A new character correspondence in groups of odd order

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Journal of Algebra 268 (2003) 8–21

www.elsevier.com/locate/jalgebr

A new character correspondence in groupsof odd order✩

Gabriel Navarro

Departament d’Àlgebra, Universitat de València, 46100 Burjassot, Spain

Received 27 September 2001

Communicated by George Glauberman

1. Introduction

If p is a prime andG is a finitep-solvable group, we have associated to every comirreducible characterχ ∈ Irr(G) of G a pair(Q, δ), whereQ is a p-subgroup ofG andδ ∈ Irr(Q), which is uniquely determined byχ up to G-conjugacy (see [8]). We sathat everyG-conjugate of(Q, δ) is a vertex of χ and we denote by Irr(G|Q,δ) theset of irreducible characters ofG with vertex (Q, δ). This partitions the set of compleirreducible characters of a finitep-solvable groupG into natural families. (Theorem A o[8] gives a list of the most relevant properties of thesevertices.) Let us denote by Irr0(G)

the set ofp-defect zero characters ofG. Also, let NG(Q,δ) be the set of elements oNG(Q) stabilizingδ.

Theorem A. Suppose thatG is a group of odd order. LetQ be ap-subgroup ofG and letδ ∈ Irr(Q). Then there is a natural injection

∗ : Irr(G|Q,δ) → Irr0(NG(Q,δ)/Q

).

In the very special case whereδ = 1Q, the set Irr(G|Q,δ) is a canonical set of liftings othe irreducible Brauer characters ofG with “classical” vertexQ (see [8, Theorem (6.3)])Hence, the Alperin Weight Conjecture, as proved in [5], implies that|Irr(G|Q,δ)| =|Irr0(NG(Q,δ)/Q)|. Therefore, Theorem A provides a canonical bijection betweenirreducible Brauer characters and the Alperin weights for groups of odd order (whicthe main theorem in [7]).

✩ Research partially supported by Ministerio de Ciencia y Technologia, Grant BFM 2001-1667-C03-02E-mail address:[email protected].

0021-8693/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0021-8693(03)00420-4

G. Navarro / Journal of Algebra 268 (2003) 8–21 9

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[4]). Ifthe

d,

Now, let us denote by Irrp′(G) the set of thoseχ ∈ Irr(G) of degree not divisible byp.If χ ∈ Irr(G) has vertex(Q, δ), by Theorem A(b) of [8] we have thatχ ∈ Irrp′(G) if andonly if Q ∈ Sylp(G) andδ is linear. In fact, it is the main result of [6], that in this case∣∣Irr(G|Q,δ)

∣∣= ∣∣Irr(NG(Q)|δ)∣∣,where Irr(NG(Q)|δ) is the set of irreducible characters ifNG(Q) lying aboveδ. Byelementary character theory,|Irr(NG(Q)|δ)| = |Irr0(NG(Q,δ)/Q)|, and we see thaTheorem A provides a canonical bijection


).

(This is a stronger form of the natural correspondence Irrp′(G) → Irrp′(NG(Q)) con-structed in [2] by M. Isaacs which proved the McKay conjecture for groups of odd or

Although we have seen two important cases in which the map∗ is a bijection, it is notdifficult to find examples in which∗ is not surjective. If we do not assume that|G| is odd,it is the main result in [9] to prove that∣∣Irr(G|Q,δ)

∣∣� ∣∣Irr0(NG(Q,δ)/Q

)∣∣for anyp-solvable groupG.

2. Reviewing character correspondences

If P acts coprimely on a groupM, let IrrP (M) be the set of irreducible characters ofM

which areP -invariant. We writeC = CM(P) for the fixed points subgroup.

Theorem 2.1 (Glauberman–Isaacs).Suppose thatP acts coprimely on a groupM. Thenthere is a natural bijection

∗ : IrrP (M) → Irr(C).

Proof. If P is solvable, this is the Glauberman correspondence (see Chapter 13 ofP is not solvable, then|M| is odd, by the Feit–Thompson theorem. In this case, this isIsaacs correspondence (Theorem (10.8) of [2]). Finally, if|M| is odd andP is solvable,both character correspondences coincide by the main result of [11].✷

This natural bijection, in the case where|M| is odd on which we are mainly focusewas constructed by repeatedly using the following key result.

