World Applied Sciences Journal 16 (11): 1631-1637, 2012 ISSN 1818-4952 © IDOSI Publications, 2012

Isologism, Schur-Pair Property and Baer-Invariant of Groups

Roozbeh Modabbernia

Islamic Azad University, Shoushtar Branch, Iran

Abstract: This paper is devoted to study the connection between the concepts of Schur-pair and the Baer- invariant of groups with respect to a given variety ν of groups. It is shown that if a group G belongs to a certain class of groups then so does its ν-covering group G*. Among other results, a theorem of E.W.Read in 1977 is being generalized to arbitrary varieties of groups. In addition, I consider covering groups and marginal extensions of a ν-perfect group with respect to a subvariety of abelian groups ν and show that any ν-marginal extension of a ν-perfect group G is a homomorphic image of a ν-stem cover of G.

Key words: Schur-pair and the baer-invariant of groups • varietal covering groups • subvarieties of abelian groups

INTRODUCTION (ii) V( G) = 1iffV*( G) = GiffG ∈ v (iii) [NV*G]= 1iffN ⊆ V*( G) Let F∞ be a free group freely generated by a countable set {x , x ,…}. GGVGN( )  V*GN( ) 1 2 (iv) V= andV*⊇ Let ν be a variety of groups defined by the set of NN N N laws V, which is a subset of F . It will be assumed that V( N) ⊆⊆[ NV*G] N V( G).Inparticular, ∞ (v) the reader is familiar with the notions of verbal V() G=[] GV*G subgroup, V(G) and of marginal subgroup, V*(G), If N V( G) = 1,thenN ⊆ V *( G) associated with a variety of groups ν and agiven group (vi) G. See also [23] for more information on the varieties andV*( G/N )= V* () G / N of groups. (vii) V*( G)  V( G) is contained in the Frattini Let G be a group with a normal subgroup N. Then subgroup of G. we define [NV*G] to be the subgroup of G generated by the elements of the following set: The following useful lemma can be proved easily [4].

−1 v( g1i ,...,g n,...,g r)(() v g 1r ,...,g ) Lemma 1.2: Let ν be a variety of groups and G be a  group. If G = H N, where H a subgroup and N is a 1≤≤ i r;v ∈ V;g,...,g1r ∈ G;n ∈ N normal subgroup of G, then V (G) = V(H) [NV*G] Let ν be a variety of groups defined by the set of It is easily checked that [NV*G] is the smallest laws V and let G be an arbitrary group with a free normal subgroup T of G contained in N, such that presentation N/T⊆V* (G/T). The following lemma gives basic properties of 1→R→F→G→1 verbal and marginal subgroups of a group G with ν respect to the variety , which are useful in our where F is a free group. Then the Baer-invariant of G investigations [4] for the proofs. with respect to the variety ν, denoted by ν M (G), is defined to be Lemma 1.1: Let ν be a variety of groups defined by the set of laws V and let N be and let N be a nornal R  V(F) subgroup of a group G. Then the following statements vM(G)= [RV*F] hold: GG = = One may check that the Baer-invariant of a group (i) V() V *() G 1and V *  VG() VG() G is always abelian and independent of the choice of Corresponding Author: Roozbeh Modabbernia, Islamic Azad University, Shoushtar Branch, Iran 1631 World Appl. Sci. J., 16 (11): 1631-1637, 2012

