Conjugacy of Involutions in Integral Group Rings of Certain Dihedral and

Semidihedral Groups

by Alexander Zimmermann

There is an extensive literature on conjugacy of torsion units in the group of augmented units in an integral group ring ZZG of a finite group G as long as one looks at conjugacy in the corresponding rational group ring QG (cf. for example [LB-83a, LT-90, LP-92, MRSW-87]). H. Zassenhaus conjectured ([Za-74]) that every torsion unit of ZZG is conjugate to an element of G∪(−G) by a unit of QG.

However, very few results are known if one restricts to conjugacy in ZZG. The principal motivation for integral conjugacy of torsion units arise from

the famous subgroup rigidity theorem of A. Weiss ([W-87]), G. Thompson ([Th-89]) and K. W. Roggenkmap and L. L. Scott ([R-92]) that states in the most simple form that subgroups of finite order and augmentation 1 of the group ring ẐZ_{p}G are conjugate in the units of ẐZ_{p}G to subgroups of G as long as G is a p-group. Here ẐZ_{p} is the ring of p-adic integers. This of course implies that every augmented unit of finite order is part of a group basis of ẐZ_{p}G.

Does this also happen if one replaces ẐZ_{p} by ZZ? To be more precise: Is every augmented unit of finite order in ZZG part of a group basis of ZZG?

For p-groups G this has been an open question. For non-p groups there are counterexamples but the subgroups under consideration are not conjugate to a subgroup of any group basis even in the units of ẐZ_{q}G for a suitably chosen prime q. Therefore essentially a local phenomenon is measured. The integral case in the above sense has been open up to now.

Out of the results treated in the literature concerning conjugacy classes of units in the integral group ring of a finite group I want to mention the discussion of group rings of abelian groups in G. Higman’s thesis [H-39], the direct calculations of Hughes and Pearson [HP-72] on the integral group ring of the dihedral group of order 6, similar calculations of C. Polcino-Milies [PM-74] concerning the dihedral group of order 8 and those of Allen and Hobby on the alternating group of degree 4 ([AH-80]). A. K. Bhandari and I. S. Luthar ([BL-83b]) counted the number of conjugacy classes of elements of finite order in the augmented units of the integral group ring of dihedral groups of order 2p for odd primes p. Neither Allen and Hobby nor Luthar and Bhandari determine which of their units are contained in group bases and which are not. The results in [HP-72, PM-74, AH-80, BL-83b] are obtained by complicate calculations and so I think another method has to be applied.

In 1989 K. W. Roggenkamp and L. L. Scott developed a theory similar to A. Fröhlich’s theory of invertible bimodules to determine the conjugacy classes of units u in V (ZZG), the group of units of augmentation 1 in the integral group ring ZZG for a finite group G, that are conjugate in V (QG) to an element of the fixed group basis G. We refer to [R-89, RT-92] for details. This theory was applied in [Zi-92] to integral group rings of dihedral groups of order 2^{n+1}, n being any natural number greater or equal to 2, and to integral group rings of dihedral groups of order 2p, p being any odd prime number. Here the difficulties mentioned above do not arise.

The results of [R-89], [RT-92, Part I, Chapter VII] are summarized as follows: Let U be a fixed subgroup of G and let C_{ZZG}(U) be the centralizer of U in ZZG. We will assume C_{ZZG}(U) to be abelian and ZZG to be Eichler over ZZ. Then the kernel Cl_{ZZG}(C_{ZZG}(U)) of the natural induction homomorphism Cl(C_{ZZG}(U)) ⟶ Cl(ZZG), Cl(Λ) being the locally free class group of the ring Λ, parametrizes up to automorphisms of U, the conjugacy classes– conjugate in V (ZZG)– of subgroups H of V (ZZG) that are in V (QG) conjugate to U. An element of this kernel is induced from an element of the class group of the center of ZZG if and only if the corresponding conjugacy class of groups H are part of some group bases.

In the author’s forthcoming dissertation [Zi-92] this theory is applied to certain dihedral groups G = D_{m} with subgroup U generated by a non central involution b in D_{m}.

First we set m = 2^{n}, n being a natural number the following result is obtained:

Theorem 1 [Zi-92] Let D_{2^{n}} be the dihedral group of order 2^{n+1}.

