2-Blocks with minimal nonabelian defect groups

2-Blocks with minimal nonabelian defect groups

Benjamin Sambale Mathematisches Institut Friedrich-Schiller-Universität

07743 Jena Germany

benjamin.sambale@uni-jena.de May 23, 2011

Abstract

We study numerical invariants of2-blocks with minimal nonabelian defect groups. These groups were classi- fied by Rédei (see [41]). If the defect group is also metacyclic, then the block invariants are known (see [43]).

In the remaining cases there are only two (infinite) families of “interesting” defect groups. In all other cases the blocks are nilpotent. We prove Brauer’sk(B)-conjecture and the Olsson-conjecture for all2-blocks with minimal nonabelian defect groups. For one of the two families we also show that Alperin’s weight conjecture and Dade’s conjecture is satisfied. This paper is a part of the author’s PhD thesis.

Keywords:blocks of finite groups, minimal nonabelian defect groups, Alperin’s conjecture, Dade’s conjecture.

Contents

1 Introduction 1

2 Fusion systems 2

3 The caser > s= 1 3

3.1 TheB-subsections . . . 3

3.2 The numbersk(B),ki(B)andl(B) . . . 4

3.3 Generalized decomposition numbers . . . 6

3.4 The Cartan matrix . . . 9

3.5 Dade’s conjecture . . . 11

3.6 Alperin’s weight conjecture . . . 13

3.7 The gluing problem . . . 14

4 The caser=s >1 14 4.1 TheB-subsections . . . 14

4.2 The gluing problem . . . 17

4.3 Special cases . . . 18

1 Introduction

LetR be a discrete complete valuation ring with quotient fieldK of characteristic 0. Moreover, let (π) be the maximal ideal ofRandF :=R/(π). We assume thatF is algebraically closed of characteristic2. We fix a finite groupG, and assume thatKcontains all|G|-th roots of unity. LetBbe a block ofRGwith defect groupD. We

denote the number of irreducible ordinary characters ofB byk(B). These characters split inki(B)characters of height i ∈ N⁰. Similarly, let kⁱ(B) be the number of characters of defect i ∈ N⁰. Finally, let l(B) be the number of irreducible Brauer characters ofB. The defect groupDis calledminimal nonabelian if every proper subgroup ofDis abelian, but notDitself. Rédei has shown thatDis isomorphic to one of the following groups (see [41]):

(i) hx, y|x^2r =y^2s= 1, xyx⁻¹=y^1+2s−1i, wherer≥1ands≥2,

(ii) hx, y|x^2r =y^2s = [x, y]² = [x, x, y] = [y, x, y] = 1i, wherer ≥s≥1, [x, y] :=xyx⁻¹y⁻¹ and [x, x, y] :=

[x,[x, y]],

(iii) Q8.

In the first and last caseDis also metacyclic. In this caseB is well understood (see [43]). Thus, we may assume thatD has the form (ii).

2 Fusion systems

To analyse the possible fusion systems onDwe start with a group theoretical lemma.

Lemma 2.1. Let z:= [x, y]. Then the following hold:

(i) |D|= 2^r+s+1.

(ii) Φ(D) = Z(D) =hx², y², zi ∼=C₂r−1×C₂s−1×C₂. (iii) D⁰=hzi ∼=C₂.

(iv) |Irr(D)|= 5·2^r+s−2.

(v) Ifr=s= 1, thenD∼=D8. For r≥2 the maximal subgroups ofD are given by hx², y, zi ∼=C₂^r−1×C2^s×C2,

hx, y², zi ∼=C2^r×C₂s−1×C2, hxy, x², zi ∼=C₂r×C₂s−1×C₂.

We omit the (elementary) proof of this lemma. However, notice that|P⁰|= 2and|P: Φ(P)|=|P : Z(P)|=p² hold for every minimal nonabelian p-group P. Rédei has also shown that for different pairs (r, s) one gets nonisomorphic groups. This gives precisely _n−1

2

isomorphism classes of these groups of order 2ⁿ. For r6= 1 (that is|D| ≥16) the structure of the maximal subgroups shows that all these groups are nonmetacyclic.

Now we investigate the automorphism groups.

Lemma 2.2. The automorphism group Aut(D)is a2-group, if and only if r6=sorr=s= 1.

Proof. If r6=s or r=s = 1, then there exists a characteristic maximal subgroup of D by Lemma 2.1(v). In these cases Aut(D) must be a 2-group. Thus, we may assume r = s ≥2. Then one can show that the map x7→y,y7→x⁻¹y⁻¹is an automorphism of order3.

Lemma 2.3. Let P ∼=C₂n1 ×. . .×C₂nk withn₁, . . . , n_k, k∈N. ThenAut(P)is a 2-group, if and only if the n_i are pairwise distinct.

Proof. See for example Lemma 2.7 in [34].

Now we are able to decide, when a fusion system onD is nilpotent.

Theorem 2.4. Let F be a fusion system onD. ThenF is nilpotent ors= 1 orr=s. Ifr=s≥2, thenF is controlled byD.

Proof. We assumes6= 1. LetQ < Dbe anF-essential subgroup. SinceQis alsoF-centric, we getCP(Q) =Q.

This shows thatQis a maximal subgroup of D. By Lemma 2.1(v) and Lemma 2.3, one of the following holds:

(i) r= 2 ( =s)andQ∈ {hx², y, zi,hx, y², zi,hxy, x², zi}, (ii) r > s= 2andQ∈

hx, y², zi,hxy, x², zi , (iii) r=s+ 1andQ=hx², y, zi.

In all cases Ω(Q) ⊆ Z(P). Let us consider the action of AutF(Q) on Ω(Q). The subgroup 1 6= P/Q = NP(Q)/CP(Q)∼= AutP(Q)≤AutF(Q)acts trivially on Ω(Q). On the other hand every nontrivial automor- phism of odd order acts nontrivially onΩ(Q)(see for example 8.4.3 in [19]). Hence, the kernel of this action is a nontrivial normal2-subgroup ofAut_F(Q). In particular O₂(Aut_F(Q))6= 1. But then Aut_F(Q)cannot contain a strongly2-embedded subgroup.

This shows that there are noF-essential subgroups. Now the claim follows from Lemma 2.2 and Alperin’s fusion theorem.

Now we consider a kind of converse. Ifr=s= 1, then there are nonnilpotent fusion systems onD. In the case r=s≥2 one can construct a nonnilpotent fusion system with a suitable semidirect product (see Lemma 2.2).

