In this paper, we shall define new subgroups which are called pure−1−2−3 in abelian groups Gn×Gm for all n,m∈N which are a family of pure subgroups.
In[1],[2]H.M.A.Abdullah &nbsp;gave the some general properties of&nbsp;pure− 1−2−3 in abelian group&nbsp;G, but here, we shall prove more than properties on this subgroups in ModGn×Gm, which are not valid for pure subgroups

Abdullah, H. M. A.

Hameed, Saif Zuhair

English

In this paper, we shall define new subgroups which are called pure−1−2−3 in abelian groups Gn×Gm for all n,m∈N which are a family of pure subgroups.

In [1],[2] H.M.A.Abdullah  gave the some general properties of pure− 1−2−3 in abelian group G, but here, we shall prove more than properties on this subgroups in Mod Gn×Gm, which are not valid for pure subgroups

H. M. A. Abdullah

Saif Z. Hameed

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

ON DIRECT PRODUCT pure − 1 − 2 − 3 SUBGROUPS IN ABELIAN GROUP Gn×Gm

Scitech Research Journals

    Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                   ISSN: 2395-0218      Volume 12, Issue 2  available at www.scitecresearch.com/journals/index.php/jprm                                           1821|        SCITECH                           Volume 12, Issue 2 RESEARCH ORGANISATION        Published online: July 08, 2017| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals On direct product pure-1-2-3 subgroups in abelian group mn GG   H. M. A. Abdullah and Saif Z. Hameed* Al Mustansiriyah University,college of basic education, Department of Mathematics. *Email: saifzuhair@yahoo.com  Abstract.   In this paper, we shall define new subgroups which are called 321 pure  in abelian groups mn GG   for all Nmn ,  which are a family of pure subgroups. In [1], [2] H.M.A.Abdullah gave the some general properties of 321 pure  in abelian group G , but here, we shall prove more than properties os this subgroups in Mod mn GG  , which are not valid for pure subgroups.  Keywords: Subgroup; Abelian group; Direct product; Pure; Pure-1; Pure-2; Pure-3.  1. Introduction  we shall use following definitions, to get the results. Definition 1.1 A subgroup S  of G  said to be 1pure  in G  if for all Sx  and for all prime p , xpx = .   Remark 1.2      1.  If S  is 1pure  in G  then Sx  we have Zknp xnk ,|  in S .      2.  Every 1pure  subgroup is pure  in G .      3.  Every 1pure  subgroup is divisible.   Definition 1.3. Let BA,  are 1pure  subgroups in mn GG , . Then we shall called a direct product BA  is 1pure  in mn GG   if A  and B  are 1pure  in mn GG , . Which means that BbAaBAbax  ,),(= then ),(=),(=),( bapbpabap . Now, we are ready to show some results relating to the 1 puredirect . Theorem 1.4 Let 11 BA   and 22 BA   are two direct product 1pure  in mn GG  . Then       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                   ISSN: 2395-0218      Volume 12, Issue 2  available at www.scitecresearch.com/journals/index.php/jprm                                           1822|            1.  2211 BABA    is 1pure  in mn GG  .      2.  2211 BABA   is 1pure  Proof. 1.  Let 2211 BABAx    so 11 BAx   is 1pure , thus p  1111 ,),(==),( BbAaBAbaxbap  . And 22 BA   is 1pure , then 220000 ),(==),( BAbaxbap  . Therefore, we get  ),(=),(==),(=),( 0000 babaxbapbap  Thus,    ),(== 0 baxandaa 0= bb  so 2211=),(=),(= BABAxbabappx   . Thus, 2211 BABA    is 1pure .  2.  Let x  be any element belong to 2211 BABA   So, 222211112211 ),(,),()),(),,((= BAbaBAbababax   but, both 11 BA   and 22 BA   are 1pure . Thus p   ),(=),( 1111 babap  (1.1)  ),(=),( 2222 babap  (1.2)  By (1.1) and (1.2) we get xbababapbap =)),(),((=),(),( 22112211   xbabap =)),(),(( 2211  , but 22112211 ),(),( BABAbaba   So, xpx = .We obtain 2211 BABA   is 1pure .   Theorem 1.5 If BA  is any 1pure  in mn GG  , and KH   any subgroup of mn GG  , thenKHBA    is pure  in mn GG  .  Proof. Let x  any element in KHBA   . So KHxandBAx  . Since BA  is a 1pure  in mn GG  , so p  ( p  is prime number) xpx = .   But KHBAx   . Hence, KHBA    is 1pure .       