Integral group rings of solvable groups with trivial central units
aa r X i v : . [ m a t h . G R ] N ov INTEGRAL GROUP RINGS OF SOLVABLE GROUPS WITHTRIVIAL CENTRAL UNITS
ANDREAS B ¨ACHLE
Abstract.
The integral group ring Z G of a group G has only trivial centralunits, if the only central units of Z G are ± z for z in the center of G . We showthat the order of a finite solvable group G with this property, can only have 2,3, 5 and 7 as prime divisors, by linking this to inverse semi-rational groups andextending one result on this class of groups. We also classify the Frobeniusgroups whose integral group rings have only trivial central units. Introduction By G we always denote a finite group. The integral group ring of G is denotedby Z G and its group of units by U( Z G ). Clearly, all elements of ± G are units in Z G , the so-called trivial units . Already G. Higman determined in his PhD thesisin 1940 the groups such that all units in Z G are trivial: all of them are Dedekindgroups of small exponent. If π ( G ) denotes the prime spectrum of G , i.e. the set ofprime divisors of the order of G , then π ( G ) ⊆ { , } in all these cases and clearlyall of these groups are solvable.It remains a fundamental problem to describe generators for U( Z G ) if not allunits are trivial. It is well-known that the units of reduced norm 1 together withthe central units generate a subgroup of finite index in U( Z G ). For many finitegroups it has been shown by E. Jespers, G. Leal, J. Ritter and S. Sehgal that the so-called bicyclic units generate a subgroup of finite index in the subgroup of reducednorm 1 elements. These together with the so-called Bass units then generate asubgroup of finite index in the central units, see e.g. the recent monograph ofE. Jespers and ´A. del R´ıo [JdR16], Chapter 11, for a detailed treatment. It isrelevant to know when the Bass units can be omitted, in other words when allcentral units are trivial, i.e. belong to ± Z( G ). Ritter and Sehgal studied groups G with this property [RS90]. They obtained a description of the centers of theWedderburn components of the rational group algebra of such groups togetherwith a group theoretic characterization in terms of conditions on conjugacy classes,see Proposition 2.2 below. This class of groups has recently again been extensivelystudied by G.K. Bakshi, S. Maheshwary and I.B.S. Passi [BMP17, Mah16]. Weadopt their notation for these groups. Definition 1.1.
A group G is said to be a cut group (“central units trivial”) if Z G only contains trivial central units, that is, if u ∈ U( Z G ) is central, then u = ± z forsome z ∈ Z( G ), the center of G .In [Mah16, Theorems 1 and 2] Maheshwary shows that π ( G ) ⊆ { , , , } for G a cut group of odd order or a solvable cut group of even order such that all elementsare of prime power order. We show that this is true for all solvable groups: Mathematics Subject Classification.
Key words and phrases. integral group ring, central units, inverse semi-rational, solvablegroups, prime spectrum, Frobenius groups.The author is a postdoctoral researcher of the FWO (Research Foundation Flanders).
Theorem 1.2.
Let G be a solvable cut group. Then π ( G ) ⊆ { , , , } . This is best possible in the sense that none of the primes can be removed as canbe seen from the groups in Theorem 1.3. Note that for non-solvable cut groups sucha restriction on the prime spectrum cannot exist as all Weyl groups, in particularall symmetric groups S n , are cut groups. There is also a group theoretical interpre-tation of and interest in cut groups. By Proposition 2.2, a cut group is nothing butan inverse semi-rational group as introduced by D. Chillag and S. Dolfi in [CD10].And this is also the path we will take to prove Theorem 1.2: we will improve oneof the results of Chillag and Dolfi on the prime spectrum of such groups that aresolvable.As a second result, we classify the Frobenius groups with only trivial central unitsin their integral group rings. Theorem 1.3.
Let K be a Frobenius complement.(1) If | K | is even and K is the complement of a cut Frobenius group G , then G is isomorphic to a group in the series in (1a) – (1f) with b, c, d ∈ Z ≥ or G is one of the groups in (1 α ) – (1 δ ) .(a) C b ⋊ C (b) C b ⋊ C (c) C b ⋊ Q (d) C c ⋊ C (e) C d ⋊ C (f ) C d ⋊ ( Q × C ) ( α ) C ⋊ Q ( β ) C ⋊ ( C ⋊ C ) ( γ ) C ⋊ SL(2 , ( δ ) C ⋊ SL(2 , Conversely, for each of the above structure descriptions there is a cut
Frobe-nius group of that form and it is unique up to isomorphism.(2) If | K | is odd, then there is a cut Frobenius group G with complement K andFrobenius kernel F if and only if K ≃ C and one of the following holds(a) F is a cut -group admitting a fixed point free automorphism of order .In particular, F has order a for some a ∈ Z ≥ and is an extensionof an abelian group of exponent a divisor of by an abelian group ofexponent a divisor of .(b) F is an extension of an elementary abelian -group by an elementaryabelian -group, exp F = 7 and F admits a fixed point free automor-phism of order fixing each cyclic subgroup of F . Theorem 1.2 is proved in Section 3, while Theorem 1.3 is proved in Section 4.We give corollaries describing cut
Frobenius groups with abelian Frobenius kernel(Corollary 4.6) and cut
Camina groups (Corollary 4.8), see the end of Section 4.The notation is mainly standard. C n denotes a cyclic group of order n , Q thequaternion group of order 8. F q denotes a finite field with q elements. Euler’stotient function is indicated by φ . Standard references are [Hup67, Sco87].2. Preliminaries
We need the following group theoretical notion of (inverse) semi-rational groupsthat was coined by Chillag and Dolfi [CD10]. By ∼ we denote conjugacy (in agroup that should always be clear from the context). Definition 2.1.