the free presentation of G [7, 8]. In particular, if ν is the Conversely, if V(G) satisfies the descending chain variety of abelian or nilpotent groups of class at most c G condition on normal subgroups and Gv , then (c≥1), then the Baer-invariant of the group G will be N RF ′ N V(G)=<> 1 . (the so called Schur-multiplicator of G) or it []R,F In section 2, we deal with the connection between the Schur pair property and the Baer-invariant of R γc1+ (F) will be , respectively (here γc+1 (F)stands [R , F] groups, In fact, it will be shown that if (ν,x) is a Schur- c * for the (c+1) st term of the lower central series of F pair and G is a ν-covering group of a group G then G∈X if and only if G*∈X (see Corollary 2.3). and [R,c F] = [R,F,…,F], where F is repeated c times [8-10, 12]. In sectin 3, we study the varietal covering groups and among the other results, a theorem of E.W. Read An exact sequence 1→A→G* →G→1 is said to be [24] is being generalized, extensively. Section 4 is a ν-stem cover of G, if (i) A⊆V(G*)∩V*(G*) and (ii) devoted to study the ν-covering groups and ν-marginal A≅νM (G). In this case G* is called a ν-covering group extensions of a ν-perfect group, when ν is taken to be a of G. Note that if ν is taken to be the variety of abelian subvariety of abelian groups. groups, then we have the usual definition of covering group. SCHUR-PAIR AND THE Let ν be a variety of groups defined by the set of BAER-INVARIANT OF GROUPS laws V and let G and H be groups. Then (α,β) is said to be a ν-isologism between G and H, if there exist Let ν be a variety of groups defined by the set of isomorphisms laws V and let X be a class of groups. Then (ν,x) is said to be a Schur-pair, when G is any group with GH α→: G V*(G) V*(H) ∈X it implies that V(G)∈X. V* (G) In particular, if x is the class of all finite groups β → and ; V(G) V(H), such that for all then the above property is known as a Hall’s first conjecture [3]. v(x ,...,x)∈∈ Vandallg,...,g G 1 r 1r In this section we give some equivalent conditions we have that (ν,x) has Schur-pair property if and only if, when β=(v(g,...,g)) v(h,...,h) 1 r 1r G is in x then its Baer-invariant ν M (G) is also in x. For the class of finite groups, we have the whenever hi∈α(giV*(G)), i = 1,…,r. In this case we remarkabletheorem of C.R.Leedham-Green and McKay write G v H and say that G is ν-isologic to H. In [7], which reads as follows : particular, if ν is the variety of abelian groups we obtain the notion of isoclinism due to P. Hall [3, 17, 18]. Theorem 2.1: ([7; Theorem 1.17]) Let ν be a variety of The following lemma of H.N. Hekster [4] is groups defined by the set of laws V and let x be the needed, in our investigation. class of finite groups. Then the following conditions are equivalent: Lemma 1.3: Let ν be a variety of groups defined by the set of laws V and let G be a group with a subgroup H (a) (ν,x) is a Schur-pair ; and a normal subgroup N. (b) For any finite group G, the order of the Baer- Then the following statements hold. invariant of G, |νM(G)|, divides a power of |G|.

* * (i) H v HV (G). In particular, if G= HV (G) then Let x be an arbitary class of groups, which is G G v H. Conversely, if satisfies the extension, quotient and normal subgroup closed, i.e. x= V*(G) PQSnX. Then we are able to prove a result similar to descending chain condition on subgroups and Theorem 2.1 for the class x, which is much larger than G v H, then G = HV* (G). the class of finite groups. G G (ii) v . In particular, if N V(G)=<> 1 N  V(G) N Theorem 2.2: Let ν be a variety of groups defined by G the set of laws V and let x be a class of groups with x = then G v . N PQSnX. Then the following conditions are equivalent: 1632 World Appl. Sci. J., 16 (11): 1631-1637, 2012

(a) (ν,x) is a Schur-pair; Now, considering this notion we have been able to (b) If G is any group in X, then so is ν M (G). prove a result similar to Theorem 1.3.