We show that there is also a nonnilpotent fusion system in the caser > s= 1.

Proposition 2.5. If s= 1, then there exists a nonnilpotent fusion system on D.

Proof. We may assumer≥2. LetA₄ be the alternating group of degree4, and letH :=hexi ∼=C₂r. Moreover, let ϕ : H → Aut(A₄) ∼= S₄ such that ϕ

ex ∈ Aut(A₄) has order 4. Write ey := (12)(34) ∈ A₄ and choose ϕ such that ϕ

xe(ey) := (13)(24). Finally, let G:=A₄oϕH. Since all 4-cycles in S₄ are conjugate,Gis uniquely determined up to isomorphism. Because [ex,ey] = (13)(24)(12)(34) = (14)(23), we get hx,e eyi ∼= D. The fusion systemFG(D)is nonnilpotent, sinceA4(and thereforeG) is not2-nilpotent.

3 The case r > s = 1

Now we concentrate on the caser > s= 1, i.e.

D:=hx, y|x^2r =y²= [x, y]²= [x, x, y] = [y, x, y] = 1i

withr≥2. As beforez:= [x, y]. We also assume thatB is a nonnilpotent block. By Lemma 2.2,Aut(D)is a 2-group, and the inertial indext(B)ofB equals1.

3.1 The B-subsections

Olsson has already obtained the conjugacy classes of so called B-subsections (see [34]). However, his results contain errors. For example he missed the necessary relations[x, x, y]and[y, x, y]in the definition ofD.

In the next lemma we denote byBl(RH)the set of blocks of a finite groupH. IfH≤Gandb∈Bl(RH), then b^G is the Brauer correspondent of b (if exists). Moreover, we use the notion of subpairs and subsections (see [36]).

Lemma 3.1. Let b∈Bl(RDCG(D))be a Brauer correspondent ofB. ForQ≤D letbQ∈Bl(RQCG(Q))such that (Q, bQ)≤(D, b). SetT := Z(D)∪ {xⁱy^j :i, j∈Z, iodd}. Then

[

a∈T

n

a, b^C_C^G(a)

D(a)

o

is a system of representatives for the conjugacy classes of B-subsections. Moreover,|T |= 2^r+1.

Proof. If r = 2, then the claim follows from Proposition 2.14 in [34]. For r ≥ 3 the same argument works.

However, Olsson refers wrongly to Proposition 2.11 (the origin of this mistake already lies in Lemma 2.8).

From now on we writeba:=b^C_C^G(a)

D(a)fora∈ T.

Lemma 3.2. Let P ∼=C2^s ×C₂² with s∈N, and let αbe an automorphism of P of order 3. Then CP(α) :=

{b∈P :α(b) =b} ∼=C2^s.

Proof. We write P = hai × hbi × hci with |hai| = 2^s. It is well known that the kernel of the restriction map Aut(P) → Aut(P/Φ(P)) is a 2-group. Since |Aut(P/Φ(P))| = |GL(3,2)| = 168 = 2³·3·7, it follows that

|Aut(P)|is divisible by3only once. In particular every automorphism ofP of order3is conjugate toαorα⁻¹. Thus, we may assumeα(a) =a,α(b) =candα(c) =bc. ThenCP(α) =hai ∼=C2^s.

3.2 The numbers k(B), k

(B) and l(B )

The next step is to determine the numbersl(ba). The caser= 2needs special attention, because in this caseD contains an elementary abelian maximal subgroup of order8. We denote the inertial group of a blockb∈Bl(RH) withHEGbyTG(b).

Lemma 3.3. There is an element c∈Z(D) of order2^r−1 such that l(ba) = 1for all a∈ T \ hci.

Proof.

Case 1:a∈Z(D).

Then b_a = b^C_D^G(a) is a block with defect group D and Brauer correspondent b_D ∈ Bl(RDC_CG(a)(D)). Let M := hx², y, zi ∼= C₂r−1×C₂². Since B is nonnilpotent, there exists an element α ∈ T_NG(M)(b_M) such that αC_G(M)∈T_NG(M)(b_M)/C_G(M)has order q∈ {3,7}. We will exclude the caseq= 7. In this caser= 2and T_NG(M)(b_M)/C_G(M)is isomorphic to a subgroup ofAut(M)∼= GL(3,2). Since

(M,^db_M) =^d(M, b_M)≤^d(D, b_D) = (D, b_D)

for all d ∈D, we have D ⊆T_NG(M)(b_M). This impliesT_NG(M)(b_M)/C_G(M)∼= GL(3,2), because GL(3,2) is simple. By Satz 1 in [2], this contradicts the fact that T_NG(M)(b_M)/C_G(M) contains a strongly 2-embedded subgroup (of course this can be shown “by hand” without invoking [2]). Thus, we have shownq= 3. Now

T_NG(M)(bM)/CG(M)∼=S3

follows easily. By Lemma 3.2 there is an elementc:=x²ⁱy^jz^k∈C_M(α)(i, j, k∈Z) of order2^r−1. Let us assume thatj is odd. Sincexαx≡xαx⁻¹≡α⁻¹ (modC_G(M))we get

α(x²ⁱy^jz^k+1)α⁻¹=αx(x²ⁱy^jz^k)x⁻¹α⁻¹=xα⁻¹(x²ⁱy^jz^k)αx⁻¹

=x(x²ⁱy^jz^k)x⁻¹=x²ⁱy^jz^k+1.

But this contradicts Lemma 3.2. Hence, we have proved thatj is even. In particularc∈Z(D). For a /∈ hciwe have α /∈CG(a)and l(ba) = 1. While in the case a∈ hciwe get α∈CG(a), andba is nonnilpotent. Thus, in this casel(ba)remains unknown.

Case 2:a /∈Z(D).

Let CD(a) = hZ(D), ai =: M. Since (M, bM) is a Brauer subpair, bM has defect group M. It follows from (M, bM)E(D, bD)that also ba has defect groupM and Brauer correspondent bM. In case M ∼=C2^r ×C2 we getl(ba) = 1. Now let us assumeM ∼=C2^r−1×C₂². As in the first case, we chooseα∈TN_G(M)(bM)such that αC_G(M)∈T_NG(M)(b_M)/C_G(M)has order3. Sincea /∈Z(D), we deriveα /∈C_G(a)andt(b_a) =l(b_a) = 1.