Theorem 2.2. Suppose thatP acts coprimely onM of odd order. Suppose thatH is aP -invariant subgroup ofM such that[M,P ]′C ⊆ H ⊆ M. Letχ ∈ IrrP (M). Then there is(a unique) χ [H ] ∈ IrrP (H) such that

χH = χ [H ] + 2∆ + Ξ,

10 G. Navarro / Journal of Algebra 268 (2003) 8–21

s

.rigid

Given

where∆ andΞ are characters ofH , and no irreducible constituent ofΞ is P -invariant.Also, the mapIrrP (M) → IrrP (H) given byχ �→ χ [H ] is a bijection.

Proof. This is Corollary (4.3) of [11]. ✷The characterχ [H ] is the character written asχσ(M,H,P) in [11].Now, if P acts coprimely onM, a group of odd order, andC = CM(P), then the Isaac

correspondence on coprime action is constructed by induction on|M|. LetH = [M,P ]′C.By using the solvability ofM and coprime action, it is easy to check thatM > H (ifM > 1). Then, the Isaacs correspondence is the composition of the natural map IrrP (M) →IrrP (H) (from Theorem 2.2) and the natural map IrrP (H) → Irr(C) obtained by induction

When proving properties of this correspondence, one readily needs to relax itsdefinition. This was done by T. Wolf in [11].

Theorem 2.3. Suppose thatP acts coprimely onM of odd order. Suppose thatH is anyP -invariant subgroup with[M,P ]′C ⊆ H ⊆ M. Letχ ∈ IrrP (M). Thenχ∗ = (χ [H ])∗.

Proof. This is Theorem (4.6) of [11]. ✷In [2], Isaacs found another character correspondence for groups of odd order.

a primep, we denote by Irrp′(G) the irreducible characters ofG of degree not divisibleby p.

Theorem 2.4. Suppose thatG is a group of odd order and letP ∈ Sylp(G). Then thereexists a natural bijection

∗ : Irrp′(G) → Irrp′(NG(P)

).

Proof. This is Theorem (10.9) of [2]. ✷In the coprime action correspondence, we saw that the subgroups of the form[M,P ]′C

played an important role. In thep′-degree character correspondence, the subgroups

Op′p(G)′NG(P)

are going to play the corresponding part.

Lemma 2.5. Suppose thatG is a finite group and letP ∈ Sylp(G). LetK �G be such thatG/K has a normal Sylowp-subgroup. Suppose thatK ′NG(P) ⊆ H ⊆ G. LetL = H ∩K.ThenL � G, K/L is abelian,H = LNG(P) andKH = G.

Proof. We have thatKP � G. By the Frattini argument, we have thatKNG(P) = G.Hence,G = KH . Now, K ′ ⊆ L ⊆ K, and thereforeL � G. Now, H/L is isomorphicto G/K, and thenH/L has a normal Sylowp-subgroupPL/L. Hence, H/L =NH/L(PL/L) = NH (P )L/L = NG(P)L/L. ThusH = LNG(P). ✷


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The next theorem tells us how thep′-degree correspondence is constructed. Recallif χ ∈ Irrp′(G) andL �G, thenχL has aP -invariant irreducible constituent and that evetwo of them areNG(P)-conjugate. This easily follows from the Clifford corresponde(Theorem (6.11) of [4]) and Sylow theory.

Theorem 2.6. Suppose thatG is a finite group of odd order and letP ∈ Sylp(G). Suppose

that Op′p(G)′NG(P) ⊆ H ⊆ G. If χ ∈ Irrp′(G), then there is(a unique) χ(H) ∈ Irrp′(H)

such that

χH = χ(H) + 2∆ + β,

where∆ and β are characters ofH , and βH∩Op′p(G)

has noP -invariant irreducible

constituents. Moreover the mapχ �→ χ(H) is a bijection betweenIrrp′(G) and Irrp′(H).