Proof: Left 1→R→F→G→1 be a free presentation of VARIETAL COVERING GROUPS the group G. Then This section is devoted to study the covering RF groups of a group G, with respect to a given variety of 1→ → →→G1 [RV** F] [ R V F ] groups ν. One should note that, in general, groups might not possess ν-covering groups. In [19], we is a ν-marginal extensionof G. Now if is a Schur-pair presented a class of groups lacking ν-covering groups, and G∈X, then using the property x = Qx and with respect to a certain given variety ν [7]. However, I.Schur in [25] showed the existence of RF covering groups for finite groups and then Jones [26] ⊆ V* [ RV** F ] [RV F] generalized it to every group in the variety of abelian groups. In [14] we have also shown that every group we have F has a ν-covering group with respect to the variety of abelian groups of exponent m, when m is a square-free [RVF]* ∈X positive integer. Now, assuming the existence of a F V*  covering group of a given group G with respect to a [RVF]* variety ν, we are able to give the structure of such covering groups. V(F) Hence ∈x and so ν M (G)∈x. [RV* F] Theorem 3.1: Let ν be a variety of groups defined by Conversely, with the same notation, let E be a the set of laws V and let G be a group with a free E group with marginal factor group ≅ G. By the presentation 1→R→F→G→1. Then V* (E) assumption G∈X and hence (i) If S is a normal subgroup of F such that

VF( ) → → →→ RSR VF( ) 1 vM() G * V() G 1 = →× [RV F] [ R V** F ] [ R V F ] [RV * F]

VF( ) is a ν-marginal extension of V(G)∈X, then is then G*= F/S is a ν-covering group of G. [RVF]* also in x. It is easily cheched that V(E) is a (ii) Every ν-covering group of G is a homomrphic VF( ) F homomrophic image of Therefore V(E)∈X, i.e. image of . [RVF]* [RV* F] (ν,x) is a Schur-pair. (iii) For any ν-covering group G* of the group G with a The following interesting corollary states that a ν-stem cover group G in the above mentioned class of groups x has the same structure as its covering group and its proof 1→A→G* →G→1 follows from the above theorem. there exists a normal subgroup S of F as in part (i), Corollary 2.3: Let ν be a variety of groups defined by satisfying also F/S≅G* and R/S≅A. the set of laws V and let x be a class of groups * * with x = PQSn X. Let (ν,x) be a Schur-pair and G be a Proof: (i) Put A=R/S. Then G /A≅F/R≅G and ν-covering group of G. Then G∈X if and only if G*∈X. A≅νM(G). From the assumption we have R⊆V(F) S. Clearly Remark: J.A. Hulse and Lennox in [5] did introduce a generalized version of the Schur-pair property as RF A=⊆= V(* ) V(**G) follows: (ν,x) is said to be an ultra Schur-pair, if for any SS group G with a normal subgroup N such that and N * VFS ∈ X, it holds that [NV G]∈X [13, 20]. RF( )  * * A =⊆==V V(G) N V (G) SS S 1633 World Appl. Sci. J., 16 (11): 1631-1637, 2012

Hence G*=F/S is a ν-covering group of G. using the above theorem we deduce that any two covering groups are ν-isologic, which generalizes a (ii) Let F be the free group freely generated by the theorem of Bioch and van der Waall [2]. set X and let π: F→G be an epimorphism with R= kerπ. Let G* be a ν-covering group of G, with Corollary 3.2: [15] Let ν be a variety of groups defined the ν-stem cover 1AG→ →* φ → G1 → . Cleary for by the set of laws V and G be a group. Then all ν ν any x∈X, there exists g ∈G* such that -covering groups of G are -isologic. x In [16], by imposing some condition on φ=π(g ) (x). Now, we put H =< g ∈GxX,* ∈> x x homomorphisms we give a sufficient condition for two hence G*=HA. But using Lemma 1.2, ν-covering groups of a given to be isomorphic. Also we ⊆ **=* = A V (G ) V(H),soG H. We consider the deduce that all ν-covering groups of a group in ν are homomorphism ψ:F→G* given by Hopfian.