Let G be a group and x ∈ G .(1) x is rational in G if x j ∼ x for all j ∈ Z with ( j, o ( x )) = 1.(2) The element x is called inverse semi-rational in G if x j is conjugate to x or x − for all j ∈ Z with ( j, o ( x )) = 1. NTEGRAL GROUP RINGS WITH TRIVIAL CENTRAL UNITS 3 (3) x is semi-rational in G if each if x j is conjugate to x or x m for all j ∈ Z with ( j, o ( x )) = 1 for some fixed m .A group is called rational (respectively inverse semi-rational , semi-rational ) if everyof its elements is respectively of that type.For a character ψ (ordinary or Brauer), Q ( ψ ) denotes the field of character valuesof ψ , that is, the field extension of Q generated by { ψ ( g ) : g ∈ G } (respectively by { ψ ( g ) : g ∈ G p ′ } if ψ is a p -Brauer character and G p ′ denotes the elements of G with an order not divisible by p ). It is well-known that a group G is rational if andonly if for each χ ∈ Irr( G ), Q ( χ ) = Q , whence the name.The following characterization of cut groups is essentially due to Ritter andSehgal. We link it to inverse semi-rational groups. Proposition 2.2.
Let G be a group. The following are equivalent.(1) G is a cut group.(2) G is inverse semi-rational.(3) For every x ∈ G and k ∈ Z with ( k, | G | ) = 1 , x k ∼ x or x k ∼ x − .(4) If Q G ≃ L mk =1 M n k ( D k ) is the Wedderburn decomposition ( m, n k ∈ Z ≥ , D k rational division algebras), then for each k , Z( D k ) ≃ Q ( √− d ) for some d ∈ Z ≥ , where Z denotes the center.(5) For each χ ∈ Irr( G ) , Q ( χ ) = Q ( √− d ) for some d ∈ Z ≥ , i.e. the field ofcharacter values of χ is Q or an imaginary quadratic extension of Q foreach absolutely irreducible character of G .Proof. The equivalence of (1), (3), (4), and (5) is proved in [RS90]. Clearly, (2)implies (3). To see that (3) ⇒ (2), let x ∈ G and j ∈ Z with ( j, o ( x )) = 1 begiven. We can choose k = o ( x ) · s + j for some s ∈ Z such that ( k, | G | ) = 1(this is possible, for instance, by Dirichlet’s Theorem on the infinitude of primes inarithmetic progressions). Then x j = x k is conjugate to x or x − , by (3). (cid:3) We will use these equivalences freely without further mention. From (4) it easilyfollows that each quotient of a cut group is again a cut group. Note that Q ( G ), thefield extension of the rationals generated by the values of all irreducible charactersof the cut group G might have degree larger than 2 over Q as can already be seenby the non-abelian group of order 21. We have { G : G is rational } ⊆ { G : G is cut } ⊆ { G : G is semi-rational } . In case of groups of odd order, the first class only contains the trivial group, andthe last two classes coincide, cf. [CD10, Remark 13].We will use the following notation introduced by Chillag and Dolfi together withtheir observations we state here.
Definition 2.3.
Let G be a group, x ∈ G . Set B G ( x ) = N G ( h x i ) / C G ( h x i ), theautomorphisms of h x i induced by elements of G . We identify B G ( x ) with thecorresponding subgroup of Aut( h x i ). Lemma 2.4 ([CD10, Lemma 5 (2), (3)]) . Let G be a group.(2) An element x is inverse semi-rational in G if and only if Aut( h x i ) = B G ( x ) h τ i , where τ ∈ Aut( h x i ) is the inversion automorphism of h x i de-fined by τ ( y ) = y − . In particular for an inverse semi-rational element x in G , we have [Aut( h x i ) : B G ( x )] ≤ .(3) If p is a prime of the form p ≡ , then a p -element x is inversesemi-rational in G if and only if x is rational in G . ANDREAS B¨ACHLE Solvable Groups
It is a classical result of R. Gow [Gow76] that the orders of solvable rationalgroups are divisible only by the primes 2, 3 and 5. In such results, the treatment ofthe smallest primes usually requires the most effort; Gow used local Schur indicesto show that 7 cannot divide the order of a solvable rational group. The primesspectra of semi-rational groups were studied by Chillag and Dolfi [CD10]. Theyshowed that π ( G ) ⊆ { , , , , , } for a solvable semi-rational group G . Itseems to be still unknown whether 17 can divide the order of a solvable semi-rationalgroup. For solvable groups G that are inverse semi-rational (i.e. cut ), they prove π ( G ) ⊆ { , , , , } , however it remained open whether 13 can be omitted fromthe list. In [Mah16, Theorems 1 and 2] Maheshwary shows that π ( G ) ⊆ { , , , } for G a (solvable) cut groups of odd order or a solvable cut groups of even ordersuch that each element has order a prime power (with different techniques thanwe use here). To prove that this is best possible even for all solvable cut groups,we improve the above mentioned result of Chillag and Dolfi [CD10, Theorem 2] onsolvable inverse semi-rational groups, and show that 13 can be removed from thelist.In the investigation of the prime spectra of classes of solvable groups, the notionof the k -eigenvalue property, as introduced in [FeS86, (2.8) Definition] by E. Fariase Soares, has proved to be very useful. Definition 3.1.