We denote by IBr(b_u) := {ϕ_u} for u ∈ T \ hci the irreducible Brauer character of b_u. Then the generalized decomposition numbersd^u_χϕ

u forχ∈Irr(B)form a columnd(u). Let2^k be the order ofu, and letζ:=ζ₂k be a primitive2^k-th root of unity. Then the entries ofd(u)lie in the ring of integersZ[ζ]. Hence, there exist integers a^u_i(χ)∈Zsuch that

d^u_χϕ

u =

2^k−1−1

X

i=0

a^u_i(χ)ζⁱ.

We expand this by

a^u_i+2k−1 :=−a^u_i for alli∈Z.

Let|G|= 2^amwhere 2- m. We may assume Q(ζ_|G|)⊆K. ThenQ(ζ_|G|)|Q(ζm) is a Galois extension, and we denote the corresponding Galois group by

G:= Gal(Q(ζ_|G|)|Q(ζm)).

Restriction gives an isomorphism

G ∼= Gal(Q(ζ₂a)|Q).

In particular|G|= 2^a−1. For everyγ∈ G there is a numberγe∈N such thatgcd(eγ,|G|) = 1, eγ≡1 (modm), andγ(ζ_|G|) =ζ_|G|^eγ hold. ThenG acts on the set of subsections by

γ(u, b) := (u^eγ, b).

For everyγ∈ G we get

d(u^eγ) =X

s∈S

a^u_sζ₂^s_k^eγ

for every systemS of representatives of the cosets of2^k−1Zin Z. It follows that a^u_s = 2^1−aX

γ∈G

d u^eγ

ζ₂⁻k^eγs (1)

fors∈ S.

Now let u∈ T \Z(D)and M := CD(u). Then b_u and b^T_M^{NG(M)(bM)∩NG(hui)} have M as defect group, because D*NG(hui). By (6B) in [6] it follows that the2^r−1 distinctB-subsections of the form^γ(u, bu)withγ∈ G are pairwise nonconjugate. The same holds foru∈Z(D)\ {1}. Using this and equation (1) we can adapt Lemma 3.9 in [33]:

Lemma 3.4. Let c∈Z(D)as in Lemma 3.3, and letu, v∈ T \ hciwith|hui|= 2^k and|hvi|= 2^l. Moreover, let i∈ {0,1, . . . ,2^k−1−1}andj∈ {0,1, . . . ,2^l−1−1}. If there existγ∈ G andg∈Gsuch that^g(u, b_u) =^γ(v, b_v), then

(a^u_i, a^v_j) =



















2^d(B)−k+1 if u∈Z(D) andjeγ−i≡0 (mod 2^k)

−2^d(B)−k+1 if u∈Z(D) andjeγ−i≡2^k−1 (mod 2^k) 2^d(B)−k if u /∈Z(D) andjeγ−i≡0 (mod 2^k)

−2^d(B)−k if u /∈Z(D) andjeγ−i≡2^k−1 (mod 2^k)

0 otherwise

.

Otherwise(a^u_i, a^v_j) = 0. In particular (a^u_i, a^v_j) = 0 ifk6=l.

Using the theory of contributions we can also carry over Lemma (6.E) in [20]:

Lemma 3.5. Let u∈Z(D)withl(bu) = 1. If uhas order2^k, then for everyχ∈Irr(B)holds:

(i) 2^h(χ)|a^u_i(χ)fori= 0, . . . ,2^k−1−1, (ii)

2^k−1−1

P

i=0

a^u_i(χ)≡2^h(χ) (mod 2^h(χ)+1).

By Lemma 1.1 in [39] we have

k(B)≤

∞

X

i=0

2²ⁱk_i(B)≤ |D|. (2)

In particular Brauer’sk(B)-conjecture holds. Olsson’s conjecture

k0(B)≤ |D:D⁰|= 2^r+1 (3)

follows by Theorem 3.1 in [39]. Now we are able to calculate the numbersk(B),ki(B)andl(B).

Theorem 3.6. We have

k(B) = 5·2^r−1=|Irr(D)|, k0(B) = 2^r+1=|D:D⁰|, k1(B) = 2^r−1, l(B) = 2.

Proof. We argue by induction onr. Let r= 2, and letc∈Z(D)as in Lemma 3.3. By way of contradiction we assumec=z. If αandM are defined as in the proof of Lemma 3.3, then αacts nontrivially onM/hzi ∼=C₂². On the other handxacts trivially onM/hzi. This contradictsxαx⁻¹α∈CG(M).

This showsc∈ {x², x²z}and D/hci ∼=D8. Thus, we can apply Theorem 2 in [8]. For this let M1:=

(hx, zi ifc=x² hxy, zi ifc=x²z.

ThenM 6=M₁∼=C₄×C₂andM :=M/hci ∼=C₂²∼=M₁/hci=:M₁. Letβbe the block ofRC_G(c) :=R[C_G(c)/hci]

which is dominated byb_c. By Theorem 1.5 in [33] we have 3

|

CG(c)(M)(β_M)/C_C

G(c)(M)|

and

3

-

G(c)(M1)(β_M

1)/C_C

G(c)(M1)|, where (M , β_M) and (M1, β_M

1) are β-subpairs. This shows that case (ab) in Theorem 2 in [8] occurs. Hence, l(bc) =l(β) = 2. Now Lemma 3.3 yields

k(B)≥1 +k(B)−l(B) = 9.

It is well known thatk₀(B)is divisible by 4. Thus, the equations (2) and (3) implyk₀(B) = 8. Moreover, d^z_χϕ

z =a^z₀(χ) =±1

holds for everyχ∈Irr(B)withh(χ) = 0. This shows4k1(B)≤ |D|−k0(B) = 8. It follows thatk1(B) =l(B) = 2.

Now we consider the caser≥3. Sincez is not a square inD, we have z /∈ hci. Leta∈ hcisuch that|hai|= 2^k. Ifk=r−1, thenl(ba) = 2as before. Now letk < r−1. ThenD/haihas the same isomorphism type asD, but one has to replacerbyr−k. By induction we getl(ba) = 2fork≥1. This shows

k(B)≥1 +k(B)−l(B) = 2^r+1+ 2^r−1−1.

Equation (2) yields

2^r+2−4 = 2^r+1+ 4(2^r−1−1)≤k₀(B) + 4(k(B)−k₀(B))

≤

∞

X

i=0

2²ⁱki(B)≤ |D|= 2^r+2.

Now the conclusion follows easily.

As a consequence, Brauer’s height zero conjecture and the Alperin-McKay-conjecture hold forB.

