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M.Suzuki, Structure of a group and the structure of its lattice of subgroups, Erg.
` CA’ FOSCARI DI VENEZIA UNIVERSITA Dipartimento di Informatica Technical Report Series in Computer Science
Rapporto di Ricerca CS-2005-13 Ottobre 2005
G. Busetto On projective images of normal subgroups with abelian quotient
Dipartimento di Informatica, Universit`a Ca’ Foscari di Venezia Via Torino 155, 30172 Mestre–Venezia, Italy
On projective images of normal subgroups with abelian quotient G. Busetto*
*Indirizzo dell'autore: Dipartimento di Informatica Universita' di Venezia Ca' Foscari Via Torino 155. 30173 MESTRE
1
1. Introduction.
Since in 1951 Zappa [19] proved that if G and G are groups, π :G →G is a projectivity andG is finite and soluble, then G is soluble, a lot of work has been done in order to have a better understanding of the structure of a projective image of a soluble group of given derived length. In particular many papers are concerned in giving an upper bound for the derived length of a group which is the projective image of a soluble group with given derived length. Here by a projectivity π :G →G we mean a lattice isomorphism from the lattice of subgroups L (G ) of G to that of G , and by a projective image of a group G , we mean a group G whose lattice of subgroups is isomorphic to the lattice of subgroups of G . In 1970 Yakovlev proved that a projective image of a group of derived length n is soluble with derived length bounded by a function of n, which, in Yakovlev case, was a cubic polynomial [17]. More recently the upper bound for the derived length of G has been brought firstly to 6n-4 [3] and then to 3n-1 [4]. In Schmidt's book [13], the author poses the question of finding the best upper bound, pointing out that no example in known where the derived length ofG exceeds n+1. We also like to point out that no example is known where the derived length of G exceeds that of G , if the derived length of G is at least 3. Thus, although the best answer might be n+1 or, even better, n if n is sufficiently large, already a reasonably good improvement of the upper bound is bringing it down to 2n; for n=1, this is the real upper bound. However, if the problem of bringing the upper bound down to 2n could possibly be solved with the present knowledge of the structure of subgroup lattices of soluble groups, further improvements seem to need more information and more machinery; for example, no lattice-theoretical characterization is known of the class of soluble groups, although it is known in some special cases, like, for example, for the class of finitely generated soluble groups [11]. This shows that some important information on the lattice of subgroups of soluble groups are still missed. We mention the fact that the upper bound 2n is already established if G is a finite group of odd order, and the fact that it has not been obtained yet in total generality, depends on the peculiar behavior of 2-groups; for, the method used to obtain the upper bounds quoted above, reduces the problem to study the image of a normal subgroup N of a group G when G N is cyclic, and this case reduces easily to consider the following situation: G = N
is a finite p-group, where N is a normal subgroup of G , π :G →G is a projectivity and N π is core free in G . In this situation it is known that, if p is odd, then N is abelian and G is metabelian ([8]); if p=2 than N need not be abelian, and G need not be metabelian ([5]); however, N ′ , the derived subgroup of N , has order at most 2 and G has derived length at most 3 ([4]); using induction on the derived length of G and the information on the structure of G and G in this special case , it is easy to find the upper bound 3n-1 for the derived length of G if n is the derived length of G . However, it seems that, in order to improve such upper bound, it is necessary to obtain further information on the behavior of a normal subgroup N and its image N π via a projectivity π , in a more general situation, namely when G N is not necessarily cyclic. In this paper we consider, in the second section, such more general situation with a particular care to the case when the normal subgroup N is G ′ , the derived subgroup of G ; in the third section, as an
2 application of the obtained results, although we have not been able to obtain the upper bound 2n in general, we prove that a projective image of a metabelian group has derived length at most 4. In what follows π :G → G will always denote a projectivity from the group G to the group G ; we shall denote by H π or H indifferently the image via π of a subgroup H of G . Also, following Schmidt ([13]), we shall denote by H G and H G respectively the preimages via π of the normal closure and of the core of H in G , that is π G π π −1
H G = ((H ) )
G π
= (H )
−1
π
and
H G = ((H )G )
π −1
−1
= (H G ) π .
If g ∈G we denote by H g the preimage via π of the conjugate of H by g , that is H g = H πg π
−1
3
1. Considerations on projective images of normal subgroups with abelian quotient. Note: in all the statements of the present section, G and G will always be finite pgroups, p a prime. In the next lemma we collect some well known results on the following situation: Lemma 1.1.Let G = N , where N ∧M = G ′ . The set of elements like y generate p
α −α 1
3
p
α −α 2
3
; this completes the proof of ii). iii) Proposition 1.6, ii). iv) Proposition 1.6, iii) 2.On the derived length of a projective image of a metabelian group.
The following lemma is a particular case of a result by Menegazzo([9]); we state this particular case since Menegazzo's proof is quite long and complicated. Lemma 2.1. Let G be a group and N 