Let a group G act on a finite dimensional vectorspace V over afinite field F . We say that V , or the action of G on V , has the k -eigenvalue property ,for some k ∈ Z > provided that (a) k divides | F × | and (b) for every v ∈ V , thereexists g ∈ G such that v g = λv , where λ is some fixed element of order k in F × .A key point in the proof of Theorem 1.2 is the following result due to Farias eSoares, so that we will restate it here. By ζ n we denote a primitive n th root ofunity. Theorem 3.2 ([FeS86, (2.9) Theorem B (c)]) . Let H be a solvable group actingon a finite-dimensional vector space V over F p , where p ∤ | H | . Assume that theaction has the k -eigenvalue property. Let ϕ be a Brauer character afforded by V and K = Q ( ϕ ) . If k = | K ( ζ p ) : K | and | K : Q | ≤ , then p ∈ { , , , } .Proof of Theorem 1.2. By [CD10, Theorem 2] we have π ( G ) ⊆ { , , , , } forsolvable cut groups G . Assume that G is a solvable cut group of minimal order suchthat 13 ∈ π ( G ). Let 1 = G n +1 E G n E ... E G E G = G be a chief series of G . Note that each factor group G/G j is again a cut group. Hence G n is the only chief factor of G divisible by 13 and, as a minimal normal subgroup,elementary abelian. Thus V = G n ∈ Syl ( G ) and G = V ⋊ α H , where H ≃ G/V is also a cut group of order not divisible by 13. Now, the group homomorphism α : H → Aut( V ) ≃ GL( d,
13) turns V into an F H -module, which is irreducibleas V is a minimal normal subgroup of G . We will write V additively. Let ϕ be theBrauer character of H afforded by V . Then ϕ is actually an ordinary character as13 ∤ | H | . By [Ten12, Lemma 3] and the remark following it, the field of charactervalues of ϕ is contained in the field of character values of each of its irreducibleconstituents. But then [ Q ( ϕ ) : Q ] ≤ ⇒ (2). All elementsof order 13 in G are rational in G by Lemma 2.4, so for x ∈ V we have that j · x ∼ x for each j ∈ Z with ( j,
13) = 1, hence there exists h ∈ H such that x h = j · x . Inparticular, the H -module V has the 12-eigenvalue property. But then we are in thesituation of Theorem 3.2: H is solvable, K = Q ( ϕ ) has degree at most 2 over Q NTEGRAL GROUP RINGS WITH TRIVIAL CENTRAL UNITS 5 and k = [ K ( ζ ) : K ] = 12, as K ∩ Q ( ζ ) = Q (the unique quadratic subfield of Q ( ζ ) is real) and H (and hence G ) cannot exist. (cid:3) Remark 3.3.
By looking a bit closer at the group structures, one can see that aset S is the prime spectrum of a solvable cut -group if and only if S is one of thefollowing sets: { } , { } , { , } , { , } , { , } , { , , } , { , , } , { , , , } . By the bound on the prime spectrum in Theorem 1.2, the description of cut -groupsof odd order in [CD10, Theorem 3] and the examples in [BMP17, Theorem 5] it onlyremains to consider { , } , { , , } and { , , , } . For the first two this can bedone by purely arithmetic conditions: note that the generators of a cyclic subgroupof order 7 decompose into one or two orbits under the conjugation action. But thismeans that the group has a section of order 6 or 3 which is impossible. The solvable cut -groups ( C ⋊ Q ) × ( C ⋊ C ) and S × ( C ⋊ Q ) × ( C ⋊ SL(2 , { , , , } . 4. Frobenius Groups
Frobenius groups that are rational have been determined in [DS04] (there are twoinfinite series together with “Markel’s group”), and the structure of semi-rationalFrobenius groups has been studied in [ADD16], though a complete classification hasnot been obtained. We will now prove the classification of cut
Frobenius groupsgiven in Theorem 1.3 in the introduction. To this end we will first show thatthere are only eight isomorphism types of potential Frobenius complements in cut
Frobenius groups (Proposition 4.2) and use this to determine the structure of all cut
Frobenius groups. The general structure of Frobenius groups is fairly wellunderstood, standard references include [Hup67, Pas68, Sco87].In the following proposition we strengthen the conditions of [ADD16, Lemma 2.5]for cut groups. Note that we can in particular prove that cut
Frobenius groups aresolvable, which is not the case for semi-rational Frobenius groups (e.g. the Frobeniusgroup C ⋊ SL(2 ,
5) is indeed semi-rational but not cut ). Small parts of the proofsare similar to those in [ADD16], however we include the complete proofs to makethe exposition self-contained.
Proposition 4.1.
