Torsion Groups with the Norm of pd-Subgroup of Finite Index

. The authors study the relations between the properties of torsion groups and their norms of pd -subgroups. The norm N pdIG of pd - subgroups of a group G is the intersection of the normalizers of all its pd -subgroups or a group itself, if the set of such subgroups is empty in a group. The structure of the norm of pd -subgroups in torsion groups is described and the conditions of Dedekindness of this norm is proved (Dedekind group is a group in which all subgroups are normal). It is proved that a torsion group is a (cid:28)nite extension of its norm of pd -subgroups if and only if it is a (cid:28)nite extension of its center. By this fact and the structure of the norm of pd -subgroups, we get that any torsion group that is a (cid:28)nite extension of this norm is locally (cid:28)nite.

Êëþ÷îâi ñëîâà: ïåðiîäè÷íà ãðóïà, íîðìàëiçàòîð, íîðìà ãðóïè, íîðìà pd-ïiäãðóï, íåäåäåêiíäîâiñòü, iíäåêñ ïiäãðóïè â ãðóïi, öåíòð ãðóïè MSC2020: 20E07 IF sntrodution sn group theoryD n importnt ple is oupied y results relted to the study of groups in whih ll sugroups of some nonEempty system Σ re normlF © F hF vueryeD wF qF hrveuD eF F shyryeD PHPR 110 ysyx qy sr ri xyw yp P DEfqy yp psxsi sxhi st is ler tht eh group with this property n e de(ned s group tht oinides with the intersetion of the normlizers of ll sugroups from the system ΣF sf the system Σ ontins ll sugroups of group G tht hve given theoretil group propertyD then this intersetion is lled the ΣEnorm of group GF hen studying ΣEnorms nd the impt of their properties on the properties of groupD mny questions rise relted to the study of the properties of groupsD depending on the hoie of the system Σ of sugroups nd the restritions tht the de(ned ΣEnorm stis(esF imilr prolems were onsidered y mny lgeristsD ut in the mjority of (ndingsD ΣEnorm oinides with groupF sn other wordsD groups with the ondition of normlity for ll sugroups of the system Σ were studiedF qroups with di'erent systems Σ of norml sugroups were tively studied in the seond hlf of the PHth enturyF hereforeD for mny systems Σ of sugroupsD the struture of groups oiniding with their ΣEnorm is knownD nd it nturlly rises questions out the properties of the groups in whih the ΣEnorm stis(es some restritions nd is proper sugroup of groupF por the (rst timeD suh prolem ws onsidered y F fer in the IWQHs I for the system ΣD whih onsists of ll sugroups of this groupF F fer lled the orresponding ΣEnorm the norm N (G) of group G nd de(ned it s the intersetion of the normlizers of ll sugroups of group GF st is ler tht the norm N (G) is ontined in ll other ΣEnormsD whihD in turnD n e onsidered s generliztionsF purtherD fer9s ide of the group norm ws trnsferred to other systems Σ of sugroups @seeD for exmpleD Q !IPD IR !IUA nd found pplition in ring theory in the study of the fer kernelD whih is n nlogue of the group norm for rings PF he gol of this rtile is to study the properties of other ΣEnorm of group ! the norm N pdI G of pdEsugroups in the lss of torsion groups nd to estlish the onditions under whih this norm hs (nite index in groupF es will e shown elowD ll in(nite torsion groups tht re (nite extensions of the norm N pdI G re entrlEyE(niteF his result grees with the results of pF wF vymn IQ out the properties of torsion pdEgroups with the ondition G = N pdI G @tht isD nonEeelin groups in whih ll pdEsugroups re normlAF

PF reliminry results
e group G is lled pdEgroup if it ontins elements of prime order pF e nonEeelin group ontining nonEidentity pEelementsD in whih every pdE sugroup is normlD is lled pdIEgroupF st is known @vemm I nd P IQA tht ny torsion nonErmiltonin pdIEgroup ontins unique sugroup of prime order p nd does not ontin yli sugroups of order p 2 F roposition I presents omplete desription of nonErmiltonin pdIE groupsF Proposition 1 @ IQD heorem IA. eny torsion nonErmiltonin group G is pdIEgroup if nd only if it n e represented s semidiret produt G = a DD where a is ylow pEsugroup of order p = 2 of group G nd D is torsion hedekind groupF ell tht hedekind group is group in whih every sugroup is norml @tht isD suh group is either eelin or rmiltoninAF Denition 1. he intersetion of the normlizers of ll pdEsugroups of group G @provided tht the set of suh sugroups in group is nonEemptyA is lled the norm of pdEsugroups or the pdEnorm of this group nd denoted y N pdI G F sn other wordsD the pdEnorm of group G is the mximl sugroup of this group tht normlizes every pdEsugroup in GF por groups with n empty set of pdEsugroupsD let9s ssume tht G = N pdI G F xote tht ny pdIEgroup is metErmiltoninD iFeFD nonEeelin group in whih eh nonEeelin sugroup is normlF oD the pdEnorm is relted to the norm of nonEeelin sugroups or the metnorm of group @ QD RAF fy the de(nition of pdEnormD in the se G = N pdI G the set of pdEsugroups is empty or ll suh sugroups re norml in group G @iFeFD G is eelin or pdIEgroupAF vet us onsider the properties of the pdEnorm N pdI G nd group GD in whih this norm is proper sugroupF purtherD only groups with nonEempty set of pdEsugroups will e onsiE deredD nd y the term 4group4we will men pdEgroupF fy the de(nition of the pdEnorm N pdI G D we get the following resultF Lemma 1.Let N pdI G be the norm of pd-subgroups of a group G. Then the following propositions take place: where N (G) is the Baer`s norm and Z(G) is the center of a group G;