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                   ISSN: 2395-0218      Volume 12, Issue 2  available at www.scitecresearch.com/journals/index.php/jprm                                           1823|        Easily to show the following:  Theorem 1.6  1.  If BA  is any subgroup of 1pure  subgroup BA  of mn GG  . Then BA  is a 1pure .  2.  If BA  is a 1pure  in mn GG   and BA  any subgroup of BA  then BABA is 1pure  in BAGG mn  Proof. 1.  Let BAx  , so BAx   but BA  is 1pure  in mn GG  , then p , we have BAxbabap =),(=),(  for some BbAa  , . So, BA  is 1pure . 2.  Let BABABAba),( . Since BAba ),(  and BA  is a 1pure  in mn GG  . Thus, we have p , ),(=),( babaP . Clearly BAbaBAbapBAbap  ),(=)),((=),( . Thus, BABA is 1pure  in BAGG mn.  Definition 1.7 A subgroup BA  is said to be 2pure  in mn GG   if BAxx  ,  and pp,  is prime number, then )(= pxppx .  Remark 1.8      1.  It is clear that any 1pure  is a 2pure .      2.  BAxx  ,  and if BA  is 2pure  in mn GG  , then we are ready to prove the following results of 2pure .  Theorem 1.9 Any 2pure  subgroup of Torsion-free group is 1pure .  Proof. Let timesBA |  be any 2pure  in Torsion-free mn GG   and BAx  , so )(=, pxppxp , put BAbax ),(= , thus )),((=),( bappbap , which implies that (0,0)=)),((),( bappbap   so (0,0)=))),((),(( bapbap  . But mn GG   is a Torsion-free. Then ),(=),( babaP , we obtain the result.   Theorem 1.10  Any 2pure  subgroup of Torsion-free group is       1.  pure .      2.  1pure .  Proof. 1.  Let BA  be any 2pure  in mn GG  , we claim that BA  is a neat in mn GG  .     Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                   ISSN: 2395-0218      Volume 12, Issue 2  available at www.scitecresearch.com/journals/index.php/jprm                                           1824|        Let BAx  , and suppose that xpp |,  in mn GG  , so BAmGnGpx. Thus,   .,),(= mmnnmn GgGgforsomeggpx   (1.3) Since BA  is a 2pure , so we can write )(= pxppx , so xpggpxppggpp mnmn22 =),()(=)),((   but mn GG   is Torsion-free we get xgg mn =),(  by (1.3) we can obtain )(),(=),(= BApggpggx mnmn  . Which means that BAp x | , 2).([3] qpby  We get the result. 2.  Let BA  be any 2pure  in a Torsion-free mn GG   for all BAx  , so for all prime p we have )(= pxppx , and let ),(= bax  for some BbAa  ,  then, )),((=),( bappbap , But mn GG   is a Torsion-free group. Thus, 0=)),(),(( bapbap   which implies that )),(=),( bapba  and we we get . But BAba  ),(  and for all prime p , ),(| bap  in BA . Moreover, BA  is a Neat in mn GG  .  By the above theorems we get the main results:  Theorem 1.11 Let mn GG   be a Torsion-free group then the following statements are equivalents:       1.  BA  is a 1pure ;      2.  BA  is a 2pure ;      3.  BA  is a pure .   Theorem 1.12  If BA  is 2pure  and a Neat subgroup of mn GG   then BA  is pure .  Proof. Since BA  is a Neat in mn GG  , then we have ppBApBAGGp mn ,),(=)(    is prime number. we shall prove the statement by induction. So let p  and  Zk  )(= BAPBAGGP kmnk    is true.  We have to show that  )(= 1 BAPBAGGP kmnk   .  The induction BAGGPBAP mnkk   11 )(  is obvious. Remained to show that )(11 BAPBAGGP kmnk    . Let BAGGPba mnk   1),(  and we may write ),( ba  in the form ),(=),( 1 mnk ggPba   for some mnmn GGgg ),( . Thus, mnkmnk GGPBAggpPba  )),((=),(  but BA  is a Neat. Thus, )(=)),((=),( BAPGGPBAggPPba kmnkmnk   . Therefore, ),(=),( 00 baPbak for some BAba ),( 00 . Since BA  is 2pure , then we get     Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                   ISSN: 2395-0218      Volume 12, Issue 2  available at www.scitecresearch.com/journals/index.php/jprm                                           1825|        BAPbaPbaPPPbaPPbaPba kkkkk   100100100100 ),(=))),(((=)),((=),(=),( . Assuming 1k . So )(=)( 11 BAPGGPBA kmnk   . Consequently BA  is a pure   Now we shall prove some properties of 2pure  subgroups, which are valid in 1pure .  Theorem 1.13  If BA  is a 2pure  subgroup of mn GG   then:       1.  DCBA    is also 2pure  for any subgroup DC  of mn GG  .      