3.3 Generalized decomposition numbers

Now we will determine some of the generalized decomposition numbers. Again letc ∈Z(D)as in Lemma 3.3, and letu∈Z(D)\hciwith|hui|= 2^k. Then(a^u_i, a^u_i) = 2^r+3−kand2|a^u_i(χ)forh(χ) = 1andi= 0, . . . ,2^k−1−1.

This gives

|{χ∈Irr(B) :a^u_i(χ)6= 0}| ≤2^r+3−k−3|{χ∈Irr(B) :h(χ) = 1, a^u_i(χ)6= 0}|.

Moreover, for every characterχ∈Irr(B)there existsi∈ {0, . . . ,2^k−1−1}such thata^u_i(χ)6= 0. Hence, k(B)≤

2^k−1−1

X

i=0

X

χ∈Irr(B), a^u_i(χ)6=0

1≤

2^k−1−1

X

i=0

2^r+3−k−3 X

χ∈Irr(B), h(χ)=1, a^u_i(χ)6=0

1

!

=|D| −3

2^k−1−1

X

i=0

X

χ∈Irr(B), h(χ)=1, a^u_i(χ)6=0

1

≤ |D| −3k1(B) =k(B).

This shows that for everyχ∈Irr(B)there exists i(χ)∈ {0, . . . ,2^k−1−1}such that d^u_χϕu =

(±ζ₂^i(χ)k ifh(χ) = 0

±2ζ₂^i(χ)k ifh(χ) = 1. In particular

d^u_χϕu =a^u₀(χ) =

(±1 ifh(χ) = 0

±2 ifh(χ) = 1 fork= 1.

By Lemma 3.4 we have(a^u_i, a^u_i) = 4foru∈ T \Z(D)and i= 0, . . . ,2^r−1−1. Ifa^u_i has only one nonvanishing entry, thena^u_i would not be orthogonal toa^z₀. Hence,a^u_i has up to ordering the form

(±1,±1,±1,±1,0, . . . ,0)^T,

where the signs are independent of each other. The proof of Theorem 3.1 in [39] gives

|d^u_χϕu|= 1

foru∈ T \Z(D)and χ∈Irr(B)withh(χ) = 0. In particulard^u_χϕu = 0for characters χ∈Irr(B)of height1.

By suitable ordering we get a^u_i(χj) =

(±1 ifj−4i∈ {1, . . . ,4}

0 otherwise andd^u_χjϕu = (

±ζ^[

j−1 4 ]

2^r if1≤j≤k0(B) 0 ifk₀(B)< j≤k(B) fori= 0, . . . ,2^r−1−1, whereχ₁, . . . , χ_k0(B) are the characters of height0.

Now letIBr(b_c) :={ϕ₁, ϕ₂}. We determine the numbersd^c_χϕ

1, d^c_χϕ

2 ∈Z[ζ₂r−1]. By (4C) in [6] we haved^c_χϕ

1 6= 0 ord^c_χϕ2 6= 0for allχ∈Irr(B). As in the proof of Theorem 3.6,bc dominates a blockbc ∈Bl(R[CG(c)/hci])with defect groupD₈. The table at the end of [14] shows that the Cartan matrix ofb_c has the form

8 4 4 3

or 4 2

2 3

.

We label these possibilities as the “first” and the “second” case. The Cartan matrix ofb_c is 2^r−1

8 4 4 3

or2^r−1 4 2

2 3

respectively. The inverses of these matrices are 2^−r−2

3 −4

−4 8

and2^−r−2

3 −2

−2 4

.

Letm^(c,b_χψ^c) be the contribution ofχ, ψ∈Irr(B)with respect to the subsection(c, bc)(see [6]). Then we have

|D|m^(c,b_χψ^c)= 3d^c_χϕ

1d^c_ψϕ

1−4(d^c_χϕ

1d^c_ψϕ

2+d^c_χϕ

2d^c_ψϕ

1) + 8d^c_χϕ

2d^c_ψϕ

2

or

|D|m^(c,b_χψ^c)= 3d^c_χϕ

1d^c_ψϕ

1−2(d^c_χϕ

1d^c_ψϕ

2+d^c_χϕ

2d^c_ψϕ

1) + 4d^c_χϕ

2d^c_ψϕ

2 (4)

respectively. For a characterχ∈Irr(B)with height 0we get 0 =h(χ) =ν |D|m^(c,b_χχ^c)

=ν(3d^c_χϕ1d^c_χϕ1) =ν(d^c_χϕ1) by (5H) in [6]. In particulard^c_χϕ1 6= 0. We definec^j_i ∈Z^k(B)by

d^c_χϕ

j =

2^r−2−1

X

i=0

c^j_i(χ)ζ₂ⁱr−1

forj= 1,2. Then

(c¹_i, c¹_j) =

(δij16 first case

δij8 second case, (c¹_i, c²_j) =

(δij8 first case

δij4 second case, (c²_i, c²_j) =δ_ij6

as in Lemma 3.4. (Since the2^r−2 B-subsections of the form ^γ(c, b_c)for γ∈ G are pairwise nonconjugate, one can argue like in Lemma 3.4.) Hence, in the second case

d^c_χiϕ1= (

±ζ^[

i−1 8 ]

2^r−1 if1≤i≤k0(B)

0 ifk0(B)< i≤k(B) (second case)

holds for a suitable arrangement. Againχ1, . . . , χ_k0(B)are the characters of height0. In the first case 1 =h(ψ) =ν |D|m^(c,b_χψ^c)

=ν(3d^c_χϕ

1d^c_ψϕ

1) =ν(d^c_ψϕ

1)

by (5G) in [6] for h(ψ) = 1 and h(χ) = 0. As in Lemma 3.5 we also have 2 | c¹_i(ψ) for h(ψ) = 1 and i= 0, . . . ,2^r−2−1. Analogously as in the caseu∈Z(D)\ hciwe conclude

d^c_χϕ1 =

(±ζ₂^i(χ)r−1 ifh(χ) = 0

±2ζ₂^i(χ)r−1 ifh(χ) = 1 (first case) (5) for suitable indicesi(χ)∈ {0, . . . ,2^r−2−1}. Since(c²_i, c²_j) =δij6, in both casesc²_i has the form

(±1,±1,±1,±1,±1,±1,0, . . . ,0)^Tor (±2,±1,±1,0, . . . ,0)^T.