Proof. Write L = H ∩ Op′p(G). By Lemma 2.5, we have thatL � G. Let θ ∈ Irr(L) beP -invariant belowχ . By Theorem (10.6) of [2], we have that

χH = χ(H) + 2∆ + β,

where χ(H) lies aboveθ , has degree not divisible byp and none of the irreduciblconstituents ofβ lie aboveθ . It suffices to show that no irreducible constituent ofβ liesabove aP -invariant irreducible character ofL. If τ ∈ Irr(L) is P -invariant lying belowβ ,thenτ lies belowχ . Thenτ = θn for somen ∈ NG(P) ⊆ H . Thenθ lies belowβ , and thisis impossible. The rest of the theorem easily follows from Theorem (10.6) of [2].✷

Now, the Isaacs correspondence in Theorem 2.4 is constructed by induction o|G|as follows. LetH = Op′p(G)′NG(P). By using the solvability ofG, it is easy to checkthatG > H (if G > 1). Then, the Isaacs correspondence is the composition of the nmap Irrp′(G) → Irrp′(H) (obtained from Theorem 2.6) and the natural map Irrp′(H) →Irrp′(NG(P)) obtained by induction.

The next result is the analogous of Theorem 2.3 for Isaacsp′-degree charactecorrespondence.

Theorem 2.7. Suppose thatH is a subgroup ofG containing Op′p(G)′NG(P). Letχ ∈ Irrp′(G). Then(χ(H))∗ = χ∗.

Proof. This is Theorem (2.3) of [10]. ✷Finally, it is important to notice that if a groupA acts on an odd order groupG fixing

P ∈ Sylp(G) (setwise), then (χa)∗ = (

χ∗)afor everya ∈ A andχ ∈ Irrp′(G). This easily follows from the uniqueness in Theorem 2Of course, the analogous fact is true for the Isaacs correspondence in Theorem 2.1


(13.1)

main

he

t

t

same argument, or for the Glauberman correspondence (in this case, use Theoremof [4]).

3. More character correspondences

From now on, we assume that the reader is familiar with the definition andproperties of the Gajendragadkar special characters [1]. Let us denote byX p′(G) the setof p′-special characters ofG. By definition, recall thatXp′(G) ⊆ Irrp′(G). If we restrictthe Isaacs correspondence in Theorem 2.4 to the set ofp′-special characters, we have tfollowing.

Theorem 3.1. Suppose that|G| is odd and letP ∈ Sylp(G). Then

∗ :Xp′(G) → Irr(NG(P)/P

)is a bijection.

Proof. This is Corollary (3.3) of [7]. ✷The main objective in this section is to prove the following result.

Theorem 3.2. Let Q be ap-subgroup of a group of odd orderG. For everyN � G withQ ∩ N ∈ Sylp(N), there is a natural bijection

˜:Xp′,Q(N) → Irr(NN(Q)/Q ∩ N

),

whereXp′,Q(N) is the set ofQ-invariant p′-special characters ofN . In fact, if M is anormal subgroup ofG contained inN and θ ∈ Xp′,Q(N) and η ∈ Xp′,Q(M), then η isbelowθ if and only ifη is belowθ .

We need a lemma.

Lemma 3.3. Suppose thatG is a group of odd order and letP be a Sylowp-subgroupof G. Suppose thatK ⊆ M ⊆ G are normalp′-subgroups ofG. Suppose thatG/K has anormal Sylowp-subgroup and letK ′NG(P) ⊆ H ⊆ G. Letχ ∈ Irrp′(G) andθ ∈ IrrP (M).Thenχ lies aboveθ if and only ifχ(H) lies aboveθ [M∩H ].

Proof. Write N = NG(P). As in Lemma 2.5, letL = H ∩ K � G. Then, we know thaH = LN , G = KH andK ∩ H = L. Let C = CM(P). Notice that

M ∩ H = M ∩ LNG(P) = LNM(P) = LC.