Let G be a cut Frobenius group. Then G is solvable. If K is aFrobenius complement of G , then the Sylow p -subgroups of K are cyclic of order atmost p if p is odd and cyclic of order at most or quaternion of order for p = 2 .Proof. Assume by way of contradiction that the cut
Frobenius group G is notsolvable. Then a Frobenius complement K of G is not solvable and, by a theoremof Zassenhaus [Pas68, 18.6.], K has a normal subgroup of index at most 2 isomorphicto SL(2 , × M , where M is of order coprime to 30. By [JdR16, Theorem 11.4.10] G has a quotient isomorphic to SL(2 ,
5) or to the covering group of S havingSmallGroup ID [240, 89] in GAP [GAP16]. It is readily checked that the lastmentioned group has fields of character values isomorphic to Q ( √
2) and Q ( √ ,
5) has a character field Q ( √ cut groups. Asone of the two necessarily appears as a quotient of G , G is also not cut , implyingthat cut Frobenius groups are necessarily solvable.Let p be an odd prime and P ∈ Syl p ( K ) with P = 1. Then P = h x i ≃ C p f by[Pas68, 18.1] and | Aut( P ) | = ( p − p f − . As P is abelian, p ∤ | B K ( x ) | . But then[Aut( P ) : B K ( x )] ≤ f ≤ p = 2 the Sylow 2-subgroups of K are cyclic or generalized quaternion by[Pas68, 18.1]. If P ∈ Syl ( K ) is cyclic, then K has a normal 2-complement byCayley’s normal 2-complement theorem, i.e. K ≃ M ⋊ P with M a group of odd ANDREAS B¨ACHLE order. Then P is a cyclic 2-group that is a cut group as a quotient of a cut groupand hence of order at most 4. Now assume that P = h x, y | x n − = 1 , y = x n − , x y = x − i is a (generalized) quaternion Sylow 2-subgroup of K of order 2 n , n ≥
3. Then h x i E P is a cyclic normal subgroup of order 2 n − . Clearly B P ( x ) = h τ i , where τ denotes the inversion-automorphism of h x i . But as P ∈ Syl ( K ), the 2-part ofthe order of B K ( x ) and B P ( x ) coincide. As | Aut( h x i ) | = φ (2 n − ) = 2 n − , a powerof 2, we have by Lemma 2.4, that x is inverse semi-rational in K if and only if2 n − = 2, hence n = 3 and P is the quaternion group of order 8. (cid:3) From the fact that the Frobenius complement K is a solvable cut group, we inferwith Theorem 1.2 that π ( K ) ⊆ { , , , } , so in particular there are only finitelymany isomorphism types of Frobenius complements in Frobenius groups that are cut . Proposition 4.2. If K is a Frobenius complement that is cut , then K is isomorphicto one of the following groups: C , C , C , C , Q , C ⋊ C , SL(2 , , Q × C . Proof.
To filter the correct complements, we use π ( K ) ⊆ { , , , } , the structureof the Sylow subgroups obtained in Proposition 4.1 together with the followingadditional results on Frobenius groups which eliminate another 7 possibilities :(1) If the Frobenius complement is of even order, then it has a unique (hencecentral) involution [Pas68, Theorem 18.1].(2) A Frobenius complement cannot contain a subgroup which is itself a Frobe-nius group [Sco87, 12.6.11].This can be done by working out the possible structures of such groups or by asearch in GAP of the groups meeting these requirements. By the bound of the orderobtained by the structures of the Sylow subgroups this is a finite problem. (cid:3)
In a large part of the proof of Theorem 1.3 we will have to deal with Frobeniusgroups G that have an elementary abelian p -group as Frobenius kernel F . If, inthis situation, K denotes a Frobenius complement of G , then we can consider F as an F p K -module, which is semi-simple. Hence F ≃ V ⊕ ... ⊕ V s for irreducible F p K -modules V j , which are necessarily faithful. Each V j then corresponds to anordinary character of K . If V is an F p K -module, such that V ⋊ K is a Frobeniusgroup with complement K , then we can define a new group (a subdirect product)by taking V s ⋊ K with “diagonal action”, i.e. K acts on all s copies of V in thesame way, that is if ( x , ..., x s ) ∈ V s , then ( x , ..., x s ) y = ( x y , ..., x ys ) for y ∈ K (and x yj is defined in V ⋊ K ). Clearly, this is again a Frobenius group. Definition 4.3.
Let V be a F p K -module such that G = V ⋊ K is a Frobeniusgroup. Let β : K → GL F p ( V ) be the corresponding homomorphism. For x ∈ V denote by Z G ( x ) those automorphisms of h x i that are induced by elements of β ( K ) ∩ Z(GL F p ( V )).Thus the elements of Z G ( x ) are exactly those automorphisms of h x i induced bythose multiples of the identity of V that come from elements of K . Actually, Z G ( x )is equal for each x ∈ V \{ } . Note that if in the previous definition the F p K -module V is absolutely irreducible, then Z G ( x ) are those automorphisms of h x i that areinduced by elements of β (Z( K )). For later use we record the following observation. Namely S , C ⋊ C with faithful action, C ⋊ C , C ⋊ C with faithful action, C × ( C ⋊ C ), C ⋊ C with faithful action and C × ( C ⋊ C ). NTEGRAL GROUP RINGS WITH TRIVIAL CENTRAL UNITS 7
Lemma 4.4.