1. The V (QD_{2^{n}})–orbit of a non central involution b ∈ D_{2^{n}} splits into 2^{n−1} · ∏_{k≤n} h_{k}^{+} conjugacy classes in ZZD_{2^{n}}, h_{k}^{+} being the class number of the maximal real subfield of the field of 2^{k}–th roots of unity over Q.

2. Out of these conjugacy classes 2^{n−1} classes consist of involutions being member of group bases.

3. Cl(C_{ZZD_{2^{n}}}(b)) ≃ C_{2^{n−1}} × ∏_{k≤n} Cl(ZZ[ζ_{2^{k}} + ζ_{2^{k}}^{−1}])^{2}, with ζ_{2^{k}}^{2^{k−1}} + 1 = 0.

4. Cl_{ZZD_{2^{n}}}(C_{ZZD_{2^{n}}}(b)) ≃ C_{2^{n−1}} × ∏_{k≤n} Cl(ZZ[ζ_{2^{k}} + ζ_{2^{k}}^{−1}]).

So H. Cohn’s conjecture that h_{k}^{+} = 1 for all numbers k (cf. [ACH-65]) is true if and only if every involution in V (ZZD_{2^{n}}) is part of a group basis for all n ∈ IN. The conjecture is proved by J. van der Linden [vdL-82] for k ≤ 7 and for k = 8 using the generalized Riemann hypothesis.

We apply this result to describe the central automorphism group of ZZD_{2^{n}} explicitly:

Theorem 2 [Zi-92] The outer central automorphism group Outcent(ZZD_{2^{n}}) of the integral group ring of the dihedral group D_{2^{n}} of order 2^{n+1} is generated by conjugation by a − b + 1, an automorphism of order 2^{n−1} modulo inner automorphisms, and conjugation by 1 + b · (a + a^{−1}), an automorphism of order 2^{n−2} modulo inner automorphisms. Here a is an element of order 2^{n} in D_{2^{n}} and b is a non central involution in D_{2^{n}}.

For an abelian group A we denote by A_{[2]} the kernel of the endomorphism of A sending each element to its square. We define η(p) := ζ_{p} + ζ_{p}^{−1} for a primitive p^{th} root of unity ζ_{p}. If m is a prime p we obtain the following result:

Theorem 3 [Zi-92] Let D_{p} be the dihedral group of order 2p, p being an odd prime.

1. In ZZD_{p} there are exactly

\frac{|Cl(ZZ[η(p)]C_{2})|}{|Cl(ZZ[η(p)])|}

conjugacy classes of involutions, locally and rationally conjugate to b.

2. An involution b ∈ D_{p} is contained in exactly \frac{p−1}{2} representatives of conjugacy classes of group bases.

3. Out of the conjugacy classes of involutions rationally and locally conjugate to b in ZZD_{p} exactly |(Cl(ZZ[η(p)]))_{[2]}| conjugacy classes consist out of involutions, contained in a group basis.

4. If p ∈ {(2nq)^{2} + 1|n ∈ ZZ, n ≥ 2, qZZ ∈ Spec(ZZ)} is prime, there is a conjugacy class of involutions whose elements are conjugate to b in V (ẐZ_{r}D_{p}) for all primes r, but they are not part of any group basis of ZZD_{p}.

The last part is an immediate application of the result of Ankeney, Chowla and Hasse of [ACH-65].

Remark 1 In [BL-83b] A. K. Bhandari and I. S. Luthar determined the total number of conjugacy classes of involutions in V (ZZD_{p}) without minding if a conjugacy class is locally conjugate to the fixed involution b of D_{p}. A. K. Bhandari and I. S. Luthar do not discuss at all questions concerning the relation of elements of finite order and group bases they lie in. They do, however, also look at elements of order p in V (ZZD_{p}) and they are able to give the generators explicitely if the class number of ZZ[η(p)] is equal to 1.

I am very grateful to Professor Dr. J. Ritter to bring this result of A. K. Bhandari and I. S. Luthar to my attention at the Erfurt conference.

Remark 2 The proof of Theorem 3 is incomparatively easier than that of Theorem 1 since in case of dihedral groups of order 2p, p being an odd prime, there are only three Wedderburn components for QD_{p} and two of them are abelian. In contrary the Wedderburn decomposition of QD_{2^{n}} is much more involved.