We show that the latter possibility does not occur. In the second case for every characterχ∈Irr(B)with height 1there exists i∈ {0, . . . ,2^r−2−1}such thatc²_i(χ)6= 0. In this case we get

d^c_χ

iϕ₂ =











±ζ^[

i−1 4 ]

2^r−1 if1≤i≤2^r 0 if2^r< i≤k0(B)

±ζ^[

i−k0 (B)−1

2 ]

2^r−1 ifk0(B)< i≤k(B)

(second case),

whereχ₁, . . . , χ_k0(B)are again the characters of height0. Now let us consider the first case. Since(c¹_i, c²_j) =δ_ij8, the value±2must occur in every columnc¹_i fori= 0, . . . ,2^r−2−1at least twice. Obviously exactly two entries have to be±2. Thus, one can improve equation (5) to

d^c_χ

iϕ₁ =







±ζ^[

i−1 8 ]

2^r−1 if1≤i≤k₀(B)

±2ζ^[

i−k0 (B)−1

2 ]

2^r−1 ifk0(B)< i≤k(B)

(first case).

It follows

d^c_χiϕ2=











±ζ^[

i−1 4 ]

2^r−1 if1≤i≤2^r 0 if2^r< i≤k₀(B)

±ζ^[

i−k0 (B)−1

2 ]

2^r−1 ifk0(B)< i≤k(B)

(first case).

Hence, the numbersd^c_χϕ2are independent of the case. Of course, one gets similar results ford^u_χϕiwithhui=hci.

3.4 The Cartan matrix

Now we investigate the Cartan matrix ofB.

Lemma 3.7. The elementary divisors of the Cartan matrix ofB are2^r−1 and|D|.

Proof. Let C be the Cartan matrix of B. Since l(B) = 2, it suffices to show that 2^r−1 occurs as elementary divisor ofC at least once. In order to proof this, we use the notion of lower defect groups (see [35]). Let (u, b) be a B-subsection with |hui| = 2^r−1 and l(b) = 2. Let b₁ :=b^NG(hui). Then b₁ has also defect group D, and l(b₁) = 2holds. Moreover, u^2r−2 ∈Z(N_G(hui)). Letb₁ ∈Bl(R[N_G(u)/hu^2r−2i]) be the block which is covered by b1. Then b1 has defect group D/hu^2r−2i. We argue by induction on r. Thus, let r = 2. Then b = b1 and D/hu^2r−2i=D/hui ∼=D8. By Proposition (5G) in [8] the Cartan matrix ofbhas the elementary divisors1and 8. Hence,2 = 2^r−1 and16 =|D|are the elementary divisors of the Cartan matrix ofb. Hence, the claim follows from Theorem 7.2 in [35].

Now assume that the claim already holds for r−1 ≥2. By induction the elementary divisors of the Cartan matrix ofb₁ are2^r−2 and|D|/2. The claim follows easily as before.

Now we are in a position to calculate the Cartan matrixC up to equivalence of quadratic forms. Here we call two matricesM1, M2 ∈Z^l×l equivalent if there exists a matrixS ∈GL(l,Z)such thatA=SBS^T, where S^T denotes the transpose ofS.

By Lemma 3.7 all entries ofCare divisible by2^r−1. Thus, we can considerCe:= 2^1−rC∈Z^2×2. ThendetCe= 8 and the elementary divisors ofCe are1 and8. If we write

Ce=

c1 c2

c2 c3

,

thenCe corresponds to the positive definite binary quadratic formq(x1, x2) :=c1x²₁+ 2c2x1x2+c3x²₂. Obviously gcd(c1, c2, c3) = 1. If one reduces the entries ofCe modulo2, then one gets a matrix of rank 1 (this is just the multiplicity of the elementary divisor1). This shows thatc1 orc3 must be odd. Hence,gcd(c1,2c2, c3) = 1, i. e.

qis primitive (see [10] for example). Moreover,∆ :=−4 detCe=−32is the discriminant ofq. Now it is easy to see thatq(andC) is equivalent to exactly one of the following matrices (see page 20 in [10]):e

1 0 0 8

or 3 1

1 3

.

The Cartan matrices for the blockbc with defect groupD8 (used before) satisfy 1 −1

0 1

8 4 4 3

1 −1

0 1

^T

=

0 1

−1 1

4 2 2 3

0 1

−1 1 ^T

= 3 1

1 3

.

Hence, only the second matrix occurs up to equivalence. We show that this holds also for the blockB.

Theorem 3.8. The Cartan matrix ofB is equivalent to 2^r−1

3 1 1 3

.

Proof. We argue by induction on r. The smallest case was already considered bybc (this would correspond to r = 1). Thus, we may assume r ≥ 2 (as usual). First, we determine the generalized decomposition numbers d^u_χϕ foru∈ hci \ {1} with |hui|= 2^k <2^r−1. As in the proof of Theorem 3.6, the group D/hui has the same isomorphism type asD, but one has to replace rby r−k. Hence, by induction we may assume thatbu has a Cartan matrix which is equivalent to the matrix given in the statement of the theorem. LetC_u be the Cartan matrix ofb_u, and letS_u∈GL(2,Z)such that

C_u= 2^r−1S_u^T 4 2

2 3

S_u,

i. e. with the notations of the previous section, we assume that the “second case” occurs. (This is allowed, since we can only compute the generalized decomposition numbers up to multiplication withSu anyway.) As before we write IBr(bu) = {ϕ1, ϕ2}, Du := (d^u_χϕi) and (de^u_χϕi) := DuS_u⁻¹. The consideration in the previous section carries over, and one gets

de^u_χϕ

1 = (

±ζ^[

i−1 2r+2−k]

2^k if1≤i≤k0(B) 0 ifk₀(B)< i≤k(B) and

de^u_χϕ

2 =











±ζ^[

i−1 2r−k+1]

2^k if1≤i≤2^r 0 if2^r< i≤k0(B)

±ζ^[

i−k0 (B)−1 2r−k ]

2^k ifk₀(B)< i≤k(B) ,

whereχ1, . . . , χ_k0(B)are the characters of height 0. But notice that the ordering of those characters forϕ1 and ϕ2is different.

Now assume that there is a matrixS ∈GL(2,Z)such that C= 2^r−1S^T

1 0 0 8

S.

IfQdenotes the decomposition matrix ofB, we set(deχϕi) :=QS⁻¹forIBr(B) ={ϕ1, ϕ2}. Then we have

|D|m^(1,B)_χψ = 8deχϕ1deψϕ1+deχϕ2deψϕ2 forχ, ψ∈Irr(B).