Let K be a group, let p an odd prime and let V be an F p K -modulesuch that G = V ⋊ K is a cut Frobenius group. Then the following are equivalent:(1) V s ⋊ K (with diagonal action) for all s ≥ is a cut Frobenius group.(2) V ⋊ K (with diagonal action) is a cut Frobenius group.(3) Z G ( x ) h τ i = Aut( h x i ) for each x ∈ V , where τ is the inversion automor-phism of h x i .Proof. We will only give a proof in case p ≡ x of order p is inverse semi-rational in a group X if andonly if B X ( x ) = Aut( h x i ), by Lemma 2.4.Clearly, (1) implies (2). Now assume that (2) holds and we want to show (3).Hence assume that H = V ⋊ K is a cut Frobenius group. We have to showthat all automorphisms of h x i for x ∈ V ≤ G are induced by scalar multiplesof the identity that come from an element of K . Let x ∈ V \ { } and λ ∈ F × p .Then there is a unique k ∈ K such that x k = λx , because G is a cut Frobeniusgroup. Now for any y ∈ V , there is an ℓ ∈ K such that ( x, y ) ∈ V ≤ H getsconjugate by ℓ to λ ( v, w ), as V ⋊ K is a cut group. Now as the action is diagonal,( x ℓ , y ℓ ) = ( x, y ) ℓ = λ ( x, y ) = ( λx, λy ) = ( x k , λy ). By uniqueness, k = ℓ and theaction of k multiplies all elements of V by the same scalar λ , hence multiplicationby λ is an element of Z G ( x ) for every x ∈ V .For the implication (3) ⇒ (1), assume that Z G ( x ) = Aut( h x i ) for each x ∈ V and set L = V s ⋊ K for some s ≥ L is a Frobeniusgroup. For each y ∈ V s , B L ( y ) = Aut( h y i ) and y is (inverse semi-) rational in L .As L is the union of conjugates of the Frobenius complement K (which is cut as aquotient of the cut group G ) and the Frobenius kernel V s , it follows that L is a cut group. (cid:3) Proof of Theorem 1.3.
Let G always be a Frobenius group with Frobenius kernel F and Frobenius complement K .(1) If K has even order, then the Frobenius kernel F is abelian [Hup67, V, 8.18].Let x ∈ F \ { } . Then C G ( x ) = F and hence B G ( x ) = N G ( h x i ) / C G ( h x i ) isisomorphic to a subgroup of K . Note that if x ∈ F is an element of order r n for an odd prime r , then we need that φ ( r n ) / ( r − r n − divides | K | ,by Lemma 2.4; if in this situation the 2-part of ( r −
1) is at least 4, thenwe even have that φ ( r n ) divides | K | , as Aut( C r n ) is cyclic. From this itfollows that n ≤
1, i.e. all Sylow subgroups of F are elementary abelian.Going through the possibilities for even order Frobenius complements in cut groups given in Proposition 4.2, we obtain the candidates for Frobeniuskernels stated in Table 1 sorted according to their Frobenius complements. Table 1.
Possible isomorphism types of Frobenius complements K together with their Frobenius kernels F of cut Frobenius groups. K Candidates for FC C b C C b , C c , C b × C c C C d Q C b , C c , C b × C c C ⋊ C C c , C d , C c × C d SL(2 , C c , C d , C c × C d Q × C C c , C d , C c × C d For each of these possibilities, we will now prove that either there is aunique cut
Frobenius group of that form or that no cut
Frobenius group ofthat form exists. We will first handle the cases where the Frobenius kernel F is an elementary abelian p -group, as we have representation theory atour disposal in these situations. When considering F as an F p K -module,it will be written additively. For the convenience of the reader we providethe GAP
SmallGroup ID of the group whenever this is appropriate.(i) F = C d , K = C : We provide details in this case. We will prove thatthere is a unique cut Frobenius group with kernel F and complement K for each d . Note that F is a splitting field for C and there are two non-isomorphic faithful irreducible F C -modules W and W . A generator t of C acts as multiplication by ¯3 and ¯5 on W ≃ F and W ≃ F , respectively.Clearly W ⋊ C and W ⋊ C are cut Frobenius groups and so are W d ⋊ C and W d ⋊ C for all d ≥ V ⊕ ... ⊕ V d ) ⋊ C , (1)where each V j is either isomorphic to W or W , is a cut Frobenius group,then all V j ’s have to be isomorphic. For if not, then for the element x = ( ¯1 , ¯1 ,..., ¯1 ) ∈ V ⊕ V ⊕ ... ⊕ V d , the subgroup h x i contains only two G -conjugates of x , i.e. B G ( x ) ≃ C , and x is not inverse semi-rational in G .Note that the isomorphism type of G does not depend on whether all V j ’sin (1) are isomorphic to W or to W as t t induces an automorphism of C , which in turn induces an isomorphism W d ⋊ C → W d ⋊ C . Thus, upto isomorphism, we get a unique cut Frobenius group of the form C d ⋊ C for each d ∈ Z ≥ as stated in (1e) of Theorem 1.3.(ii) F = C c , K = C : With the same argument as in the previous case, wesee that there is a unique cut Frobenius group as given in (1d).(iii) F = C b , K = C : There is a unique faithful irreducible F C -module,where C acts by inversion on F . Giving us case (1a) of Theorem 1.3.