The method used in the proofs of the Theorems 1 and 3 are illustrated by another example, the semidihedral group of order 16. In fact we will prove here the

Theorem 4 Let S be the semidihedral group of order 16. In the group of augmented units of ZZS there is a conjugacy class C of involutions such that each representative of C is not part of a group basis of ZZS.

Remark 3 Because of the subgroup rigidity theorem every involution of ZZS is conjugate by a unit of the 2-adic group ring ẐZ_{2}S to an involution in S (cf. [W-87, Th-89, RT-92, R-92]).

Remark 4 Since S is a 2-group this solves an open problem: In case of p-groups G the coefficient domain ZZ is definitively worse than ẐZ_{p} in spite of subgroup rigidity!

We have to consider various pullback diagrams arising all in the same manner: If A is a K-algebra for a field K of characteristic 0 and if Γ is an R-order in A for R being the ring of algebraic integers in K. Then let e^{2} = e ∈ Z(A) be a central idempotent. It is well known (see [CR-82,7, (2.12)] that

Γ ⟶^{·e} Γe
↓ ·(1 − e)  ↓
Γ(1 − e) ⟶ Γe/(Γ ∩ Γe)

is a pullback diagram. We call this diagram induced by e.

Let S =< a, b|a^{8} = b^{2} = baba^{−3} = 1 > be the semidihedral group of order 16. We prove that there is an involution u in V (ZZS), the group of augmented units, not contained in a group basis.

Let C′ := C_{ZZS}(b) be the centralizer of the involution b in ZZS. Then obviously C′ is additively generated by the set

{1, b, a + a^{3}, (a + a^{3})b, a^{2} + a^{6}, (a^{2} + a^{6})b, a^{5} + a^{7}, (a^{5} + a^{7})b, a^{4}, a^{4}b}.

Since C′ is abelian we have to calculate the kernel Cl_{ZZS}(C′) of the homomorphism Cl(C′) to Cl(ZZS) and prove that there is an element (A) in Cl_{ZZS}(C′) not being an image of an element of Cl(Z(ZZS)) under the induction map, Z(ZZS) being the center of ZZS.

We use Reiner-Ullom’s version of Milnor’s Mayer-Vietoris-Sequence (cf. [RU-74], [CR-82,7, 49.28)]). The idempotent e_{4} = (1 + a^{4})/2 induces a pullback diagramm

C′ ⟶^{·e_{4}} B_{2}
↓ ·(1 − e_{4})  ↓
ZZ[\sqrt{−2}] < b > ⟶ \overline{B}_{2}

with B_{2} (cf. [Zi-92]) being the image of the residue modulo (1 − a^{4})C′ in ZZD_{4}, the group ring of the dihedral group of order 8. Moreover \overline{B}_{2} is the image of B_{2} under the map ZZD_{2} ⟶ IF_{2}D_{2} ^{1} induced by reduction modulo 2. Since in ZZ[\sqrt{−2}] only 1 and −1 are units as is well known, we see that in ZZ[\sqrt{−2}] < b > only ±1, ±b are units (an element of ZZ[\sqrt{−2}] < b > is a unit if and only if it is a unit in its maximal order ZZ[\sqrt{−2}] ⊕ ZZ[\sqrt{−2}] induced by the two one-dimensional representations (cf. [R-83, Lemma 3])). Since B_{2} is a subring of ZZ(C_{4} × C_{2})^{2}, it has only trivial units ±1, ±b (cf.[H-39]). Next we see that U(\overline{B}_{2}) is generated by

{b, 1 + a + a^{3}, b + a + a^{3}, }

and is elementary abelian of order 8. Therefore

U(\overline{B}_{2})/(U^{∗}(ZZ[\sqrt{−2}] < b >) · U^{∗}(B_{2})) ≃ C_{2} × C_{2},

if we denote by ^{∗} the images of the unit groups in \overline{B}_{2} under the natural map. Hence the following sequence of groups is exact:

1 ⟶ C_{2} × C_{2} ⟶ Cl(C′) ⟶ Cl(B_{2}) ⊕ Cl(ZZ[\sqrt{−2}] < b >) ⟶ 1

We are therefore concerned with the calculation of the group Cl(B_{2}) and also of Cl(ZZ[\sqrt{−2}] < b >).