In particular |D|m^(1,B)χχ ≡ 1 (mod 4) for a character χ ∈ Irr(B) of height 0. For u ∈ T \Z(D) we have

|D|m^(u,bχχ ^u) = 2, and for u ∈ Z(D)\ hci we have |D|m^(u,bχχ^u) = 1. Let u ∈ hci \ {1}. Equation (4) and the considerations above imply|D|m^(u,bχχ^u)≡3 (mod 4). Now (5B) in [6] reveals the contradiction

|D|=X

u∈T

|D|m^(u,b_χχ^u)≡ |D|m^(1,B)_χχ + 2^r+1+ 2^r−1+ 3·(2^r−1−1)≡2 (mod 4).

With the proof of the last theorem we can also obtain the ordinary decomposition numbers (up to multiplication with an invertible matrix):

d_χϕ1 =

(±1 ifh(χ) = 0

0 ifh(χ) = 1, d_χiϕ2=







±1 if0≤i≤2^r 0 if2^r< i≤k0(B)

±1 ifk0(B)< i≤k(B) .

Againχ₁, . . . , χ_k0(B)are the characters of height0.

Since we know howG acts on theB-subsections, we can investigate the action ofG onIrr(B).

Theorem 3.9. The irreducible characters of height0 of B split in 2(r+ 1)families of 2-conjugate characters.

These families have sizes 1,1,1,1,2,2,4,4, . . . ,2^r−1,2^r−1 respectively. The characters of height 1 split in r families with sizes 1,1,2,4, . . . ,2^r−2 respectively. In particular there are exactly six 2-rational characters in Irr(B).

Proof. We start by determining the number of orbits of the action of G on the columns of the generalized decomposition matrix. The columns{d^u_χϕu:χ∈Irr(B)}withu∈ T \Z(D)split in two orbits of length2^r−1. For i= 1,2the columns{d^u_χϕi :χ∈Irr(B)} withu∈ hcisplit inr orbits of lengths1,1,2,4, . . . ,2^r−2respectively.

Finally, the columns {d^u_χϕu : χ ∈ Irr(B)} with u∈Z(D)\ hci consist of r orbits of lengths1,1,2,4, . . . ,2^r−2 respectively. This gives3r+ 2orbits altogether. By Theorem 11 in [3] there also exist exactly 3r+ 2families of 2-conjugate characters. (SinceGis noncyclic, one cannot conclude a priori that also the lengths of the orbits of these two actions coincide.)

By considering the column{d^x_χϕx :χ∈Irr(B)}, we see that the irreducible characters of height0split in at most 2(r+ 1)orbits of lengths1,1,1,1,2,2,4,4, . . . ,2^r−1,2^r−1respectively. Similarly the column{d^c_χϕ2 :χ∈Irr(B)}

shows that there are at mostrorbits of lengths1,1,2,4, . . . ,2^r−2of characters of height1. Since2(r+ 1) +r= 3r+ 2, these orbits do not merge further, and the claim is proved.

LetM =hx², y, zias in Lemma 3.3. ThenD⊆T_NG(M)(bM). Since e(B) = 1, Alperin’s fusion theorem implies that T_NG(M)(bM)controls the fusion of B-subpairs. By Lemma 3.3 we also have T_NG(M)(bM) ⊆CG(c) for a c∈Z(D). This shows thatBis a so called “centrally controlled block” (see [22]). In [22] it was shown that then the centers of the blocksB andbc (regarded as blocks ofF G) are isomorphic.

3.5 Dade’s conjecture

In this section we will verify Dade’s (ordinary) conjecture for the blockB (see [12]). First, we need a lemma.

Lemma 3.10. Let Be be a block of RG with defect group De ∼=C2^s ×C₂² (s ∈N0) and inertial index 3. Then k(B) =e k0(B) =e |D|e = 2^s+2 andl(B) = 3e hold.

Proof. Letα be an automorphism of De of order 3 which is induced by the inertial group. By Lemma 3.2 we haveC

De(α)∼=C2^s. We choose a system of representativesx1, . . . , xk for the orbits ofDe\C

De(α)underα. Then k= 2^s. Ifbi ∈Bl(RCG(xi))fori= 1, . . . , k andbu∈Bl(RCG(u))foru∈C

De(α)are Brauer correspondents of B, thene

k

[

i=1

(xi, bi) ∪ [

u∈C

fD(α)

(u, bu)

is a system of representatives for the conjugacy classes of B-subsections. Sincee α /∈CG(xi), we havel(bi) = 1 fori= 1, . . . , k. In particulark(B)e ≤2^s+2 holds. Now we show the opposite inequality by induction ons.

For s = 0 the claim is well known. Let s ≥ 1. By induction l(bu) = 3 for u ∈ C

De(α)\ {1}. This shows k(B)e −l(B) =e k+ (2^s−1)3 = 2^s+2−3andl(B)e ≤3. An inspection of the numbersd^x_χϕ¹ impliesk(B) =e k0(B) =e 2^s+2=|D|e andl(B) = 3. (This would also follow from Theorem 1 in [46].)e

Now assumeO₂(G) = 1(this is a hypothesis of Dade’s conjecture). In order to prove Dade’s conjecture it suffices to consider chains

σ:P1< P2< . . . < Pn

of nontrivial elementary abelian 2-subgroups of G (see [12]). (Note that also the empty chain is allowed.) In particularPiEPn andPnENG(σ)fori= 1, . . . , n. Hence, for a blockb∈Bl(RNG(σ))withb^G=B and defect group Qwe havePn ≤Q. Moreover, there exists ag ∈Gsuch that^gQ≤D. Thus, by conjugation with g we may assumePn≤Q≤D (see also Lemma 6.9 in [12]). This showsn≤3.

In the case|Pn|= 8 we havePn =hx^2r−1, y, zi=:E, because this is the only elementary abelian subgroup of order8inD. Let b∈Bl(RNG(σ))withb^G=B. We choose a defect groupQofBe:=b^NG(E). SinceΩ(Q) =Pn, we get N_G(Q) ≤ N_G(E). Then Brauer’s first main theorem implies Q =D. Hence, Be is the unique Brauer correspondent ofB inRN_G(E). ForM :=hx², y, zi ≤D we also haveN_G(M)≤N_G(Ω(M)) = N_G(E). Hence, Be is nonnilpotent. Now consider the chain

σe:







∅ ifn= 1 P1 ifn= 2 P₁< P₂ ifn= 3 for the groupGe:= NG(E). ThenN_G(σ) = N

Ge(eσ)and X

b∈Bl(RNG(σ)), b^G=B

kⁱ(b) = X

b∈Bl(RN

Ge(σ)),e b^Ge=Be

kⁱ(b).