SincePK/K � G/K is ap-subgroup andM/K � G/K is ap′-subgroup, it follows tha[M,P ] ⊆ K. Hence[M,P ]′ ⊆ K ′ ⊆ H ∩K = L, and thus[M,P ]′C ⊆ LC = M ∩H . ByTheorem 2.2, we have thatθ [M∩H ] ∈ IrrP (M ∩ H) is defined.


: apply

Now, sinceχ hasp′-degree andMP � G, it easily follows thatχM contains aP -invariant irreducible constituentγ ∈ Irr(M). SinceMN = G, we may write

χM = d(γ1 + · · · + γr),

whered is odd and{γ1, . . . , γs} are the differentN -conjugates ofγ . In particular, all ofthese characters areP -invariant. Now, by Theorem 2.2, we may write

(γi)LC = (γi)[LC] + 2∆i + Ξi,

where no irreducible constituent ofΞi is P -invariant. We claim thatγ [LC] lies aboveβ .If β1 is P -invariant lying belowγ [LC], thenβ1 andβ areP -invariant lying belowγ , sothey areC-conjugate (this is an standard argument in coprime action and charactersTheorem (13.27) and Corollary (13.9) of [4]). This proves the claim.

NowG/K has a normal Sylowp-subgroup, soOp′p(G) ⊆ K. Hence,Op′p(G)′N ⊆ H

and we haveχ(H) ∈ Irr(H) defined by Theorem 2.6. We have that

χH = χ(H) + 2∆ + Ξ,

where no irreducible constituent ofΞ lies aboveβ (otherwise,ΞH∩Op′p(G) would haveP -invariant irreducible constituents, contrary to Theorem 2.6). Hence,

χLC = (χ(H)

)LC

+ 2∆LC + ΞLC = d

(r∑i

(γi)[LC] + 2∆i + Ξi

).

Now, by using injectivity in Theorem 2.2, we have that[χLC,γ [LC]]

is odd. Sinceγ [LC] lies aboveβ , it follows that[ΞLC,γ [LC]]= 0.

Hence [(χ(H)

)LC

,γ [LC]]is odd. In particular,γ [LC] lies belowχ(H).

Now, if θ lies belowχ , it follows thatθ = γ n for somen ∈ N . Then

θ [LC] = (γ [LC])n

lies belowχ(H). Conversely, ifθ [LC] lies belowχ(H), by Clifford’s theorem, we will havethat

θ [LC] = (γ [LC])n


byose

for somen ∈ N . Now,

(γ [LC])n = (

γ n)[LC]

and by uniqueness in Theorem 2.2, we will have thatθ = γ n. This proves the lemma.✷Corollary 3.4. Suppose that|G| is odd and letP be a Sylowp-subgroup ofG. Letχ ∈ Irr(G) be ofp′-degree. LetM � G be ap′-group, and assume thatθ ∈ Irr(M) is P -invariant. Letχ∗ ∈ Irrp′(NG(P)) andθ∗ ∈ Irr(CM(P)) be the correspondents ofχ andθ ,respectively. Thenχ lies aboveθ if and only ifχ∗ lies aboveθ∗.

Proof. We argue by induction on|G|. Write N = NG(P) andC = CM(P). Let L/M =Op′p(G/M)′. Write H = LN . We have thatOp′p(G)′N ⊆ H . Assume thatH < G.

Suppose thatχ lies aboveθ . We claim thatχ(H) also lies aboveθ . Sinceχ(H) hasp′-degree, we may find aP -invariant irreducible constituentγ ∈ Irr(M) of (χ(H))M . Sinceχ lies aboveχ(H), we have thatγ is also an irreducible constituent ofχM . Therefore,γandθ areN -conjugate. Hence they areH -conjugate and the claim easily follows. Now,induction, we have that(χ(H))∗ lies aboveθ∗, and by Theorem 2.7, we are done. Suppconversely thatχ∗ lies aboveθ∗. By induction and Theorem 2.7, we will have thatχ(H)

lies aboveθ . Thenχ lies aboveθ (becauseχ lies aboveχ(H)), proving the corollary ifHis proper inG. So we may assume thatH = G. If R/M = Op′p(G/M), by Lemma 2.5, wededuce thatR = L. By the solvability ofG, we deduce thatR = L = M. ThusMP � G.Now, letK = [M,P ] � G, and notice thatK = Op′p(G). Let V = K ′. If U = V NG(P),we may apply Lemma 3.3, and we know thatχ lies aboveθ if and only if χ(U) lies aboveθ [M∩U ]. Thus, by induction and Theorems 2.3 and 2.7, we may assume thatU = G. Hence,K = V and by solvability, we have thatK = 1. Hence,M ⊆ C. ThenP � G, and thetheorem is trivial in this case.✷

We also need the following nontrivial result.