ψ=(x) gx ,x ∈X. Then ψ is onto and π=φoψ. Covering groups have been studied for the abelian case, by several authors. See for instance [6, 10, 26]. It is easily seen that ψ(R)⊆A, so In the following we deal with the property of covering groups in an arbitrary variety of groups ν, ψ([RV* F]) ⊆ψ [ ( R ) V** G ]1 = which generalizes the work of E.W. Read [24], extensively. Thus ψ induces a homomorphism ψ from Theorem 3.3: Let ν be a variety of groups and let G1 F * onto G , which is the required assertion. and G be two ν-covering groups of a given group [RVF]* 2 G. Let

(iii) Let a∈A and a∈ψ(x), for some a∈F. Then 1AGG1,i1,2→→→→ii = 1 = φ(a) = π(x). So x∈R and hence A⊆ψ(R). One can easily see be a ν-stem cover of G. Then that A=ψ(R). Now observe that

** * V (G ) V(G) ψ(R V(F))⊆ ψ (R) ψ (V(F))= A V(G ) = A 12≅ AA12 To prove the converse, suppose that Ζ=ψ(x) =ψ(y,) for some x∈V(F) and y∈R, Proof: Let 1→R→F→G→1 be a free presentation for * Hence x−1 y∈ψ ker . Thus π(x-1y) = 1 and xy−1 ∈R. It the group G and G be a fixed covering group of G with respect to a given variety ν. By the definition, there is follows that x∈R and Ζ∈ψ(R V(F) ) which shows an exact sequence 1→A→G*→G→1 such that A that A⊆ψ (R  V(F)). Hence A= ψ (R V(F)). ⊆V*(G*)∩V(G*) and A≅νM(G). Therefore ψ restricts to an isomorphis from To prove the result it suffices to show that R  V(F) S V** (G ) onto A. Let S= kerψ. Then is the isomorphism class of groups are determined [RVF]* [RVF]* A R uniquely by the presentation F/R≅G. Using Theorem kernel of the restriction of ψ to and the image * [RVF]* 3.1, we may assume that G ≅F/S, A≅R/S of this restriction is A. Thus one may conclude that for some normal subgroup S of F such that

R R V(F) S RS = × ≅×vM(G) [RVF]*** [RVF] [RVF] [ RV** F ] [RV F]

ν Now, part (i) implies that F/S is a -covering group * FL * Put V = , then clearly of G. Let θ: F/S→G be the homomrphism induced by [RV** F] [RV F] ψ. Using the fact that ψ is onto and ψ (R) =A, it follows LV** F ⊆⊆ RV F  S and hence L/S⊆V*(F/S). Now, if that θ(R/S) = A, which completes the proof.    * In general, it is not true that any two covering xS∈V (F/S) then for every v∈V and f1,…,fr∈F, we −1* groups of a given group G are isomorphic [6]. However have v(f1 ,...,f i x,...,f r1r )v(f,..,f )∈= S V(F) [RV F] 1634 World Appl. Sci. J., 16 (11): 1631-1637, 2012

F such that the following diagram commutes So x[RVF]**∈ V ( ). This implies that [RVF]* ** FF− * * V (G ) 1G→→π→→ V (F/S)⊆L/S. Thus V (F/S)=L/S. Hence ≅ L/R. ** A [RVF] [RVF]

But the factor group L/R is only determined by the free ↓β1 ↓β ↓α presentation F/R≅G and hence the result follows. 1A→ → H →→G1

Remark: In [7], Leedham-Greem and McKay Where π is the natural homomorphism induced by π introduced the generalized version of the Baer-invariant and β is the restriction of β. of a group G with respect to two varieties of groups. 1 We have proved, a result similar to Theorem 3. 3 in this ν generalized version [21, 22]. Theorem 4.3: Let be a variety contained in the variety of abelian groups and let SUBVARIETIES OF ABELIAN GROUPS 1A→→ H→→G1