The chainsσandeσaccount for all possible chains ofG. Moreover, the lengths ofσandeσhave opposite parity.

Thus, it seems plausible that the contributions ofσ and eσin the alternating sum cancel out each other (this would imply Dade’s conjecture). The question which remains is: Can we replace (G,e B,e eσ) by (G, B,eσ)? We make this more precise in the following lemma.

Lemma 3.11. Let Q be a system of representatives for the G-conjugacy classes of pairs(σ, b), where σ is a chain (of G) of length n with Pn < E andb ∈Bl(RNG(σ)) is a Brauer correspondent ofB. Similarly, let Qe be a system of representatives for theG-conjugacy classes of pairse (eσ,eb), whereeσis a chain (ofG) of lengthe n withPn< E andeb∈Bl(RN

Ge(eσ))is a Brauer correspondent ofBe. Then there exists a bijection betweenQand Qe which preserves the numberskⁱ(b).

Proof. LetbD∈Bl(RNG(D))be a Brauer correspondent ofB. We consider chains of B-subpairs σ: (P1, b1)<(P2, b2)< . . . <(Pn, bn)<(D, bD),

where thePi are nontrivial elementary abelian2-subgroups such thatPn< E. Then σis uniquely determined by these subgroupsP1, . . . , Pn (see Theorem 1.7 in [36]). Moreover, the empty chain is also allowed. LetU be a system of representatives forG-conjugacy classes of such chains. For every chainσ∈ U we define

eσ: (P1,be1)<(P2,be2)< . . . <(Pn,ben)<(D, bD) with bei ∈Bl(RC

Ge(Pi))for i= 1, . . . , n. Finally we set Ue :={eσ: σ∈ U }. By Alperin’s fusion theoremUe is a system of representatives for theG-conjugacy classes of corresponding chains for the groupe B. Hence, it sufficese to show the existence of bijections f (resp. fe) betweenU (resp. Ue) and Q (resp. Q) such that the followinge property is satisfied: Iff(σ) = (τ, b)andfe(eσ) = (eτ ,eb), thenkⁱ(b) =kⁱ(eb)for alli∈N⁰.

Letσ∈ U. Then we define the chainτ by only considering the subgroups ofσ, i. e.τ :P1< . . . < Pn. This gives CG(Pn)⊆NG(τ), and we can define

f :U → Q, σ7→ τ, b^N_n^G(τ) .

Now let (σ, b) ∈ Q arbitrary. We write σ : P₁ < . . . < P_n. By Theorem 5.5.15 in [29] there exists a Brauer correspondent β_n ∈Bl(RC_G(P_n))ofb. Since (P_n, β_n)is aB-subpair, we may assume(P_n, β_n)<(D, b_D)after a suitable conjugation. Then there are uniquely determined blocksβ_i∈Bl(RC_G(P_i))fori= 1, . . . , n−1such that

(P1, β1)<(P2, β2)< . . . <(Pn, βn)<(D, bD).

This shows thatf is surjective.

Now letσ₁, σ₂∈ U be given. We write

σ_i: (P₁ⁱ, β₁ⁱ)< . . . <(P_nⁱ, β_nⁱ)

fori= 1,2. Let us assume thatf(σ1) = (τ1, b1)and f(σ2) = (τ2, b2)are conjugate inG, i. e. there is a g∈G such that

τ₂,(^gβ_n¹)^NG(τ2)

=^g(τ₁, b₁) = (τ₂, b₂) = τ₂,(β_n²)^NG(τ2) .

Since^gβ_n¹ ∈Bl(RCG(P_n²))andβ_n² are covered byb2, there ish∈NG(τ2)with^hgβ_n¹=β_n². Then

hg(P_n¹, β_n¹) = (P_n², β²_n).

Since the blocksβ_jⁱfori= 1,2andj= 1, . . . , n−1are uniquely determined byP_jⁱ, we also have^ghσ1=σ2=σ1. This proves the injectivity off. Analogously, we define the mapfe.

It remains to show that f and fesatisfy the property given above. For this let σ ∈ U with σ : (P1, b₁) <

. . . < (P_n, b_n), eσ : (P1,be₁) < . . . < (P_n,be_n), f(σ) = τ, b^Nn^G(τ)

and fe(σ) =e τ,be_n^NGe(τ)

. We have to prove kⁱ b^Nn^G(τ)

=kⁱ ben N

Ge(τ)

fori∈N0.

Let Q be a defect group of b^Nn^G(τ). Then QC_G(Q) ⊆ N_G(τ), and there is a Brauer correspondent β_n ∈ Bl(RQCG(Q))of b^Nn^G(τ). In particular (Q, βn)is a B-Brauer subpair. As in Lemma 3.1 we may assume Q∈ {D, M,hx, zi,hxy, zi}. The same considerations also work for the defect groupQe ofben

N

Ge(τ)

. Sinceb^Dn^CG(Pn)= b^D_D^CG(Pn)=be_n^DCG(Pn), we get:

Q=D⇐⇒D⊆NG(τ)⇐⇒D⊆N

Ge(τ)⇐⇒Qe=D.

Let us consider the case Q = D ( = Q). Lete bM ∈ Bl(RCG(M)) such that (M, bM) ≤ (D, bD) and α ∈ T_NG(M)(bM)\DCG(M)⊆NG(M)⊆G. Then:e

b^N_n^G(τ) is nilpotent⇐⇒α /∈NG(τ)⇐⇒α /∈N

Ge(τ)⇐⇒ben N

Ge(τ)

is nilpotent.

Thus, the claim holds in this case. Now letQ < D (andQ < D). Then we havee QCG(Q) = CG(Q)⊆CG(Pn).

Sinceβ^Cn^G(Pn) is also a Brauer correspondent ofb^Nn^G(τ), the blocks βn^CG(Pn)and bn are conjugate. In particular b_n (andbe_n) has defect group Q. Hence, we obtainQ=Q. Ife Q∈ {hx, zi,hxy, zi}, then b^Nn^G(τ) andbe_n^NGe(τ) are nilpotent, and the claim holds. Thus, we may assumeQ=M. Then as before:

b^N_n^G(τ) is nilpotent⇐⇒α /∈NG(τ)⇐⇒α /∈N

Ge(τ)⇐⇒ben N_Ge(τ)

is nilpotent.

We may assume that the nonnilpotent case occurs. Then t b^Nn^G(τ)

= t ben N

Ge(τ)

= 3, and the claim follows from Lemma 3.10.

