A generalisation of Schenkman's theorem
AA GENERALISATION OF SCHENKMAN’S THEOREM
STEFANOS AIVAZIDIS † , INA N. SAFONOVA ∗ AND ALEXANDER N. SKIBA (cid:93)
A b s t r ac t .
Let G be a finite group and let F be a hereditary saturated formation.We denote by Z F ( G ) the product of all normal subgroups N of G such that every chieffactor H/K of G below N is F -central in G , that is,( H/K ) (cid:111) ( G/ C G ( H/K )) ∈ F . A subgroup A (cid:54) G is said to be F -subnormal in the sense of Kegel , or K - F -subnormal in G , if there is a subgroup chain A = A (cid:54) A (cid:54) . . . (cid:54) A n = G such that either A i − (cid:69) A i or A i / ( A i − ) A i ∈ F for all i = 1 , . . . , n .In this paper, we prove the following generalisation of Schenkman’s Theorem on thecentraliser of the nilpotent residual of a subnormal subgroup: Let F be a hereditarysaturated formation and let S be a K - F -subnormal subgroup of G . If Z F ( E ) = 1 for everysubgroup E of G such that S (cid:54) E then C G ( D ) (cid:54) D , where D = S F is the F -residual of S . I n t r o d u c t i o n
Throughout this paper, all groups are assumed to be finite and G will always denote afinite group.Schenkman [Sch55] proved that if S is a subnormal subgroup of the group G and if C G ( S ) = 1, then C G ( S N ) (cid:54) S N , where S N is the nilpotent residual of S . A particular caseof Schenkman’s Theorem is G = S : if G is a group with trivial centre then G N (cid:62) C G ( G N ).A normal subgroup of a group G which contains its own centraliser in G we call large .Thus the particular case of Schenkman’s Theorem we mentioned can be restated thus: ina centreless group the nilpotent residual is a large subgroup. Our goal in this note is to state and prove a generalisation of Schenkman’s Theorem validfor all hereditary saturated formations.Before continuing, we need some concepts from the theory of formations. Recall that G F denotes the F -residual of G , that is, the intersection of all normal subgroups N of G with G/N ∈ F . A non-empty class of groups F is said to be a formation if every homomorphicimage of G/G F belongs to F for every group G . A formation F is called saturated if G ∈ F whenever G F (cid:54) Φ ( G ) and hereditary (A.I. Mal’cev [Mal70]) if H ∈ F whenever H (cid:54) G ∈ F .Now let R/S be a normal section of G (meaning that S, R (cid:69) G ) and let K be a normalsubgroup of G such that K (cid:54) C G ( R/S ). Then we can form the semidirect product[
R/S ]( G/K ) , Mathematics Subject Classification.
Key words and phrases.
Nilpotent residual, hereditary saturated formation, subnormal subgroup, K - F -subnormal subgroup, F -residual. a r X i v : . [ m a t h . G R ] S e p here ( rS ) gK = g − rgK for all rS ∈ R/S and gK ∈ G/K . Moreover, we will suppress the (cid:111) symbol when theimplied action is the one above. The section
R/S is F -central in G [SS89] if for somenormal subgroup K of G such that K (cid:54) C G ( R/S ) we have [
R/S ]( G/K ) ∈ F ; otherwise R/S is called F -eccentric in G . If R/ F -central in G then we say that R is F -central in G .The importance of these concepts stems from a result of Barnes and Kegel [BK66] whichasserts that if F is a formation then every chief factor of a group G ∈ F is F -central in G . Moreover, if F is any non-empty saturated formation and every chief factor of G is F -central in G then G ∈ F (see Lemma 2.5 below).A subgroup A (cid:54) G is said to be F -subnormal in the sense of Kegel [Keg78] or K - F -subnormal in G [BBE06, Defn. 6.1.4] if there is a subgroup chain A = A (cid:54) A (cid:54) . . . (cid:54) A n = G such that either A i − (cid:69) A i or A i / ( A i − ) A i ∈ F for all i = 1 , . . . , n .Note that the K - F -subnormal subgroups play an important role in many branches offormation theory and their study is related to the investigations of many authors (cf.[BBE06, Chap. 6]).The main result of the work reported here is the following. Theorem A.