First we have a pullback diagram

ZZ[\sqrt{−2}] < b > ⟶  ZZ[\sqrt{−2}]
↓  ↓
ZZ[\sqrt{−2}]  ⟶ ZZ[\sqrt{−2}]/2ZZ[\sqrt{−2}]

by the idempotent (1 + b)/2. Second we observe that ZZ[\sqrt{−2}]/2ZZ[\sqrt{−2}] ≃ IF_{2}C_{2}. And third we realize that

1 ⟶ C_{2} ⟶ Cl(ZZ[\sqrt{−2}] < b >) ⟶ Cl(ZZ[\sqrt{−2}]) ⊕ Cl(ZZ[\sqrt{−2}]) ⟶ 1

is an exact sequence of abelian groups. Since Cl(ZZ[\sqrt{−2}]) = 1, a result known to Gauß, we see that

Cl(ZZ[\sqrt{−2}] < b >) ≃ C_{2}.

1IF_{2} is the field with 2 elements

2C_{n} denotes the cyclic group of order n

Though Cl(B_{2}), and more generally Cl(B_{n}), is calculated in [Zi-92] we repeat the few necessary arguments for the special case Cl(B_{2}). The idempotent (1 + a^{2})/2 induces the pullback diagram

B_{2}  ⟶  B_{1}
↓  ↓
ZZ < b > ⟶ ZZ/4ZZ < b >

which in turn induces the Mayer-Vietoris-sequence

1 ⟶ C_{2} ⟶ Cl(B_{2}) ⟶ Cl(B_{1}) ⊕ Cl(ZZ) ⟶ 1.

Since B_{1} = ZZ < b > +2ZZ < a, b >⊆ ZZV_{4}, the group ring of Klein’s 4group, the following pullback diagram is induced by the idempotent (1+a)/2:

B_{1}  ⟶ ZZ < b >
↓  ↓
ZZ < b > ⟶ ZZ/4ZZ < b >

Therefore again by the Mayer-Vietoris-sequence we obtain the following exact sequence:

1 ⟶ C_{2} ⟶ Cl(B_{1}) ⟶ Cl(ZZ < b >) ⊕ Cl(ZZ < b >) ⟶ 1

Therefore Cl(B_{2}) is of order 4, it is cyclic as is shown in [Zi-92], but we don’t need this here and so we don’t prove it.

Next we have to calculate the class group of the center of ZZS. It is generated additively by

{1, a + a^{3}, a^{2} + a^{6}, a^{5} + a^{7}, (1 + a^{2} + a^{4} + a^{6})b, (1 + a^{2} + a^{4} + a^{6})ab}

The idempotent e_{4} induces the pullback diagram

Z(ZZS) ⟶^{·e_{4}}  Z_{2}(1)
↓  ↓
ZZ[\sqrt{−2}] ⟶ ZZ[\sqrt{−2}]/2ZZ[\sqrt{−2}].

Since

Z_{2}(1) = ZZ + ZZ(a + a^{3}) + 2ZZa^{2} + ZZ(1 + a^{2})b + ZZ(a + a^{3})b ⊂ ZZ(C_{4} × C_{2}), the group of units of Z_{2}(1) contains only ±1. Therefore the Mayer-Vietoris-sequence becomes there

1 ⟶ C_{2} ⟶ Cl(Z(ZZS)) ⟶ Cl(Z_{2}(1)) ⊕ Cl(ZZ[\sqrt{−2}]) ⟶ 1.

We omit the proof that Cl(Z_{2}(1)) ≃ C_{4} × C_{2}, since it is not important here, and refer to [Zi-92]. What is important is the wellknown fact that Cl(ZZ[\sqrt{−2}]) = 1. Let \overline{A} be an ideal of ZZ[\sqrt{−2}] < b > locally but not globally free. The existence of \overline{A} is guaranteed by the fact that the class group Cl(ZZ[\sqrt{−2}] < b >) = C_{2}. By [Ta-84] we see that

1 ⟶ C_{2} ⟶ Cl(ZZS) ⟶ Cl(ZZD_{2}) ⊕ Cl(( \begin{matrix}ZZ[\sqrt{−2}] & ZZ[\sqrt{−2}] \\ ZZ[\sqrt{−2}] & ZZ[\sqrt{−2}] \end{matrix} )) ⟶ 1

is the Mayer-Vietoris-sequence corresponding to the idempotent e_{4}. Now,

Cl(( \begin{matrix}ZZ[\sqrt{−2}] & ZZ[\sqrt{−2}] \\ ZZ[\sqrt{−2}] & ZZ[\sqrt{−2}] \end{matrix} )) ≃ Cl(ZZ[\sqrt{−2}]) ≃ 1 (1)

by Swan’s theorem (cf. [S-62]).