In this final section we consider ν-covering groups be a ν-marginal extension of a ν-perfect group G. Then and ν-marginal extensions of a ν-perfect group with there exists a ν-covering group G* of G such that H is a ν *. respect to a subvariety of abelian groups , say. homomorphic image of G In [15], we have shown the following theorem, yielding the existence of ν-covering groups for ν- π Proof: Let 1RF→→→→G1be a free presentation of perfect groups with repect to an arbitrary variety of the group G. By Lemma 4.2, there exists an groups ν. F epimorphism β→:H such that the following [RV* F] Theorem 4.1: Let ν be a variety of groups and let G be diagram commutes: a ν-perfect group with a free presentation G≅F/R. Then V(F) ν FFπ * is a -covering group of G. 1→→→→G1 [RVF] [RVF]** [RVF]

↓β1 ↓β || Theorem 4.1: Is also generalized Theorem 2. 1 of [11], 1A→ → H→→G1 in which the variety ν was dfined by the set of outer commutator words. where β is the restriction of β. Put ker β = ker Now in the following main result (Theorem 4.3), it 1 1 T is shown that if ν is a variety of groups contained in the β= . [RVF]* abelian variety A, say, then the ν-marginal extensions where T is a normal subgroup of R and T(R∩V(F))=R. of a ν-perfect group G are homomorphic images of a ν- stem cover. Of course, we have also proved such a Hence, since G is ν-perfect we have result in [14] for any group with respect to the variety T T(R V(F)) R RV(F) F of abelian groups of exponent m, where m is a square ≅ = ≅≅ free positive integer. T V(F) R V(F) R V(F) V(F) V(F) The following lemma shortens the proof of the main theorem considerably and its proof is Thus the following exact sequence splits straightforward. T V(F) T T → →→ → 11** Lemma 4.2: Let ν be variety of groups and G be any [RVF] [RVF] T V(F) group with a free presentation 1RF→→π→→G1 and T let 1AHG1→→→→ be a ν-marginal extension of where is an abelian group and hence [RVF]* another group G . If α→:G Gis an isomorphism, then there exists an epimorphism β: T T V(F) S ≅× [RVF]*** [RV F] [RVF] F → H [RVF]* where 1635 World Appl. Sci. J., 16 (11): 1631-1637, 2012