(iv) F = C b , K = C : There is a unique faithful irreducible F C -module W which is 2-dimensional as F -vector space and for a generator t of C , t acts as (cid:0) ¯2 .. ¯2 (cid:1) on W . Hence W ⋊ C is a cut Frobenius group and, byLemma 4.4, we get that W s ⋊ C is a cut Frobenius group for each s ≥ F = C b , K = Q : Again there is a unique faithful irreducible F Q -module W which is 2-dimensional as F -vectorspace and the unique centralinvolution of K acts as (cid:0) ¯2 .. ¯2 (cid:1) on F . We can apply the same argument asabove and see that there is a cut Frobenius group of the form C b ⋊ Q ifand only if b is even. Thus we are in case (1c)(vi) F = C c , K = Q : Also in this case there is a unique faithful irreducible F Q -module W which is 2-dimensional as F -vectorspace. Now it can bechecked that W ⋊ Q is even a rational Frobenius group (also known asMarkel’s group, GAP
SmallGroup ID [200, 44] ). Note that only the centerof Q acts by diagonal matrices on W . Hence by Lemma 4.4, W s ⋊ Q cannever be a cut Frobenius group if s ≥
2. Thus there is a cut
Frobeniusgroup of the form C c ⋊ Q if and only if c = 2; this is case (1 α ).(vii) F = C c , K = C ⋊ C : The group K has a unique faithful irreduciblecharacter, which is of degree 2 and rational valued. Hence there is a uniquefaithful F K -module W which is 2-dimensional as F -vectorspace. It canbe checked that G = W ⋊ K ( GAP
SmallGroup ID [300, 23] ) is in facta cut
Frobenius group. However Z( K ) ≃ C and hence W s ⋊ K cannot be cut for s ≥
2, by Lemma 4.4, and we just obtain the group in (1 β ). NTEGRAL GROUP RINGS WITH TRIVIAL CENTRAL UNITS 9 (viii) F = C d , K = C ⋊ C : As in the previous case there is a uniquefaithful F K -module W which is 2-dimensional as F -vector space. Howeverin this case the group W ⋊ K ( GAP
SmallGroup ID [588, 33] ) is not cut as for three of its irreducible characters the field of character values is themaximal real subfield of Q ( ζ ).(ix) F = C c , K = SL(2 , , F SL(2 , W . Then W is absolutely irreducible and dim F W = 2. It is easily seenthat W ⋊ SL(2 ,
3) is a cut group (e.g. because SL(2 ,
3) acts transitively on W \{ } ). As Z(SL(2 , cut Frobeniusgroup of the form C c ⋊ SL(2 ,
3) for c >
2, by Lemma 4.4. Hence there isa unique cut
Frobenius group of the form C c ⋊ SL(2 ,
3) and c = 2 in thiscase (with GAP
SmallGroup ID [600, 150] ). Case (1 γ ).(x) F = C d , K = SL(2 , cut Frobenius group of the form C d ⋊ SL(2 ,
3) if and only if d = 2. This group has the GAP
SmallGroup ID [1176, 215] , this is case(1 δ ).(xi) F = C c , K = Q × C : The group K has two faithful irreducibleordinary characters, they are complex conjugates of each other and of degree2. Note that their field of values is Q ( ζ ), a corresponding representation incharacteristic 5 can be realized over F and a corresponding F K -module W has dimension 4. The group W ⋊ K has 20 irreducible characters withfield of character values Q ( √ cut Frobenius group ofthe form C c ⋊ ( Q × C ).(xii) F = C d , K = Q × C : From the previous case we see that there aretwo faithful irreducible F K -modules W and W which can be realized over F . Similar to in the case (i), one can argue that in a cut Frobenius grouponly one of the isomorphism types can be involved. Note that Z( K ) ≃ C and hence these elements act in a faithful F -representation as scalarmultiples of the identity. In particular W j ⋊ K is a cut Frobenius group andthen, by Lemma 4.4, each W sj ⋊ K is a cut Frobenius group ( j ∈ { , } ).Again as in the case (i) one can see that both, W s ⋊ K and W s ⋊ K , leadto isomorphic groups. Hence there is a (unique) cut Frobenius group of theform C d ⋊ ( Q × C ) if and only if d is even, which is case (1f).It remains to handle the cases where the Frobenius kernel is divisible bytwo different primes. In what follows, we will always assume that b, c, d arepositive. Assume first that G = ( C c × C d ) ⋊ ( C ⋊ C ) is a cut Frobeniusgroup with kernel F = C c × C d . Then G/O ( G ) ≃ C d ⋊ ( C ⋊ C ) is a cut Frobenius group, contradicting case (viii) above. Similarly, the existence of( C c × C d ) ⋊ ( Q × C ) would contradict (xi). Now assume that G is a Frobenius group with kernel C b × C c and the Figure 1.
Subgroupsof Aut( C ) ≃ C × C h α i h β i h αβ i C ≃ h β i h α, β i ≃ C × C h αβ i ≃ C h α, β i ≃ C × C complement K is either C or Q .Let x ∈ G be an element of or-der 15. Then Aut( h x i ) = h α i ×h β i ≃ C × C . As the Frobe-nius kernel is abelian, C G ( x ) = F and we have that B G ( x ) =N G ( h x i ) / C G ( h x i ) is isomorphic toa subgroup of K . But it is alsoa subgroup of Aut( h x i ). As x is inverse semi-rational, we have | B G ( x ) | ≥
4, hence B G ( x ) iscyclic of order 4 and B G ( x ) = h ¯ y i ,where y acts on x as β or as αβ , cf.Figure 1. In the first case x is leftinvariant by y , hence y ∈ C G ( x ),contradicting the fact that G isFrobenius. In the second case x is stabilized by y and y ∈ C G ( x ), again a contradiction with G being aFrobenius group.The last remaining case is if G is a cut Frobenius group with kernel F = C c × C d and complement K = SL(2 , x ∈ F of order 35. Then B G ( x ) ≤ Aut( C ) hasto have order 12 or 24 = φ (35). But SL(2 ,