Let F be a hereditary saturated formation. Let S be a K - F -subnormalsubgroup of G and suppose that Z F ( E ) = 1 for every subgroup E such that S (cid:54) E . Then C G ( S F ) (cid:54) S F . In this theorem, Z F ( E ) denotes the F -hypercentre of E which is defined as the product ofall normal subgroups N of E such that either N = 1 or N > E below N is F -central in E .Observe that if for a subgroup S of G we have C G ( S ) = 1 then Z N ( E ) = Z ∞ ( E ) = 1 forevery subgroup E of G such that S (cid:54) E , where N is the class of all nilpotent groups.Note, also, that every subnormal subgroup is K - F -subnormal. Therefore, Schenkman’soriginal theorem is a direct consequence of Theorem A. Corollary 1.1 (cf. [Sch55] or [Isa08, Thm. 9.21]) . Let S be a subnormal subgroup of G and suppose that C G ( S ) = 1 . Then C G ( S N ) (cid:54) S N . We discuss some further applications of Theorem A in Section 3.The following result, which we record here separately, is an important step in the proof ofTheorem A.
Theorem B.
Let F be a saturated formation. If G has no non-trivial normal F -centralsubgroups then G F is a large subgroup of G , i.e. C G ( G F ) (cid:54) G F . Corollary 1.2.
Let F be a saturated formation and let U be the F -residual of G . If U ∩ Z F ( G ) = 1 then | G/ Z F ( G ) | (cid:54) | Hol ( U ) | , where Hol ( U ) = U (cid:111) Aut ( U ) is the familiar holomorph of the group U . e prove our results in the next section and our proofs are non-standard. In fact, we aredeveloping here a novel method for proving results within the context of formation theorythat does not use the complex machinery of the theory. This should make the presentarticle accessible to a wide audience.2. Au x i l i a ry r e s u lt s a n d P r o o f o f T h e o r e m s A a n d B
Let D = M (cid:111) A and R = N (cid:111) B . Then the pairs ( M, A ) and (
R, B ) are said to be equivalent provided there are isomorphisms f : M → N and g : A → B such that f ( a − ma ) = g ( a − ) f ( m ) g ( a )for all m ∈ M and a ∈ A .In fact, the following lemma is known (cf. [SS89, Lemma 3.27]). Lemma 2.1.
Let D = M (cid:111) A and R = N (cid:111) B . If the pairs ( M, A ) and ( R, B ) areequivalent then D ∼ = R . Lemma 2.2.
Let F be a hereditary formation and let R/S be an F -central normal sectionof G and K (cid:54) L normal subgroups of G such that L (cid:54) C G ( R/S ) and [ R/S ]( G/K ) ∈ F .Then the following statements hold. (i) [ R/S ]( G/L ) ∈ F . (ii) If E is a subgroup of G then ( E ∩ R ) (cid:14) ( E ∩ S ) is F -central in E . (iii) [ T /S ]( G/K ) ∈ F and [ R/T ]( G/K ) ∈ F for every normal subgroup T of G suchthat S (cid:54) T (cid:54) R . Proof.
To facilitate notation somewhat, set V := [ R/S ]( G/K ).(i) Let W = [ R/S ]( G/L ). Then V (cid:14) ( L/K ) = (cid:2) ( R/S )( L/K ) (cid:14) ( L/K ) (cid:3) ( G/K ) (cid:14) ( L/K ) ∈ F , where the pairs (cid:0) ( R/S )( L/K ) (cid:14) ( L/K ) , ( G/K ) (cid:14) ( L/K ) (cid:1) and ( R/S, G/L ) are equivalent,so W ∼ = V (cid:14) ( L/K ) ∈ F by Lemma 2.1.(ii) Let V := (cid:2) ( E ∩ R ) S (cid:14) S (cid:3) ( KE/K ) , where ( eS ) lK = l − elS for all eS ∈ ( E ∩ R ) S/S and lK ∈ KE/K and let W := (cid:2) ( E ∩ R ) (cid:14) ( E ∩ S ) (cid:3) (cid:0) E (cid:14) ( E ∩ K ) (cid:1) . Then V ∈ F since the class F is hereditary. On the other hand, the pairs (cid:0) ( E ∩ R ) S (cid:14) S, KE/K (cid:1) , (cid:0) ( E ∩ R ) (cid:14) ( E ∩ S ) , E/ ( E ∩ K ) (cid:1) are equivalent, so W ∼ = V ∈ F .(iii) Observe that [ T /S ]( G/S ) is a normal subgroup of [
R/S ]( G/S ), thus [
T /S ]( G/K ) ∈ F since F is normally hereditary. Now let D := [ R/T ]( G/K ), where ( rT ) gK = r g T for all rT ∈ R/T and gK ∈ G/K . Then V (cid:14) ( R/T ) = (cid:2) ( R/S ) (cid:14) ( T /S ) (cid:3) (cid:0) ( T /S )( G/K ) (cid:14) ( T /S ) (cid:1) ∈ F , here the pairs ( R/T, G/K ) , (cid:0) ( R/S ) (cid:14) ( T /S ) , ( T /S )( G/K ) (cid:14) ( T /S ) (cid:1) are equivalent, so D ∈ F by Lemma 2.1. The proof is now complete. (cid:4) Lemma 2.3.