any pd-subgroup of the norm
Proof. he proof of @IAE@UA is ovious nd follows diretly from the de(nition of the pdEnorm N pdI G of group GF vet us prove properties @VA nd @WAF vet H e norml pEsugroup of group o prove the proposition @WAD let us onsider the group nd we get the required equlityF he lemm is provedF Lemma 2. If the norm N pdI G of a pd-group G does not contain elements of order p, then it is Dedekind.
Proof.vet the norm N pdI G stisfy the onditions of the lemm nd let a e n ritrry element of order p of group GF hen tht isD the norm N pdI G is hedekindF he lemm is provedF he following exmple on(rms the existene of nonEeelin pdEgroups with the norm N pdI G whih does not ontin nonEidentity pEelementsF 1. a group G contains a pd-subgroup identity-intersected with N pdI G ; 2. a group G contains a cyclic p-subgroup of order greater than p; 3. a group G contains an elementary Abelian subgroup of order p 2 ; 4. in the case of p = 2; 5. the center Z(G) of a group G contains non-identity p-elements;

the norm
G is nonErmiltonin pdIEgroupD whih does not ontin yli sugroups of order p 2 y roposition IF husD if G ontins yli sugroup Proof. he su0ieny of the onditions of the theorem is oviousD so we will prove only their neessityF vet [G : Example I. vet us onsider the group G = a b × CD in whih |a| = 3D |b| = 2D b −1 ab = a −1 nd C is n in(nite torsion eelin group without elements of orders P nd QF sn this groupD the 2dEnorm oinides with the enter of the group N 2dI G = Z(G) = C nd 2 |N 2dI G |F yn the other hndD sine ny 3dEsugroup ontins the ommuttor G = a D G is 3dIEgroup nd oinides with the 3dEnormX N 3dI G = GF Lemma 3. The norm N pdI G of a torsion pd-group G is Dedekind or a non-Hamiltonian pdI-group of the type of Proposition 1. Proof.sf the norm N pdI G is hedekindD then the lemm is provedF vet N pdI G e nonEhedekind groupF henD y vemm PD N pdI G ontins nonEidentity pE elements nd therefore pdEsugroupsF ine N pdI G normlizes ll pdEsugroupsD ny pdEsugroup of N pdI G is norml in itF husD N pdI G is nonErmiltonin pdIEgroupF he lemm is provedF QF win results fy vemm Q nd roposition I the norm N pdI G of torsion lolly nilpotent pdEgroup is hedekindF yther onditions of hedekindness of the norm N pdI G in torsion group G re given in the following theoremF Theorem 1.The norm N pdI G of a torsion pd-group G is Dedekind if at least one of the following conditions takes place: e the enter of the normN pdI G D M = g 1 , g 2 , ..., g n F ine Z G nd [N pdI G : Z] < ∞D N pdI G = F ZD where |F | < ∞F oD G = M, F Z nd [G : Z] < ∞F gonsidering tht N = M, F is (nite pdEsugroupD we onlude tht N GF purtherD y the ondition |N | < ∞ we get [G : C G (N )] < ∞F fut C G (N ) ∩ Z ⊆ Z(G)D soD in this se [G : Z(G)] < ∞D whih is desired onlusionFvet the norm N pdI G e hedekind groupF sf it does not ontin nonEidentity pEelementsD then the (nite group M = g 1 , g 2 , ..., g n ontins suh elementsF ineM is n N pdI G Edmissile sugroupD M G nd [G : C G (M )] < ∞F king into ount tht [N pdI G : Z(N pdI G )] ≤ 4 nd C G (M ) ∩ Z(N pdI G ) ⊆ Z(G), we onlude tht [G : Z(G)] < ∞ in this seF vet N pdI G e hedekind group nd x ∈ N pdI G e n element of order pF hen M, x = x, g 1 , g 2 , ..., g n is (nite ndD thereforeD n N pdI G Edmissile pdE sugroupF husD M, x G nd [G : Z(G)] < ∞ n eproved in the sme wyF he theorem is provedF Corollary 2. Any innite torsion pd-group G with pd-norm N pdI G of a nite index is locally nite.Corollary 3. Any innite torsion pd-group G with pd-norm N pdI G of a nite index is Abelian-by-nite.
ontins unique sugroup a of order p y roposition IF ine G ontins n elementry eelin sugroup of order p 2 D there exists the sugroup x of order p suh thtx ⊂ N pdI G F fut then x ∩ N pdI G = E nd N pdI G ishedekind y the proved oveF his ontrdits the ssumptionF RA vet p = 2F sf the norm N pdI G does not ontin elements of order 2D then it is hedekind y vemm PF vet N pdI G ontin PEelementsF hen ll 2dEsugroups re norml in it nd N pdI G is either eelin or 2dIEgroupF sn the ltter seD If the norm N pdI G of a torsion pd-group G is non-Dedekind, then G does not contain subgroups of order p 2 .Lemma 4. If the norm N pdI G of a torsion pd-group G is non-Dedekind, then p = 2 and the Sylow p-subgroup of G is of order p, coincides with the Sylow p-subgroup of the norm N pdI G and is a normal subgroup of G. is ylow pEsugroup of order p = 2 of group GD D is torsion hedekind groupF fy heorem ID group G does not ontin pEsugroupsD whih is di'erent from a F hereforeD a is the mximl pEsugroup of group GD iFeFD its ylow pEsugroupF he lemm is provedF Theorem 2. Any innite torsion pd-group G is a nite extension of the norm N pdI G if and only if G is central-by-nite.