3) does not contain a subgroupof order 12 and Aut( C ) is clearly not isomorphic to SL(2 , | K | is odd, then K ≃ C by Proposition 4.2, and the Frobenius kernel F of G is a { , , } -group by Theorem 1.2. If | F | is be divisible by 5, thenthe nilpotent Frobenius kernel F contains an element z of order 5 in itscenter. Then z has to be rational by Lemma 2.4. But this is impossibleas B G ( z ) ≤ C . We infer that F is a { , } -group. First assume that | F | is divisible by both primes, 2 and 7. Then F has an element z of order14 in its center and F ≤ C G ( x ). As G is a cut group, B G ( x ) ≃ C . Butthen the unique involution in h x i is central in G , which is a contradiction.Consequently, F is a p -group for p = 2 or p = 7. By a classical result ofBurnside, see e.g. [Hup67, V, 8.8], the Frobenius kernel F is nilpotent ofclass at most 2 and thus metabelian. Let x ∈ Z( F ). Then C G ( x ) = F and B G ( x ) ≤ K ≃ C . Hence φ ( o ( x )) / | o ( x ) | p = 2) or o ( x ) | p = 7). Now F/F ′ ⋊ K is an epimorphic image of G , andtherefore a cut group with F/F ′ being abelian. Thus for each y ∈ F/F ′ wehave φ ( o ( y )) / | F/F ′ is of exponent 4 (if p = 2) or of exponent 7 (if p = 7), respectively. As F is metabelian, F ′ ≤ Z( F ), and the claim on thestructure of the group follows.Assume now that F is a 2-group and x ∈ F is a 2-element. Then theorder of Aut( h x i ) is a power of 2 and hence B F ( x ) = B G ( x ) and F has tobe cut . The order of F has to be an even power of 2 as 3 | ( | F | − cut cut Frobenius group.If F is a 7-group, then clearly exp F is a divisor of 7 , by the above, wewant to show that exp F = 7. Deny the latter and assume that exp F = 49,then there is an element y ∈ F of order 49. Then necessarily B G ( y ) ≃ C ,as G is cut . Hence there is an element x ∈ F of order k ∈ { , } normalizing h y i , but not centralizing it. But then H = h x, t i ≃ C k ⋊ C , where t is a NTEGRAL GROUP RINGS WITH TRIVIAL CENTRAL UNITS 11 generator of the Frobenius complement, and B G ( y ) is an epimorphic imageof H . But this is a contradiction since H does not have an epimorphicimage that is cyclic of order 21. Hence F has exponent 7. Each 7-element x has to be inverse semi-rational in G , thus it has to have three conjugatesin the cyclic subgroup generated by it, so the generator t of a Frobeniuscomplement has to normalize h x i . Conversely, assume that F is a 7-groupof exponent 7 such that it admits a fixed point free automorphism α of order3 mapping each cyclic subgroup to itself. Then, clearly, F ⋊ h α i defines a cut Frobenius group. The result is proved. (cid:3)
Remark 4.5.
As by [CD10, Remark 13], groups of odd order are semi-rational ifand only if they are cut (= inverse semi-rational), (2b) of the Theorem 1.3 also givesa more precise description of the Frobenius groups appearing in [CD10, Theorem3], where the semi-rational groups of odd order are classified.The arguments from the last part of the proof of (1) can also be used to reducethe list of potential semi-rational Frobenius groups in [ADD16, Theorem 1.2], e.g.groups with complement C or Q , and Frobenius kernels with an order divisi-ble by two different primes can be removed from the list as well as groups withcomplement Q and kernel being an elementary abelian 17-group. Some of the ar-guments also apply to show that the isomorphism type of semi-rational Frobeniusgroups of a certain type are unique. In general, a semi-rational Frobenius group isnot uniquely determined by its Frobenius kernel and complement. The two non-isomorphic Frobenius groups ( C × C ) ⋊ C are both semi-rational (however, onlyone of them is cut ).We now describe the cut Frobenius groups with abelian kernel.
Corollary 4.6.
Let G be a cut Frobenius group with abelian kernel. Then G appearsin Theorem 1.3 (1) or is isomorphic to ( C a × C a ′ ) ⋊ C or C d ⋊ C ( a, a ′ ∈ Z ≥ , a + a ′ > , d ∈ Z ≥ ) . For each of the above structures there is a unique cut
Frobenius group of that form.Proof.
If the order of the complement is even, then the group was handled inTheorem 1.3 (1). In the case of a complement of odd order, it follows from the proofof Theorem 1.3 (2) that the Frobenius kernel F is an abelian group of exponent 7or a divisor of 4.If exp F = 7, then we can argue as in (i) in the proof of Theorem 1.3, to see thatthere is a unique cut Frobenius group of the form and C d ⋊ C for all d ∈ Z ≥ .Assume that F is a 2-group. As all groups of the form C a × C a ′ are cut , thetask amounts to find all Frobenius groups of the form ( C a × C a ′ ) ⋊ C . But it iswell-known that there is such a group if and only if a, a ′ ∈ Z ≥ , not both zero,and this group is unique up to isomorphism. This can be extracted, for example,from the more general discussion in [BH98, Section 11]. (cid:3) Remark 4.7.