Let
N, M and
K < H (cid:54) G be normal subgroups of G . (i) If N (cid:54) K then [ H/K ] (cid:0) G (cid:14) C G ( H/K ) (cid:1) ∼ = (cid:2) ( H/N ) (cid:14) ( K/N ) (cid:3) (cid:0) ( G/N ) (cid:14) C G/N (cid:0) ( H/N ) (cid:14) ( K/N ) (cid:1)(cid:1) . (ii) If T /L is a normal section of G and H/K and
T /L are G -isomorphic then C G ( H/K ) = C G ( T /L ) and [ H/K ]( G/ C G ( H/K )) ∼ = [ T /L ]( G/ C G ( T /L )) . (iii) [ M N/N ]( G/ C G ( M N/N )) ∼ = [ M/ ( M ∩ N )]( G/ C G ( M/ ( M ∩ N )) . Proof. (i) In view of the G -isomorphisms H/K ∼ = ( H/N ) (cid:14) ( K/N ) and G/ C G ( H/K ) ∼ = ( G/N ) / ( C G ( H/K ) /N ) , the pairs (cid:0) H/K, G (cid:14) C G ( H/K ) (cid:1) , (cid:0) ( H/N ) (cid:14) ( K/N ) , ( G/N ) (cid:14) C G/N (cid:0) ( H/N ) (cid:0) K/N ) (cid:1)(cid:1) are equivalent. Hence Statement (i) is a corollary of Lemma 2.1.(ii) A direct check shows that C = C G/N ( H/K ) = C G ( T /L ) and that the pairs (
H/K, G/C )and (
T /L, G/C ) are equivalent. Hence Statement (ii) is also a corollary of Lemma 2.1.(iii) This follows from the G -isomorphism M N/N ∼ = M (cid:14) ( M ∩ N ) and Statement (ii).Thus the lemma is proved. (cid:4) We say that a normal subgroup N of G is F -hypercentral in G if either N = 1 or everychief factor of G below N is F -central in G . Lemma 2.4.
Let F be a hereditary formation and put Z := Z F ( G ) . Let A , B and N besubgroups of G , where N is normal in G . Then the following hold. (i) Z is F -hypercentral in G . (ii) If N (cid:54) Z then Z/N = Z F ( G/N ) . (iii) Z F ( B ) ∩ A (cid:54) Z F ( B ∩ A ) . (iv) If B is a normal F -hypercentral subgroup of G then BN (cid:14) N is a normal F -hypercentral subgroup of G/N . Proof. (i) It suffices to consider the case Z = A A , where A and A are normal F -hypercentral subgroups of G . Moreover, in view of the Jordan-H¨older theorem for chiefseries, it suffices to show that if A (cid:54) K < H (cid:54) A A then H/K is F -central in G . Butin this case, we have H = A ( H ∩ A ) and H/K = A ( H ∩ A ) (cid:14) K ∼ = G ( H ∩ A ) (cid:14) ( K ∩ A ) , here ( H ∩ A ) (cid:14) ( K ∩ A ) is F -central in G and hence H/K is F -central in G byLemma 2.3 (ii).(ii) This assertion is a corollary of Lemma 2.3 (i), Statement (i) and the Jordan-H¨oldertheorem for chief series.(iii) First, assume that B = G and let 1 = Z < Z < . . . < Z t := Z be a chief series of G below Z . Put C i := C G ( Z i /Z i − ). Now consider the series1 = Z ∩ A (cid:54) Z ∩ A (cid:54) . . . (cid:54) Z t ∩ A = Z ∩ A. Let i ∈ { , . . . , t } . Then, by Statement (i), Z i /Z i − is F -central in G , so ( Z i ∩ A ) (cid:14) ( Z i − ∩ A )is F -central in A by Lemma 2.2 (ii). Hence, in view of the Jordan-H¨older theorem for chiefseries and Lemma 2.2 (iii), we have Z ∩ A (cid:54) Z F ( A ).Finally, assume that B is any subgroup of G . Then, in view of the preceding paragraph,we have Z F ( B ) ∩ A = Z F ( B ) ∩ ( B ∩ A ) (cid:54) Z F ( B ∩ A ) . (iv) If H/K is a chief factor of G such that N (cid:54) K < H (cid:54)
N B , then from the G -isomorphism H/K ∼ = G ( H ∩ N ) (cid:14) ( K ∩ N )we get that H/K is F -central in G by Lemma 2.3 (ii), so every chief factor of G/N below
BN/N is F -central in G/N by Lemma 2.3 (i). Therefore,
BN/N is F -hypercentral in G/N .This concludes the proof of the lemma. (cid:4)
Lemma 2.5.
