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.