Let (A) be an arbitrary preimage of the isomorphism class (\overline{A}) in Cl(C′). (Observe that (\overline{A}) maps to 1 if it is induced to an isomorphism class of an

ideal of ( \begin{matrix}ZZ[\sqrt{−2}] & ZZ[\sqrt{−2}] \\ ZZ[\sqrt{−2}] & ZZ[\sqrt{−2}] \end{matrix} ).)

If (A) would be induced by an ideal of Z(ZZS), say B, we calculate

\overline{A} ≃ A ⊗_{C′} C′(1 − e_{4})
≃ B ⊗_{Z(ZZS)} C′ ⊗_{C′} C′(1 − e_{4})
≃ B ⊗_{Z(ZZS)} C′(1 − e_{4})
≃ B ⊗_{Z(ZZS)} Z(ZZS)(1 − e_{4}) ⊗_{Z(ZZS)(1−e_{4})} C′(1 − e_{4})
≃ Z(ZZS)(1 − e_{4}) ⊗_{Z(ZZS)(1−e_{4})} C′(1 − e_{4})
≃ C′(1 − e_{4});

the fifth isomorphism holds because of (1). This however is a contradiction to the choice of \overline{A} and of A.

Let us turn to the question if there is an isomorphism class of an ideal in Cl_{ZZS}(C′) such that none of the representatives are induced by an ideal of the center of ZZS rather than an isomorphism class of an ideal in Cl(C′).

Let B̂ be an ideal generating the class group of ZZS. It has order 2 and is isomorphic to a Swan-module by a theorem of Endo (cf. [Ta-84, 3. Theorem 2.5 & Theorem 2.6]). Those, however, are induced by an ideal of the center of ZZS: For the proof of this fact it is convenient to prove a little lemma analogous to [GRH-88, Theorem 2.7].

Lemma 1 [Zi-92] K is the quotient field of the Dedekind domain R of characteristic 0. Let Λ be an R–order in a finite dimensional K–algebra A and let Δ be an R–order in a finite dimensional K–algebra B, such that Λ ≤ Δ and A ≤ B and let J be a locally free Λ–right ideal. Then J ⊗_{Λ} Δ ≃ J · Δ evaluated in B.

Proof.

0 ⟶ Δ ⟶ B ⟶ B/Δ ⟶ 0

is an exact sequence of Δ–modules and therefore also of Λ-modules. Then an exact sequence of Λ–modules is induced:

T or_{1}^{Λ}(J, B/Δ) ⟶ J ⊗_{Λ} Δ ⟶ J ⊗_{Λ} B

Since J is projective,

T or_{1}^{Λ}(J, B/Δ) = 0.

Hence

J ⊗_{Λ} B ≃ (J ⊗_{Λ} Δ) ⊗_{R} K
≃ K ⊗_{R} J ⊗_{Λ} Δ
≃ (J ⊗_{R} K) ⊗_{Λ} Δ
≃ A ⊗_{Λ} Δ
≃ K ⊗_{R} Λ ⊗_{Λ} Δ
≃ K ⊗_{R} Δ
≃ B

and the lemma is proved if one follows the isomorphisms. q.e.d.

Since for Swan-modules one can choose idèles with coefficients in the center of ZZS representing them ([Ta-84, 1 (3.13)]), by the lemma they are induced by an ideal of the center of ZZS.

Let φ denote the map Cl(C′) ⟶ Cl(ZZS) and ψ denote the map Cl(Z(ZZS)) ⟶ Cl(C′). If φ(A) is not equal to 1, it is equal to the isomorphism class of the above Swan-module D = φ(ψ(D̂)), for an ideal D̂ of the center of ZZS, since induction from the center to the group ring factors through Cl(C′). Therefore there is an ideal E of C′ in the kernel of φ with

(A) = (E) · ψ(D̂).