Let F be a saturated formation. Then the following statements are equivalent. (i) G ∈ F . (ii) Every chief factor of G is F -central in G . (iii) G has a normal F -hypercentral subgroup N such that G/N ∈ F . Proof. (i) ⇒ (ii) This follows directly from the Barnes-Kegel result [BK66].(ii) ⇒ (iii) This implication is evident.(iii) ⇒ (i) We prove this implication by induction on | G | . Let R be a minimal normalsubgroup of G and write C = C G ( R ). In view of Lemma 2.4 (iv), the hypothesis holdsfor G/R , so
G/R ∈ F by induction. Therefore, G ∈ F if either G has a minimal normalsubgroup N (cid:54) = R or if R (cid:54) Φ ( G ).Now suppose that R (cid:10) Φ ( G ) is the unique minimal normal subgroup of G and let M bea maximal subgroup of G such that G = RM . If R is non-abelian, then C = 1 since C isnormal in G and in this case we have R (cid:10) C . Therefore, G ∼ = G/C ∈ F by hypothesis.Finally, suppose that R is abelian. Then R = C since in this case we have C = R ( C ∩ M ),where C ∩ M is normal in G . Therefore, G = R (cid:111) M ∈ F by Lemma 2.1 since R/ F -central in G and the pairs ( R, M ) and (
R, G/ C R ( R )) = ( R, G/R ) are equivalent. Ourproof is complete. (cid:4)
Lemma 2.6.
Let F be a saturated formation and assume that N is normal in G . i) If G/N ∈ F and U is a minimal supplement to N in G then U ∈ F . (ii) If U is a subgroup of G such that U ∈ F and N U = G then Z := U ∩ C G ( N ) is anormal subgroup of G such that Z (cid:54) Z F ( G ) . Proof. (i) This follows from the fact that U ∩ N (cid:54) Φ ( U ) owing to the minimality of U .(ii) Since G = N U , Z is normal in G . Moreover, if H/K is a chief factor of U below Z then H/K is F -central in U by Lemma 2.5 and H/K is a chief factor of G since N (cid:54) C G ( Z ).Also, we have N (cid:54) C G ( H/K ) and so C G ( H/K ) = N ( C G ( H/K ) ∩ U ) = N C U ( H/K ) , which in turn implies that G/ C G ( H/K ) =
N U (cid:14) N C U ( H/K ) ∼ = U (cid:14) ( U ∩ N C U ( H/K ))= U (cid:14) ( C U ( H/K )( U ∩ N ))= U (cid:14) C U ( H/K ) . Thus the pairs (cid:0)
H/K, U (cid:14) C U ( H/K ) (cid:1) and (cid:0) H/K, G (cid:14) C G ( H/K ) (cid:1) are equivalent and itfollows that H/K is F -central in G by Lemma 2.1. Therefore, Z (cid:54) Z F ( G ) as wanted. (cid:4) Proof of Theorem B.
