Answer to a question on A -groups, arisen from the study of Steinitz classes
aa r X i v : . [ m a t h . G R ] J u l ANSWER TO A QUESTION ON A -GROUPS, ARISEN FROMTHE STUDY OF STEINITZ CLASSES ALESSANDRO COBBE AND MAURIZIO MONGE
Abstract
In this short note we answer to a question of group theory from [2]. In that paperthe author describes the set of realizable Steinitz classes for so-called A ′ -groups ofodd order, obtained iterating some direct and semidirect products. It is clear fromthe definition that A ′ -groups are solvable A -groups, but the author left as an openquestion whether the converse is true. In this note we prove the converse whenonly two prime numbers divide the order of the group, but we show it to be falsein general, producing a family of counterexamples which are metabelian and withexactly three primes dividing the order. Steinitz classes which are realizable forsuch groups in the family are computed and verified to form a group.1. Introduction
Let
K/k be an extension of number fields with rings of integers O K and O k respectively. Then there exists an ideal I of O k such that O K ∼ = O [ K : k ] − k ⊕ I as O k -modules and the ideal I is determined up to principal ideals. Its class inthe ideal class group Cl( O k ) of O k is called the Steinitz class of the extension andis denoted by st( K/k ). For a fixed number field k and a finite group G one canconsider the set of classes which arise as Steinitz classes of tame Galois extensionswith Galois group G , i.e. the setR t ( k, G ) = { x ∈ Cl( k ) : ∃ K/k tame Galois, Gal(
K/k ) ∼ = G, st( K/k ) = x } . A description of R t ( k, G ) is not known in general, but there are a lot of resultsfor some particular groups. These results lead to the conjecture that R t ( k, G ) isalways a subgroup of the ideal class group, which however has not been proved ingeneral. In [2] the author defines A ′ -groups in the following way and proves theabove conjecture for all A ′ -groups of odd order. Definition 1.1.
We define A ′ -groups inductively:(1) Finite abelian groups are A ′ -groups.(2) If G is an A ′ -group and H is finite abelian of order prime to that of G , then H ⋊ µ G is an A ′ -group, for any action µ of G on H .(3) If G and G are A ′ -groups, then G × G is an A ′ -group.Clearly (see [2, Proposition 1.2]) every A ′ -group is a solvable A -group, whileit was asked whether the converse is true. In this short note we find a family ofcounterexamples for this. In the last section we show how the techniques from [2]can be applied also to the calculation of the realizable Steinitz classes for thesegroups, showing in particular that R t ( k, G ) is still a subgroup of the ideal classgroup, confirming the general conjecture. Mathematics Subject Classification.
Key words and phrases.
A-groups, Steinitz classes, solvable groups. Solvable A -groups which are not A ′ -groups We start showing a positive result when only two primes divide the order. See[5, 4] for general results about the A -groups. Proposition 2.1. An A -group G having order divisible by at most two differentprimes is an A ′ -group.Proof. Indeed, let G be an A -group with order divisible only by the primes p and q ; it is always solvable by Burnside Theorem. By Hall-Higman Theorem [4, SatzVI.14.16] a solvable A -group has derived length at most equal to the number ofdistinct prime divisors of the order, so in our case G has derived length at most 2and G ′ is abelian. If the derived length is 1 then G is abelian, so we are reduced toconsider the case of derived length exactly 2.We will consider the unique subgroup K p such that K p G ′ /G ′ is the p -Sylowof G/G ′ and K p ∩ G ′ is the q -Sylow of G ′ and we will show it to be normal in G . Further by Schur-Zassenhaus Theorem it is an A ′ -group, being the semidirectproduct of an abelian q -group by an abelian p -group. Constructing analogously K q , with p an q flipped, we have that K p ∩ K q = 1, while K p K q is all of G , so K p and K q are direct factors of G , since they are normal. Therefore G is isomorphicto K p × K q and consequently G is an A ′ -group by rule 3.To construct K p let’s quotient out the q -Sylow S q of G ′ , obtaining the group˜ G = G/S q . Its p -Sylow, ˜ P say, is clearly normal being the inverse image of the p -Sylow of G/G ′ , which is a p -group since we killed all the q -part of G ′ . So we havethe exact sequence 1 → ˜ P → ˜ G → ˜ G/ ˜ P → , and furthermore ˜ G ′ is equal to G ′ /S q being S q ⊆ G ′ , and is contained in ˜ P being˜ G/ ˜ P abelian.Now ˜ G ′ has a complementary factor in ˜ P which is invariant under the actionby conjugation of the q -group ˜ G/ ˜ P by [3, Theorem 2.3, Chap. 5], so let’s assume˜ P = ˜ G ′ × F p say. Clearly F p is a p -group which is normal in ˜ G , and F p ˜ G ′ / ˜ G ′ isthe p -Sylow of ˜ G/ ˜ G ′ = G/G ′ . So if we put K p to be the preimage of F p underthe projection G → ˜ G we have that K p is normal in G , K p G ′ /G ′ is the p -Sylow of G/G ′ , and K p ∩ G ′ is the q -Sylow S q of G ′ , being the preimage of F p ∩ ˜ G ′ = 1. (cid:3) For any triple p, q, r of distinct primes we construct now a counterexample whichis a metabelian group. For any integer n let C n be the cyclic group on n elements.Let a, b be integers such that qr | p a − , pr | q b − , or equivalently such that ord × qr ( p ) | a and ord × pr ( q ) | b . Let F p a and F q b respectivelybe the fields with p a and q b elements, then the multiplicative groups F × p a and F × q b actnaturally as automorphisms on the additive groups F + p a and F + q b . If φ : C q ֒ → F × p a and ψ : C p ֒ → F × q b are embeddings we can consider the semidirect products H = F + p a ⋊ φ C q , H = F + q b ⋊ ψ C p . Let’s also consider embeddings ρ : C r ֒ → F × p a and ρ : C r ֒ → F × q b , since F × p a and F × q b are abelian groups the actions induced by C r on F + p a and F + q b commute withthose of C q and C p , so ρ , ρ induce an action of C r on H and H which is trivialon C p and C q .We define G = ( H × H ) ⋊ ρ ,ρ C r , where C r acts on H i via ρ i , for i = 1 , N A QUESTION ON A -GROUPS 3 Proposition 2.2. G is a metabelian A -group which is not an A ′ -group.Proof. Indeed, G is metabelian because F + p a × F + q b is a normal abelian subgroupwith abelian quotient, isomorphic to C q × C p × C r .To show that G cannot be obtained applying rule 2 in the inductive definitionof the A ′ -groups we prove that no Sylow subgroup is normal. Since ( r, p ) = 1, a p -Sylow P is contained in H × H , and if normal then H ∩ P would be normalin H too, but C p in F + q b ⋊ C p is clearly not normal or it would be complementedby the normal subgroup F + q b and H would be abelian, which is not the case. Thesame holds for the q -Sylow of H , and similarly C r cannot be normal unless G =( H × H ) × C r and all elements of order r would be contained in the center of G ,which is not the case.To conclude we just need to show that G is not a direct product, so it also cannotbe obtained applying rule 3. Suppose G = G × G , then exactly one of G , G hasorder divisible by r , so assume r | | G | , and we have that G contains all r -Sylowsubgroups, so in particular C r ⊂ G . Then G is contained in the centralizer of C r , that considering the definition of G we can see to be equal to C p × C q × C r .But r ∤ | G | , and if p | G we would have C p ⊂ G and C p would be the p -Sylow,and hence a characteristic subgroup, of G , and consequently normal in G , whichis absurd. Since we can prove similarly that q ∤ | G | we obtain G = 1. (cid:3) We remark that some of the smallest counterexamples are those obtained puttingthe ( p, q, r ; a, b ) equal to (5 , ,
3; 2 ,
4) and (13 , ,
2; 1 , Realizable Steinitz classes
In [1], for all number fields k and all finite groups G , a subgroup W ( k, G ) of theideal class group Cl( k ) of k was defined. In [1, Theorem 2.10] it has been shownthat R t ( k, G ) ⊆ W ( k, G )and that there is an equality whenever G is an A ′ -group of odd order ([1, Theorem4.3]). So it is a natural question to investigate whether the equality holds for thesolvable A -groups constructed above, which are not A ′ -groups, when p, q, r are allodd prime numbers. Proposition 3.1.
Let p, q, r be odd prime numbers, let G be defined as in theprevious section and let k be a number field. Then R t ( k, G ) = W ( k, G ) . Proof.
As we have said above the inclusionR t ( k, G ) ⊆ W ( k, G )is true in general and is proved in [1, Theorem 2.10]. To show the opposite one wewill rely on the notation and the main results of [1].We note that G can be written as a semidirect product of the form H ⋊ G , where H = F + p a × F + p b and G = C p × C q × C r ; let π : G → G be the usual projection.Hence, by [1, Theorem 3.5] and [1, Proposition 4.3] (applied to G ), we obtainR t ( k, G ) ⊇ W ( k, G ) H Y ℓ | H Y τ ∈ H { ℓ } ∗ W ( k, E k,G,τ ) (( ℓ − / G/o ( τ )) , So it suffices to show that(1) W ( k, G ) ⊆ W ( k, G ) H Y ℓ | H Y τ ∈ H { ℓ } ∗ W ( k, E k,G,τ ) (( ℓ − / G/o ( τ )) . ALESSANDRO COBBE AND MAURIZIO MONGE
For any prime number ℓ dividing G , the ℓ -Sylow subgroups of G have exponent ℓ , i.e. for all τ ∈ G { ℓ } ∗ , the order of τ is exactly ℓ .So let τ ∈ G be of order ℓ . Then we have two possibilities:(a) π ( τ ) is of order ℓ . Then for any element σ of the normalizer of τ , we have στ σ − = τ i for some i . Hence also π ( σ ) π ( τ ) π ( σ ) − = π ( τ ) i and, since G isabelian, we can conclude that i = 1. Therefore the normalizer of τ is equalto its centralizer and so from the definition of E k,G,τ given in [1] it is clearthat E k,G,τ = k ( ζ ℓ ). Therefore we easily obtain W ( k, E k,G,τ ) (( ℓ − / G/ℓ ) ⊆ W ( k, G ) H . (b) π ( τ ) = 1. In this case τ ∈ H and we clearly have W ( k, E k,G,τ ) (( ℓ − / G/ℓ ) = W ( k, E k,G,τ ) (( ℓ − / G/o ( τ )) . So in any case we have shown that W ( k, E k,G,τ ) (( ℓ − / G/ℓ ) is contained in thesubgroup on the right-hand side of the inclusion (1), which is therefore proved,recalling the definition of W ( k, G ). (cid:3) In particular this proves that R t ( k, G ) is a group. It is also straightforward toverify that G is very good, according to the definition given in [1]. References [1] L. Caputo, A. Cobbe. An explicit candidate for the set of Steinitz classes of tame Ga-lois extensions with fixed Galois group of odd order.
Proc. London Math. Soc. , 2013,doi:10.1112/plms/pds067.[2] A. Cobbe. Steinitz classes of tamely ramified galois extensions of algebraic number fields.
Journal of Number Theory , 130(5):1129 – 1154, 2010.[3] D. Gorenstein.
Finite groups . Chelsea Pub. Co., 1980.[4] B. Huppert.
Endliche Gruppen I , volume 134 of