!, + D @/       # //  2  /; /  -



      !

   

                       ½           ! " 







/   



!

"  



/ 

    " . 



!





 

/







  

 



                " .   

         

"  



 







   



 



 













              !  ;  ! /  

  /   F                      

 . /   #           #  F         "   /  

  /    ! "       /    5      #        ½         #       



;

 

 # .



   ! #        









 

# 

              +  !  !

/  / /  



 

















 =   

6/?

"

 / E 

 









" 

 



 "  

!







           " 

   ".  ! 

D









 

/

 







  " 













(  " !  "  





















   "   



8

 (   "  (

 







 "  

   "









     ".  ! ".  



   



!

  

    " .  



1  ! "(       /(   /   E  













  



" .  .   









#     (    D  $  (             " ! "   "  ( (    (  " .  .    .  ! "   "   $ D (              ! !  ! / #                            /     /                   /     /   (            ½           ! " #    

 ½ ½  ½

      ! "

              ;  #        ½     $ / 

-    5  ! F "  /    "  /     







  







 





 









        

6/@

#        ! #          #                "         F  /   /      /   / A         (   "      "!#.



 # 

 





     " ! # 

 



 !

 









 "   





A   ! /     / (          "           (    /        =    

#       



½  

 



       (     

     !F   /  F ! 











 



 



   









"

    // E            ! /     / (  D ?/           (         ! /     /    !      ½           ! "(  D @/     /   ! /  /  





 











    ½  

   





  

!



  !

 / /  











 =   

6/3

1    . /     !       D @/    /   !





 !"





    





 

!





  !







(   

/ /   (  



 $   &    

 %      % 5 $ $  & (          





 %

Æ

   % $ Æ

 % $





  / $



=     ½          ½          .

  



 / /  .   / /   / / .

 





  













! !





















/ /   / 







  







 

/

    













 













   



½   ½

/

 



 "

   ½          ! " #             (    + !

/







        

6/7 





/



/

½   ½

!

 / / 

 















         ! /           - + !



/  

/



/.









"

 ( 

  /



      "

  $

  /;     $ # //                 /   /(     E      -              ½  

   



½ 







! "

  

    F !   =     ! / A   (      ! /          =    (     F   ;(  F !        (  &  /   /  F ! &  / 

!  ½ 



/

  







  

  











/



 

  ! "     =      ! "   +       !    ! (        #     '           (      ! '   (          ( ) ! 3( (       

 =   

6/B

)  Ô  -     Ô ) ! ) # Ô ! (  "( 2 G/(     5        $ 2 /  -          )  

!  ½      ½   ! "(    )  Ô(            

 











"!





 ! 



    



½













½ 



 







 

4-      -                 &         ½  /      ·½  / ! "  )  

4/! -+        &    &    (  *     (      ( ! *

 . *







 





! /  



/  

/











 



 

E     ! " A      5 )    (      ! )(    (  Æ    ! "                        3 #        (   Æ      ! "     =     #(    =       ! /  * ! /(      ! " #      ! (  * ! (  (  (             /   / #      /   /    . /  "   



! /  ( /  ( 

/ 







(

 

! "

6/6

        

.  5 ! 2 5  6                             2       E                    (         

  1   (  Ë       /              E      D                    *    *    (     +  Ë      . *     . *     (        D  (                      *    *                      *    * (   . *     . *    . *     . *   4               !   /     / ( * ! "    /   /  * ! (             (           . *     . *     (        ;(   /        D D 

7  .(   ;?  ++( 2* ?                

     -       /   /  "   D  # *            (               &   0     .( # ; # //       0  D 3/ C            # 3;   2  3?  ;  ?  2  3?     - 

 =   

6/:

 ,          1 

       

/   ! 

      ,( 

/

 





 



 



  G -(      &     ,(  D  !

Ë









 









!

,

                  , ! )       

/  .  ,             -      - ,   -  - ! 0        





 









Ë  



 





 

D  -

 - 



"

 #    .(  3

( .+

                     *    *              

    #      +  Ë      . *     . *

   5 /        0   ½        ! "  )  ;     D   ½              )  ?   

  

 

!



 

   *  *

   + / +    



 





/       

        

6;"

"

 $ D 3/(   



0   ½        D   ½       !   

 ´ µ



Ë  







   ´µ





  Æ     0   ½        D   ½       ! " (   -  +  Ë  



   

 

 ´ µ





   ´µ

! " 

    . *     . *   / $       (    D   ½       ! H  ) =( D   ½          %&   )   D ?;(   0   ½        D   ½       !  "

;  Æ     0   ½     !  " $ D&  ?/        / ! "



  



0   ½ 



)





      

!





  /( !    



 







 



 / !  "

? E      ½           ½            (  ! );) ! " /         #  Æ    

! 0   ½       

E 

 



D   ½       !  "   

! 0   ½    !    

!

 









  











0   ½       

    /    





/ !  "

 =       

 







 

     /  

   

6;/





/   





/ !  "  # //

!!, .'

                     *    *              

    #      +  Ë     . *     . *    5 /          

  

 

!



 

     

/

;      ; / . 6 . /  ?   *    *      ; / . @ . /       

      



"

 / E   0   ½ 





!

 







 

   





/



A        ) (      %  # //(      # // ; A / / 

+ !     ; /     



     / ? A        *  

!



 

 

 *

 (    

   *  *

 



        

6;;



  + !

 

/

   

.

/

 *  *

 ;    /



;

      # 3;?

   /2 E         ;"":(            I&E A    %  A  > *   *  0 & )J &A&A%  +- 89!0  E    1  )           D @/ # +          1    2     +  +        !'  6:; : ?9  &#           F   G#! +1  64; ::5 :49  ! */  > *%          (&$ > +1  6?; :5 :8   &!$/*%&          ' > "'& 1&!  699:; 4 4   &!$/*%& */ $+&!.1 (#       )       "& ##$ +1  1#* A $$H  4   *%        */ &/ #*%& 99?

   $&( */   &1%$        $"2& ./&. "'$(1&( &2 B!3   >  $(!*              > "'& 1&! 6949; 8    &*%            > "'& 1&!  6985; ?4 5            > "'& 1&!    6985; 8 4    *&0$ *     F " +       & +1 !*+1$  6999; 8? 8  1 & "*  , !         > !' 1&! &   6:; 9 :?          +1 &( &++&( 

68; 4: 54

                 

   

    !    " #

  $    % $& ' ( &    ) ( *   

 + , ) ( - 

   !"#

    !    " #

  $    % $& ' ( &    ) ( *   

 + , ) ( - 

  "#$ !"#