Let D = G F , C = C G ( D ) and Z = Z F ( G ). Let U be a minimalsupplement to D in G and write T = CU .It follows that T /C ∼ = U (cid:14) ( U ∩ C ) ∈ F by Lemma 2.6 (i), so T F (cid:54) C . On the other hand,we have T F (cid:54) D since T (cid:14) ( T ∩ D ) ∼ = T D/D = CU D/D = G/D ∈ F . Therefore, T (cid:14) ( C ∩ D ) ∈ F and so, in view of Lemma 2.6 (i) again, for some subgroup H of T we have H ∈ F and T = ( C ∩ D ) H . Since C is a subgroup of T , we have C = ( C ∩ D )( C ∩ H ). Thus G = DU (cid:54) DT = D ( C ∩ D ) H = DH.
It follows that C ∩ H (cid:54) Z by Lemma 2.6 (ii), so C (cid:54) DZ and the theorem is proved. (cid:4) We have now assembled all the necessary tools to begin the proof of our main result.
Proof of Theorem A.
Assume that the assertion is false and let G be a counterexamplewith | G | + | S | minimal. Then S < G by Theorem B. By hypothesis, there is a subgroupchain S = S (cid:54) S (cid:54) . . . (cid:54) S n = G such that either S i − (cid:69) S i or S i (cid:14) ( S i − ) S i ∈ F for all i ∈ { , . . . , n } . Since S is a propersubgroup of G , we can assume without loss of generality that M := S n − < G . Now write D = S F and C = C G ( D ), so that C (cid:10) D . (Step 1). Since F is hereditary and S (cid:14) ( S ∩ G F ) ∼ = SG F (cid:14) G F (cid:54) G (cid:14) G F , we have D (cid:54) G F . Also, from Theorem B and the hypothesis, we have C ∩ S = C S ( D ) (cid:54) Z F ( S ) D = D. Step 2).
The hypothesis holds for (
W, S ) for every proper subgroup W of G containing S by [BBE06, Lemma 6.1.7], so we have C W ( D ) (cid:54) D owing to the choice of G . Nownote that S (cid:54) N G ( C ) since S (cid:54) N G ( D ), so SC is a subgroup of G and hence SC = G (otherwise, C = C SC ( D ) (cid:54) D ). (Step 3). Observe that D (cid:69) G by virtue of D being characteristic in S and (Step 2). (Step 4). Let S (cid:54) M (cid:54) V , where V is a maximal subgroup of G . Then C V ( D ) (cid:54) D by(Step 2), so V = V ∩ SC = S ( V ∩ C ) = S C V ( D ) = SD (cid:54) S and hence S = V = M . Finally, from C ∩ S (cid:54) D it follows that G/D = [
CD/D ]( S/D ) . (Step 5). First assume that S = M is not normal in G . Then G/S G ∈ F and so D (cid:54) G F (cid:54) S G , where D is normal in G by (Step 2). Now (Step 3) implies that G/D = [
CD/D ]( S/D ) , where S/D is a maximal subgroup of
G/D and thus
CD/D is a minimal normal subgroupof
G/D . Similarly, CS G /S G is a minimal normal subgroup of G/S G owing to CS G ∩ S = S G ( C ∩ S ) (cid:54) S G D (cid:54) S G . Therefore, CS G /S G is F -central in G/S G by Lemma 2.5 and the fact that G/S G ∈ F .Then, in view of the G -isomorphisms CD (cid:14) C ∼ = C (cid:14) ( C ∩ D ) = C (cid:14) ( C ∩ S G ) ∼ = CS G (cid:14) S G , the factor CD/C of G is F -central in G by Lemma 2.3 (ii). Therefore, G/D ∈ F byLemma 2.5, which in turn implies that D (cid:54) G F (cid:54) D and thus D = G F .Finally, suppose that S is normal in G so that G/S is a cyclic group of prime order by(Step 3). It follows that
G/D = [
CD/D ]( S/D ) = (
CD/D ) × ( S/D )and so
G/D ∈ F , since [ CD/C ]( G/C G ( CD/D )) ∈ F and F is hereditary. (Step 6). From (Step 5) we have D = G F , hence C (cid:54) Z F ( G ) G F = G F = D by Theorem B, against our assumption on ( G, S ). With this final contradiction our proofis concluded. (cid:4)
Proof of Corollary 1.2.