ST REFERENCES = [RVF]* T V(F) Now we have 1. Alperin, J.L. and D. Gorenstein, 1966. The multiplicators of certain simple groups. Proc. S (R  V(F))= S  (T V(F)) =[RVF]* Amer. Math. Soc., 17: 515-519. 2. Bioch, J.C. and R.W. van der Waall, 1978. and Monomiality and isoclinism of groups. J. Reine S (R V(F))= S(T(R  V(F))  V(F)) Angew. Math., 289: 74-88. = (R V(F))(S(T V(F))) 3. Hall, P., 1940. The classification of prime power = (R V(F))T= R groups. J. Reine Angew. Math., 182: 130-141. 4. Hekster, N.S., 1989. Varieties of groups and which implies that isologisms. J. Austral. Math. Soc., (Series A) 46: 22-60. R R V(F) S 5. Hulse, J.A. and J.C. Lennox, 1976. Marginal = × [RVF]*** [RVF] [RVF] series in groups. Proc. Royal Soc. Edinburgh, 76 A: 139-154. Hence by Theorem 2.1, F/S is a ν-covering group 6. Karpilovsky, G., 1987. The Schur Multiplier. of G. Moreover London Math. Soc. Monographs New Series, Vol: 2. F 7. Leedham-Green, C.R. and S. McKay, 1976. The F / S [RV* F] Baer-invariant, isologism, varietal laws and ≅≅H T/S T homology. Acta Math., 137: 99-150. [RV* F] 8. Moghaddam, M.R.R., 1979. The Baer-invariant of a direct product. Arch. der Math. (Basel), 33: 504-511. which completes the proof. 9. Moghaddam, M.R.R., 1981. On the Schur-Baer Now, we obtain the following corollary which is of property. J. Austral. Math. Soc. (Series A), 31: interest in its own account. 43-61. 10. Moghaddam, M.R.R., 1981. Some inequalities for Corollory 4.4: Let ν be a variety contained in the the Baer-invariant of a finite group. Bull. Iranian → → → → variety of abelian groups and let (e): 1 A H G 1 Math. Soc., 9: 5-10. be a ν-marginal extension of a ν-perfect group G such 11. Moghaddam, M.R.R. and S. Kayvanfar, 1997. that every other ν-marginal 1→A→H→G→1 extension Perfect groups. Indag. Math. N.S., 8 (4): 537-542. of G is a homomorphic image of the extension (e). 12. Moghaddam, M.R.R. and M.M. Nasrabadi, 2000. Then(e) is a ν-stem cover of G. Schur-Baer property in polynilpotent groups. Italian Journal of Pure and Applied Mathematics, Proof: By Theorem 4.3, there exists a ν-stem cover 8: 49-56. * (e′): 1→A1→G* →G→1 and an epimorphism ψ: G →H 13. Moghaddam, M.R.R. and A.R. Salemkar, 1998. such that the following diagram is commutative Some remarks on generalized Schur pairs. Arch. der Math., (Basel) 71: 12-16. * 1A→→1 G→→G1 14. Moghaddam, M.R.R. and A.R. Salemkar, 1999. Characterization of varietal covering and stem ↓ψ1 ↓ψ || 1A→ →→→ H G1 groups. Communication in Algebra, 27 (11): 5575-5586. 15. Moghaddam, M.R.R. and A.R. Salemkar, 2000. where ψ1 is the restriction of ψ to A1. Now by using Varietal isologism and covering groups. Arch. der [15, Thoeorem 3. 4], we obtain that ψ is an Math., (Basel) 75: 8-15. isomorphism which gives the result. 16. Moghaddam, M.R.R. and A.R. Salemkar, 2000. In the context of ν-perfect groups, we have proved Some properties on isologism of groups. J. Austral. several other results, for instance we have shown that Math. Soc. (Series A), 68: 1-9. any automorphism of a finite ν-perfect group can be 17. Moghaddam, M.R.R. and A.R. Salemkar and A. lifted to an automorphism of its ν-covering group [15]. Gholami, 2000. Someproperties on isologism of This result is a vast generalization of Alperin and groups and Baer-invariants. Southeast Asian Gorenstein [1]. Bulletin of Mathematics, 24: 255-261. 1636 World Appl. Sci. J., 16 (11): 1631-1637, 2012

18. Moghaddam, M.R.R., A.R. Salemkar and 22. Moghaddam, M.R.R., A.R. Salemkar and M. A. Gholami, 2001. Some properties on Taheri, 2001. Baerinvariants with respect to two marginal extensions and the Baer-invariant of varieties of group. Algebra Colloquium, 8 (2): groups. Vietnam Journal of Mathematics, 29 145-151. (1): 39-45. 23. Neumann, H., 1967. Varieties of Groups. Springer- 19. Moghaddam, M.R.R., A.R. Salemkar and M.M. Verlag, Berlin. Nasrabadi, Baerinvariant: inequalities and covering 24. Read, E.W., 1977. On the centre of a representation groups. To Appear. group. J. London Math. Soc., 16 (2): 43-50. 20. Moghaddam, M.R.R., A.R. Salemkar and M.R. 25. Schur, I., 1904. Uber die Darstellung der endlichen Rismanchian, 2002. Some properties of ultra Hall Gruppen durch gebrochene lineare Substitutionen. and Schur pairs. Arch. Der Math., (Basel) To J. Reine Angew. Math., 127: 20-50. Appear. 26. Wiegold, J., 1982. The Schur Multiplier: An 21. Moghaddam, M.R.R., A.R. Salemkar and M.R. elementary approach. Groups-St. Andrews Rismanchian, 2001. Generalized covering (1981), Lecture Note Series of London Math. groups. Southeast Asian Bulletin of Mathematics, Soc., Cambridge University Press, Cambridge, 25: 485-490. 71: 137-154.

1637