2.  If BA  is a 2pure  subgroup of mn GG   then any subgroup of BA  is also 2pure .      3.  The sum of any two 2pure  subgroups of mn GG   will be 2pure .      4.  If BA  is a 2pure  subgroup of BA , then DcBA  /  is 2pure .   Definition 1.14 A subgroup BA  of mn GG   is called 3pure  if ))( p ,  Zk(  and BAba  ),(  ),(=)),((=),( 1 baPbaPPbaP kkk    Remark 1.15      1.  Any 2pure  is 3pure .      2.  If 1=k , then a 3pure  is a 2pure .   Now, we are ready to prove the following results of 3pure  subgroups.  Theorem 1.16  If BA  is 3pure  and Neat subgroup of mn GG  , then BA  is a pure .  Proof. Since BA  is 3pure , then we have p , )(=)( BAPBAGGP mn   . We will prove the statement by induction. Let )(),( kp  , )(=)( BAPBAGGP mnk    is true, so we have to showing that )(=)( 11 BAPBAGGP kmnk    . Let ),(=),( 1 mnk ggPba   for some mnmn GGgg ),( . Thus, )(=)(),(=),( 1 BAPGGPBAggPba kmnkmnk   . Therefore, ),(=),( 00 baPbak for some BAba ),( 00 . Since BA  is 3pure , then )()),((=),(=),( 10000 BAPbaPPbaPbakkk   . Consequently BA  is pure .       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                   ISSN: 2395-0218      Volume 12, Issue 2  available at www.scitecresearch.com/journals/index.php/jprm                                           1826|        Now, we shall give the generalization of the Theorem 1.10.  Theorem 1.17 Any 3pure  subgroup of Torsion-free mn GG   is pure   Theorem 1.18 Any 3pure  subgroup of a torsion-free abelian group G  is 1pure  Proof. Let BA  be a pure  in a free mn GG  , let BAba ),( , then Zkp),(  we have )),((=),( bapPbaP kk (Assume that 1k ). So 0=)),(),(( baPbaPk   and since G  is torsion-free, then0=),(),( babaP   So ),(=),( baPba , we obtain that BA  is 1pure .  We know that the intersection of divisible ( pure), (Neat) subgroup is not divisible ( Neatpure ). But we shall show that if BA  and 11 BA   are two 3pure  and Neat in mn GG   then 11 BABA    is pure . First we need the following lemma:  Lemma 1.19  Let BA  be a 3pure  in mn GG   and 11 BA  , be any subgroup of mn GG  . Then ZkkpBAPBAPBABAP kkk ,()(,=)( 1111  . Proof. Let 11),( BAPBAPbakk   . Therefore, ),(=),( 00 baPba k  for some BAba 00 ,( . Since BA  is 3pure . Then ),(=),(=),( 002 baPbaPbaP kkk . Then )(),(=),( 1100 BABAPbaPbakk    we obtain the result.   Remark 1.20 The above lemma is satisfied for 21pure  subgroups.   Theorem 1.21 Let BA  and 11 BA   be two 3pure  and Neat in mn GG  . Then 11 BABA   .  Proof. By using Theorem 1.16 we have BA  and 11 BA   are pure . We need to prove 11 BABA    is pure . Let 11),( BABAba    and let us suppose that ),(| baP k  is solvable in mn GG  . Then ),(=),( mnk ggPba  for some mnmn GGgg ),( . Since BA  and 11 BA   are pure . Then ),(=),(=),( 1100 baPbabaPkk for some 111100 ),(,),( BAbaBAba   By lemma 1.19 we have )(=)( 1111 BABAPBAPBAPkkk   .Then ),(=),( zyPba k  for some 11),( BABAzy   . Consequently, 11 BABA    is pure .   Now, we are showing the following results:      Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                                   ISSN: 2395-0218      Volume 12, Issue 2  available at www.scitecresearch.com/journals/index.php/jprm                                           1827|        Theorem 1.22 Let )( MSP   be a 2pure  in mn GG  . Then )( MSP   is pure .  Proof. Claim )( MSP   is Neat. We have to show that ))((=)()( MSPPMSPGGP mn   . Let )(),(=),(=),( MSPMSPggPba mn   since )( MSP   is 2pure , )),((=),( MSPPMSP , so ))(()),((=),(=),( MSPPMSPPMSPba  . Consequently )( MSP   is Neat. By theorems (1.12, 1.13) We obtain that )( MSP   is pure  in mn GG  .   References [1]  H.M.A. Abdullah(1992), " 321 pure  in abelain," PU.M.A, Italy,No.3-4, vol. 3, pp. 135-139.  [2]  H.M.A. Abdullah(1992), " On minimal neat subgroup,"ser PU.M.A, Italy, No.3-4, Vol. 3, pp. 141-145.  [3]  Fuchi. Laszlo (1980), " Abelian group," USA.  

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ON DIRECT PRODUCT pure − 1 − 2 − 3 SUBGROUPS IN ABELIAN GROUP Gn×Gm

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