Turning our attention to cut
Frobenius groups with non-abelianFrobenius kernels we provide examples in both situations, when the kernel is a 7-group and when it is a 2-group. For the first, we refer to [CD10, Example 1]. Thereit is shown that the Heisenberg group P of order 7 (i.e. the non-abelian group oforder 7 and exponent 7) admits a fixed-point free automorphism of order 3, whichresults in a Frobenius group P ⋊ C that is semi-rational (recall that in the case ofgroups of odd order, cut and semi-rational are the same [CD10, Remark 13]). The group X = h s, t, u, v | s = t = u = v = 1 = [ s, t ] = [ u, v ] ,s u = st , t u = s t, s v = s , t v = s t i ≃ ( C × C ) ⋊ ( C × C )has exponent 4 and admits a fixed-point free automorphism α of order 3, given by s α = s t , t α = s, u α = uv, v α = u. Hence G = X ⋊ h α i is a Frobenius group. As X has exponent 4, G is actually a cut group. Using this group, we can construct cut Frobenius groups with non-abelianFrobenius kernels of order 2 e ·
3, for each e ∈ Z ≥ .Now consider the group Y = h s, t, u, v | s = u = 1 , s = t ∈ Z( Y ) , u = v ∈ Z( Y ) , [ s, t ] = u , [ s, u ] = s , [ s, v ] = u , [ t, u ] = u , [ t, v ] = s , [ u, v ] = s i . Then Z( Y ) = Y ′ = h s , u i is isomorphic to a Klein four group C and Y / Z( Y )is isomorphic to C . The group Y also admits a fixed-point free automorphism β of order 3 and exp Y = 4, so that we get another example of a cut Frobeniusgroup Y ⋊ h β i , now with a Frobenius kernel that is not split. The GAP
SmallGroupIDs of these groups are [64, 242] , [192, 1023] , [64, 245] and [192, 1025] ,respectively.All other examples of cut Frobenius groups with indecomposable (i.e. it cannot bewritten as direct product of smaller groups) non-abelian Frobenius kernel of orderat most 2 · [768, 1083600] , [768, 1083604] , [768,1083733] , [768, 1085039] with Frobenius kernel [256, 6815] , [256, 6817] , [256, 10326] , [256, 55682] , respectively. The latter all have exponent 4 (ex-cept [256, 10326] , which has exponent 8).A group G is called a Camina group , if its derived subgroup G ′ is different from1 and G and x G = xG ′ for all x ∈ G \ G ′ , i.e. the conjugacy class and the cosetmodulo G ′ coincide for elements outside G ′ . (Note that in some definitions G ′ = 1is allowed for Camina groups.) In [BMP17, Theorem 4] it is proved that a Camina p -group has the cut property if and only if p ∈ { , } . With the classification ofall Camina groups by R. Dark and C.M. Scoppla cf. [DS96, IL15] and the aboveclassification of cut Frobenius groups we obtain the following result.
Corollary 4.8.
A Camina group G is a cut group if and only if one of the followingholds: • G is a - or a -group, • G is a Frobenius group of the form ( C n × C m ) ⋊ C , C n ⋊ C , C n ⋊ C , C n ⋊ Q , C ⋊ Q for n, m ∈ Z ≥ . • G a Frobenius group of the form C n ⋊ C , C n ⋊ C , C n ⋊ C , for n ∈ Z ≥ , where a generator of the complement raises each element ofthe Frobenius kernel to the same power. • G is a cut Frobenius group with a cyclic complement of order and non-abelian kernel as described in Theorem 1.3 (2) . Acknowledegment.
The author is grateful to Silvio Dolfi for an interesting com-munication on inverse semi-rational groups and would like to thank Eric Jespersfor stimulating discussions.
NTEGRAL GROUP RINGS WITH TRIVIAL CENTRAL UNITS 13
References [ADD16] S.H. Alavi, A. Daneshkhah, and M.R. Darafsheh,
On semi-rational Frobenius groups ,J. Algebra Appl. (2016), no. 2, 1650033, 8.[BH98] R. Brown and D.K. Harrison, Abelian Frobenius kernels and modules over number rings ,J. Pure Appl. Algebra (1998), no. 1-3, 51–86.[BMP17] G.K. Bakshi, S. Maheshwary, and I.B.S. Passi,
Integral group rings with all central unitstrivial , J. Pure Appl. Algebra (2017), no. 8, 1955–1965.[CD10] D. Chillag and S. Dolfi,
Semi-rational solvable groups , J. Group Theory (2010), no. 4,535–548.[DS96] R. Dark and C.M. Scoppola, On Camina groups of prime power order , J. Algebra (1996), no. 3, 787–802.[DS04] M. R. Darafsheh and H. Sharifi,
Frobenius Q -groups , Arch. Math. (Basel) (2004),no. 2, 102–105.[FeS86] E. Farias e Soares, Big primes and character values for solvable groups , J. Algebra (1986), no. 2, 305–324.[GAP16] The GAP Group,
GAP – Groups, Algorithms, and Programming, Version 4.8.3 , 2016, .[Gow76] R. Gow,
Groups whose characters are rational-valued , J. Algebra (1976), no. 1,280–299.[Hup67] B. Huppert, Endliche Gruppen. I , Die Grundlehren der Mathematischen Wis-senschaften, Band 134, Springer-Verlag, Berlin-New York, 1967.[IL15] I.M. Isaacs and M.L. Lewis,
Camina p -groups that are generalized Frobenius comple-ments , Arch. Math. (Basel) (2015), no. 5, 401–405.[JdR16] E. Jespers and ´A. del R´ıo, Group ring groups. Volume 1: Orders and generic construc-tions of units , Berlin: De Gruyter, 2016.[Mah16] S. Maheshwary,
Integral group rings with all central units trivial: solvable groups ,preprint (2016), 8 pages, arXiv:1612.08344v1[math.RA] .[Pas68] D. Passman,
Permutation groups , W. A. Benjamin, Inc., New York-Amsterdam, 1968.[RS90] J. Ritter and S.K. Sehgal,
Integral group rings with trivial central units , Proc. Amer.Math. Soc. (1990), no. 2, 327–329.[Sco87] W. R. Scott,
Group theory , second ed., Dover Publications, Inc., New York, 1987.[Ten12] J.F. Tent,
Quadratic rational solvable groups , J. Algebra (2012), 73–82.
Vakgroep Wiskunde, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium
E-mail address ::