Suppose first that X is a group with no non-trivial normal F -central subgroups. In other words, assume that Z F ( X ) = 1. In this case, it is a consequenceof Theorem B that C X ( X F ) (cid:54) X F . Since X F is a normal subgroup of X , however, itfollows that X (cid:14) X F embeds isomorphically as a subgroup of Aut ( X F ). Thus we see that (cid:12)(cid:12) X (cid:14) X F (cid:12)(cid:12) (cid:54) (cid:12)(cid:12) X (cid:14) C G ( X F ) (cid:12)(cid:12) (cid:54) (cid:12)(cid:12) Aut ( X F ) (cid:12)(cid:12) , and we deduce that | X | (cid:54) (cid:12)(cid:12) Hol ( X F ) (cid:12)(cid:12) . So the assertion holds for groups with trivial F -hypercentre.Now, observe that the group X = G (cid:14) Z F ( G ) has trivial F -hypercentre by Lemma 2.4 (ii).Also, X F = U Z F ( G ) (cid:14) Z F ( G ) ∼ = U (cid:14) ( U ∩ Z F ( G )) ∼ = U. he first equality is a consequence of that fact that residuals behave well with respectquotients and the last isomorphism follows from U ∩ Z F ( G ) = 1, which holds by hypothesis.Thus (cid:12)(cid:12) G (cid:14) Z F ( G ) (cid:12)(cid:12) (cid:54) (cid:12)(cid:12) Hol (cid:0) U Z F ( G ) (cid:14) Z F ( G ) (cid:1)(cid:12)(cid:12) = | Hol ( U ) | , as wanted. The corollary is proved. (cid:4) F u rt h e r a p p l i c at i o n s Let F be a hereditary saturated formation. A subgroup A of G is said to be F -subnormal in G if there is a subgroup chain A = A (cid:54) A (cid:54) . . . (cid:54) A n = G such that A i / ( A i − ) A i ∈ F for all i ∈ { , . . . , n } . It is clear that every F -subnormal subgroup is also K - F -subnormalin the group. Therefore, we get from Theorem A the following corollaries. Corollary 3.1.
Let F be a hereditary saturated formation. Let S be an F -subnormalsubgroup of G and suppose that Z F ( E ) = 1 for every subgroup E of G such that S (cid:54) E .Then C G ( S F ) (cid:54) S F . Corollary 3.2.
Let S be a K - U -subnormal subgroup of G and suppose that Z U ( E ) = 1 for every subgroup E of G such that S (cid:54) E . Then C G ( S U ) (cid:54) S U . In this corollary S U is the supersoluble residual of S and Z U ( E ) is the supersolublehypercentre of E , that is, the product of all normal subgroups N of E such that either N = 1 or N > E below N is cyclic. In recent years, the investigations of many authors have focussed on the so-called σ -subnormal subgroups (see, for example, [Ski15], [BS17], [Ski18], [Ski19], [Ski20], [GS19],[BBKPAPC20], [BBKPAY20], [KY20], [KT20b]). We explain this notion below.Let σ be a partition of the set of all primes P , that is, σ = { σ i | i ∈ I } , where P = (cid:83) i ∈ I σ i and σ i ∩ σ j = ∅ for all i (cid:54) = j . The group G is said to be σ -primary if G is a σ i -subgroupfor some i and σ -nilpotent if G = G × · · · × G t for some σ -primary groups G , . . . , G t .The symbol N σ denotes the class of all σ -nilpotent groups.Furthermore, a subgroup A of G is said to be σ -subnormal in G if there is a subgroupchain A = A (cid:54) A (cid:54) . . . (cid:54) A n = G such that either A i − (cid:69) A i or A i / ( A i − ) A i is σ -primary for all i ∈ { , . . . , n } . It is notdifficult to show that A is σ -subnormal in G if and only if it is K - N σ -subnormal in G .The results below are thus also corollaries of Theorem A. Corollary 3.3 ([DSZ20]) . Let S be a σ -subnormal subgroup of G and suppose that Z σ ( E ) = 1 for every subgroup E of G such that S (cid:54) E . Then C G ( S N σ ) (cid:54) S N σ . Corollary 3.4.
Let S be an N σ -subnormal subgroup of G and suppose that Z σ ( E ) = 1 for every subgroup E of G such that S (cid:54) E . Then C G ( S N σ ) (cid:54) S N σ . By Z σ ( E ) we denote the σ -nilpotent hypercentre of E which is defined as the productof all normal subgroups N of E such that either N = 1 or N >
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Formations of Algebraic Systems , Nauka, Moscow,1989. † S t o c k h o l m , S w e d e n .
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