As explained in the beginning of the section, the Dade conjecture follows.

Theorem 3.12. The Dade conjecture holds for B.

3.6 Alperin’s weight conjecture

In this section we prove Alperin’s weight conjecture forB. Let(P, β)be a weight forB, i. e.P is a 2-subgroup of G and β is a block of R[NG(P)/P] with defect 0. Moreover, β is dominated by a Brauer correspondent b∈Bl(RNG(P))ofB. As usual, one can assumeP ≤D. IfAut(P)is a 2-group, thenNG(P)/CG(P)is also a 2-group. ThenPis a defect group ofb, sinceβhas defect0. Moreover,βis uniquely determined byb. By Brauer’s first main theorem we haveP =D. Thus, in this case there is exactly one weight forB up to conjugation.

Now let us assume that Aut(P) is not a 2-group (in particular P < D). As usual, β covers a block β1 ∈ Bl(R[CG(P)/P]). By the Fong-Reynolds theorem (see [29] for example) also β1 has defect 0. Hence, β1 is dominated by exactly one blockb1∈Bl(RCG(P))with defect groupP. Sinceββ1 6= 0, we also havebb1 6= 0, i. e.bcoversb1. Thus, the situation is as follows:

β ∈Bl(R[NG(P)/P])oo //

OO

b∈Bl(ROONG(P))

β1∈Bl(R[CG(P)/P])oo //b1∈Bl(RCG(P))

By Theorem 5.5.15 in [29] we haveb^N₁^G(P)=bandb^G₁ =B. This shows that(P, b1)is aB-Brauer subpair. Then P =M ( =hx², y, zi)follows. By Brauer’s first main theorembis uniquely determined (independent ofβ). Now we prove that alsoβ is uniquely determined byb.

In order to do so it suffices to show thatβis the only block with defect0which coversβ1. By the Fong-Reynolds theorem it suffices to show that β1 is covered by only one block ofRT_NG(M)/M(β1) =R[T_NG(M)(b1)/M]with defect0. For convenience we writeCG(M) := CG(M)/M,NG(M) := NG(M)/M andT := T_NG(M)(b1)/M. Let χ ∈Irr(β1). The irreducible constituents ofInd^T_C

G(M)(χ) belong to blocks which covers β1 (where Ind denote induction). Conversely, every block ofRTwhich coversβ₁ arises in this way (see Lemma 5.5.7 in [29]). Let

Ind^T

CG(M)(χ) =

t

X

i=1

e_iψ_i withψ_i∈Irr(T)ande_i∈Nfori= 1, . . . , t. Then

t

X

i=1

e²_i =|T : C_G(M)|=|T_NG(M)(b₁) : C_G(M)|= 6

(see page 84 in [17]). Thus, there is some i∈ {1, . . . , t} withei = 1, i. e.χis extendible toT. We may assume e1= 1. By Corollary 6.17 in [17] it follows thatt=|Irr(T/CG(M))|=|Irr(S3)|= 3 and

{ψ1, ψ2, ψ3}=

ψ1τ:τ∈Irr(T/CG(M)) ,

where the characters in Irr(T/CG(M)) were identified with their inflations in Irr(T). Thus, we may assume e2 = 1and e3= 2. Then it is easy to see that ψ1 and ψ2 belong to blocks with defect at least 1. Hence, only the block with containsψ3is allowed. This shows uniqueness.

Finally we show that there is in fact a weight of the form(M, β). For this we chooseb,b1,β1,χandψias above.

Thenχ vanishs on all nontrivial2-elements. Moreover, ψ1 is an extension ofχ. Let τ ∈Irr(T/CG(M))be the character of degree 2. Thenτ vanishs on all nontrivial 2-elements of T/C_G(M). Hence, ψ₃ =ψ₁τ vanishs on all nontrivial 2-elements ofT. This shows thatψ₃ belongs in fact to a blockβe∈Bl(RT)with defect 0. Then

M,βe^NG(M)

is the desired weight forB.

Hence, we have shown that there are exactly two weights forB up to conjugation. Since l(B) = 2, Alperin’s weight conjecture is satisfied.

Theorem 3.13. Alperin’s weight conjecture holds for B.

3.7 The gluing problem

Finally we show that the gluing problem (see Conjecture 4.2 in [26]) for the block B has a unique solution.

We will not recall the very technical statement of the gluing problem. Instead we refer to [37] for most of the notations. Observe that the fieldF is denoted by kin [37].

Theorem 3.14. The gluing problem forB has a unique solution.

Proof. As in [37] we denote the fusion system induced byBwithF. Then theF-centric subgroups ofDare given byM1:=hx², y, zi,M2:=hx, zi,M3:=hxy, ziandD. We have seen so far that Aut_F(M1)∼= OutF(M1)∼=S3, Aut_F(Mi)∼=D/Mi ∼=C2 fori= 2,3andAut_F(D)∼=D/Z(D)∼=C₂² (see proof of Lemma 3.3). Using this, we get Hⁱ(Aut_F(σ), F^×) = 0 for i= 1,2 and every chain σ ofF-centric subgroups (see proof of Corollary 2.2 in [37]). Hence,H⁰([S(F^c)],A²_F) = H¹([S(F^c)],A¹_F) = 0. Now the claim follows from Theorem 1.1 in [37].

4 The case r = s > 1

In the section we assume thatB is a nonnilpotent block ofRGwith defect group D:=hx, y|x^2r =y^2r = [x, y]²= [x, x, y] = [y, x, y] = 1i

forr≥2. As before we definez:= [x, y]. Since|D/Φ(D)|= 4,2 and3are the only prime divisors of|Aut(D)|.

In particulart(B)∈ {1,3}. Ift(B) = 1, thenB would be nilpotent by Theorem 2.4. Thus, we havet(B) = 3.

4.1 The B-subsections

We investigate the automorphism group ofD.

Lemma 4.1. Letα∈Aut(D)be an automorphism of order3. Thenzis the only nontrivial fixed-point ofZ(D) under α.

Proof. Since D⁰ = hzi, z remains fixed under all automorphisms of D. Moreover, α(x) ∈ yZ(D)∪xyZ(D), becauseαacts nontrivially onD/Z(D). In both cases we haveα(x²)6=x². This shows thatα|Z(D)∈Aut(Z(D)) is also an automorphism of order 3. Obviously αinduces an automorphism of order 3 on Z(D)/hzi ∼=C₂²r−1. But this automorphism is fixed-point-free (see Lemma 1 in [27]). The claim follows.