Recent Advances in Mathematics, Statistics and Economics Weakly Extension Mostafa Zeriouh, M’hamed Ziane, Seddik Abdelalim and Hassane Essanouni   p Abstract—In this paper, let abelian p  group of A an A while there x  A exists x mx  if such where extension property and A1 is the first Ulm subgroup of A . Keywords—Abelian goups,  A that  B satisfies the weak II. MAIN RESULT A an abelian p  groups and  an automprphism of A which satisfies the weakly extension Theorem 1. Let p  group, order, direct sums of cyclic groups, basic subgroups, monomorphism group, automorphism group. property. If for all x  A such that: o( x)  p r where r  *       A   x   A where A is a subgroup of A I. INTRODUCTION I N 1987, P. Schupp showed, in [3], that the extension property in the category of groups, characterizes the inner automorphisms. M. R. Pettet gives, in [4], a simpler proof of Schupp’s result and shows that the inner automorphisms of a group are also characterized by the lifting property in the category of groups. The automorphisms of abelian p  groups having the extension property in the category of abelian p  groups are characterized in [1]. then there exists mx  such that  ( x)  mx x  A1 Proof x  A such that o( x )  p r where r     and A  x   A where A is a subgroup of A Since  ( x)  A then there exists mx  such that Let  a class of abelian p  groups and let A  . We say that automorphism  of A has the weak extension property if for all B   and any monomorphism of groups   A  B and if there exists an element m in  such that the restriction of  to mA is an isomorphism between mA and mB then there exists   Aut ( B) such Definition 1. Let  ( x)  mx x  a   where a  A A   B1  This work was supported in part by the Department of Mathematical and Computer Sciences , Faculty of sciences , University of Mohamed first, BP.717 60000, Oujda, Morocco. M. Zeriouh: Department of Mathematical and Computer Sciences , Faculty of sciences , University of Mohamed first, BP.717 60000, Oujda, Morocco. [email protected] M. Ziane: Department of Mathematical and Computer Sciences , Faculty of sciences , University of Mohamed first, BP.717 60000, Oujda, Morocco. S. Abdelalim: Laboratory of Mathematics, Computing and Application, Department of Mathematical and computer, Faculty of sciences University of Mohamed V Agdal, BP.1014 . Rabat, Morocco. [email protected] H. Essanouni: Laboratory of Mathematics, Computing and Application, Department of Mathematical and computer, Faculty of sciences University of Mohamed V Agdal, BP.1014 . Rabat, Morocco.  1 (1) a  ( A ) , assume the contrary: a  ( A  )1 .  Let B a basic subgroup of group A therefore by theorem 32.4 [3] we have:  Bn    xni   iI n B   Bn with  n 1 o( xni )  p n , i  I n  and with Bn   B j , n  1 , we have: n  1 , Prove that that the following diagram is commutative: ISBN: 978-1-61804-225-5 B  is a direct  ( x)  mx x  A1   Aut ( A)   . We defined weak extension property and we establish the result follows: for all summand a prime number and A  j n B  A n n where An  Bn  p n A  m  1 such that: a   b1   bm  am (2) where all the bi  Bi , am  Am and bm  0 . so there exists G such that G  y   A  r m y  A and o( y )  p . Consider the group With 172 Recent Advances in Mathematics, Statistics and Economics We define a group homomorphism to G  y   A   A  x   A  from  ( x)   ( p m y )  p m ( y )  and since  ( y) is an element of G  y   A while  ( y)  ky  b1'   bm'  am' (6) by:  ( x )  p m y   (bi )  bi , bi  Bi  (a )  a , a  A m m m  m (3) where all the bi  Bi and am  Am then by  (3) and (6) we have : is clearly a monomorphism of groups. Indeed, if a  ker ( ) then  (a)  0 and by i.e.  ( x)   ( p m y )  p m  ( y) (mx  a  )   A  such that a  mx x  a  i.e.  p m (ky  b1'   (mx x  a  )  0  (2) we have: 0   (mx x  a ) 0   (mx x  b1   bm  am ) (3) and 0  mx p m y  b1  show that: i.e. 0  mx p m y  a  then 0  mx p m y  a  which implies that p r  m  mx p m i.e. pr  mx t  thus mx  tp such that  bm  am  kp m y  p mam' (8) Consider now the projection prm of G on Bm (8) implies that : prm (mx p m y  b1   bm  am )  prm (kp m y  p mam' ) bm  0 i.e. But since p r m A p r m thus the restriction of ' A and p  to p r m while i.e. r m  is a monomorphism. G p A to p A i.e. by then   ( x)  mx p y  b1  and by A. ACKNOWLEDGMENT G REFERENCES  [1] S. Abdelalim and H. Essannouni, Characterization of the automorphisms of an Abelian group having the extension property, vol. 59, Portugaliae Mathematica. Nova Srie 59.3, 325-333, 2002. [2] L. Fuchs, Infinite Abelian Groups, vol. 1 Academic press New York, 1970. [3] P. E Schupp, A Characterizing of Inner Automorphisms, Proc of A.M.S V 101, N 2. 226-228, 1987. [4] M.R. Pettet, On Inner Automorphisms of Finite Groups, Proceeding of A.M.S. V 106, N 1, 1989. G M. Zeriouh: Department of Mathematical and Computer Sciences , Faculty of sciences , University of Mohamed first, BP.717 60000, Oujda, Morocco. [email protected] M. Ziane: Department of Mathematical and Computer Sciences , Faculty of sciences , University of Mohamed first, BP.717 60000, Oujda, Morocco.  bm  am (5) (3) we have: ISBN: 978-1-61804-225-5 satisfies the weak I would thank professor Abdelhakim Chillali for his helpful comments and suggestions.  x  A ,  ( x)  ( x) (4) (1) and (2) we have :  ( x)   (mx x  a  )   (mx x  b1   bm  am ) m   Aut ( A) extension property and A is the first Ulm subgroup of  A where 1   a   A1  ( x)  mx x  A 1.  ( x)  mx x  A1 A is an isomorphism from diagram is commutative:  ( A  )1  A 1 x  A if  x  is a direct summand of A while there exists mx  such that ' r m A a   ( A  )1 . We proved that for all r m G ( more precisely:   prm  id prm ). Using the fact That  checks the weak extension property then there exists   Aut (G) such that the following p (7) mx p m y  b1   bm  am which is absurd , hence then ker ( )  0 which implies that The other hand we have:  bm'  am' ) (4) , (5) and (7) show that : r mx x  a   0 i.e. r m  kp m y  p m am' mx x  tp r x  t  0  0 hence ' ' 173 Recent Advances in Mathematics, Statistics and Economics S. Abdelalim: Laboratory of Mathematics, Computing and Application, Department of Mathematical and computer, Faculty of sciences University of Mohamed V Agdal, BP.1014 . Rabat, Morocco. [email protected] H. Essanouni: Laboratory of Mathematics, Computing and Application, Department of Mathematical and computer, Faculty of sciences University of Mohamed V Agdal, BP.1014 . Rabat, Morocco. ISBN: 978-1-61804-225-5 174