Since im ψ is a group and ψ(D̂) is in im ψ we conclude that

(E) ∈ ker φ \ im ψ.

In either case there is an ideal, A if it is in ker φ, or E, if it is not, inducing an involution in V (ZZS) not being part of a group basis but rationally conjugate to b.

References

[ACH-65] N.C. Ankeney, S. Chowla, H. Hasse: On the class number of the maximal real subfield of a cyclotomic field; J. reine angew. Math. 217 (1965), 217–220.

[AH-80] P. J. Allen, C. Hobby: A Characterization of Units in ZZ[A_{4}]; J. of Algebra 66, 534–543, (1980).

[BL-83b] A. K. Bhandari, I. S. Luthar: Conjugacy classes of torsion units of the integral group ring of D_{p}; Comm. Alg. 11 (14), 1607–1627 (1983); Erratum in Comm. Alg. 18 (1990).

[CR-82,7] C. W. Curtis, I. Reiner: Methods of Representation theory, Vol. 1 & 2; John Wiley Interscience 1982 & 1987.

[GRH-88] W. Gustafson, K.W. Roggenkamp: Automorphisms and Picard groups for Hereditary Orders; Visiting Scholar’s Lectures (1988), Texas Tech Univ.

[H-39] G. Higman: The Units in Group Rings; Dr. phil. thesis, Oxford University (1939).

[HP-72] I. Hughes, K. E. Pearson: The group of units in the group ring ZZS_{3}; Can. Math. Bull. 15 (1972), 529–534.

[LB-83a] I. S. Luthar, A. K. Bhandari: Torsion Units of Integral Group Rings of Metacyclic Groups; J. Number Theory 17 (1983), 270– 283.

[LT-90] I.S. Luthar, P. Trama: Zassenhaus-conjecture for certain integral group rings; J. Indian Math. Soc. 55 (1990) no. 1–4, 199– 212.

[LP-92] I.S. Luthar, I.B.S. Passi: Torsion Units in Matrix Group Rings; Comm. Alg. 20 (4) (1992), 1223–1228.

[MRSW-87] Z. Marciniak, J. Ritter, S.K. Sehgal, A. Weiss: Torsion Units in Integral Group Rings of Some Metabelian Groups, II; J. Number Theory 25 (1987), 340–352.

[PM-74] C. Polcino Milies: The Group of Units of the Integral Group Ring ZZD_{4}; Bol. Soc. Brasil. Math. 4 (1974), 85–92.

[RU-74] I. Reiner, S. Ullom: A Mayer-Vietoris Sequence for Class Groups; J. Algebra 31 (1974), 305–342.

[R-83] K. W. Roggenkamp: Units in metabelian group rings: non splitting examples for normalized units; J. of Pure and Appl. Algebra 27 (1983), 299–314.

[R-89] K.W. Roggenkamp: The isomorphism problem and related topics; Bayreuther Mathematische Schriften 33 (1989), 173–196.

[R-92] K. W. Roggenkamp: Subgroup rigidity of p–adic group rings (Weiss’ arguments revisited); to appear J. Lon. Math. Soc. (1992).

[RT-92] K.W. Roggenkamp, M.J. Taylor: Group Rings and Class Groups; Basel, 1992.

[S-62] R.G. Swan: Projective modules over group rings and maximal orders; Annals of Mathematics 76 (1962), 55-61.

[Ta-84] M. J. Taylor: Classgroups of Group Rings; Cambridge 1984.

[Th-89] G. Thompson: On the conjugacy of group bases, Ph.D. theses, Univ. of Virginia, 1989.

[vdL-82] F. J. van der Linden: Class Number Computations of Real Abelian Number Fields; Math. Comp. 39 no.160 (1982), 693– 707.

[W-87] A. Weiss: Rigidity of p–adic p–torsion; Annals of Mathematics (1987), 317–332.

[Za-74] H. Zassenhaus: On the torsion units of finite group rings, Estudos de Mathematica, Institut de alta cultura, Lisboa (1974), 119–126.

[Zi-92] A. Zimmermann: Endliche Untergruppen der Einheitengruppe ganzzahliger Gruppenringe; Dissertation, Stuttgart 1992.