Lemma 3.5. Let N be a normal subgroup of a group of odd orderG. Letχ ∈ Irr(G) andθ ∈ Irr(N) be p′-special. LetP be a Sylowp-subgroup ofG. Let ν ∈ Irr(NG(P ∩ N))

be the uniquep′-special character withν∗ = χ∗ (by using Theorem3.1 and the fact thatNG(P) ⊆ NG(P ∩ N)). Thenθ lies belowχ if and only if θ∗ ∈ Irr(NN(P ∩ N)) liesbelowν.

Proof. This is Theorem (3.6) of [7]. ✷Now, we are ready to prove Theorem 3.2.

Proof of Theorem 3.2. First, we prove that there is a natural bijection

˜:Xp′,Q(N) → Irr(NN(Q)/Q ∩ N

).


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By Theorem 3.1, we have a natural bijection

∗ :Xp′(N) → Irr(NN(Q ∩ N)/Q ∩ N

).

Now, Q acts onN stabilizingQ ∩ N . By the last paragraph in Section 2, we have tα ∈Xp′(N) is Q-invariant if and only ifα∗ is. So we deduce that

∗ :Xp′,Q(N) → IrrQ(NN(Q ∩ N)/Q ∩ N

)is a bijection. Now,Q acts coprimely on thep′-groupNN(Q∩N)/Q∩N with fixed points

CNN(Q∩N)/Q∩N(Q) = NN(Q)/Q ∩ N.

By the Glauberman correspondence, we have a natural bijection

IrrQ(NN(Q ∩ N)/Q ∩ N

)→ Irr(NN(Q)/Q ∩ N

).

The composition of these two bijections is the natural bijection that we were lookinIt remains to show that is satisfies the desired property.

Suppose now thatM is a normal subgroup ofG contained inN and letθ ∈ Xp′,Q(N)

andη ∈ Xp′,Q(M). We want to prove thatη is belowθ if and only if η is belowθ . We dothis by induction on|G|. Write P = Q ∩ N ∈ Sylp(N). We have that

NN(Q) ⊆ NN(P ) ⊆ NN(P ∩ M).

We wish to apply Lemma 3.5 to the groupN with the normal subgroupM � N . Letν ∈ Irr(NN(M ∩ P)) be such thatν∗ = θ∗. By Lemma 3.5, we know thatθ lies aboveη if and only if ν lies aboveη∗. Now, notice thatQ ⊆ NG(P ∩ M). We have thaNN(P ∩ M) � NG(P ∩ M) suchQ ∩ NN(P ∩ M) ∈ Sylp(NN(P ∩ M)). Also, notice

that ν = θ , and η∗ = η. Hence, arguing by induction on|G|, we may assume thaNG(P ∩M) = G. Hence,P ∩M �G. Now, since eachp′-special character contain normp-subgroups in its kernel (Corollary (4.2) of [1]), by working in the factor groupG/P ∩M,and applying induction, we may easily assume thatM is ap′-group. In this case,η is theQ-Glauberman correspondent ofη.

Now, letη0 ∈ Irr(CM(P)) be theP -Glauberman correspondent ofη. By Corollary 3.4applied inN , we have thatθ lies aboveη if and only if θ∗ lies aboveη0. SinceP � Q, wehave that theQ-Glauberman correspondent ofη0 is η (by Theorem (13.1) of [4]). Now, wework in the factor groupX = NN(P )/P . This group is acted coprimely byQ. Also,X has anormal subgroupY = CM(P)P/P . Sinceθ∗ hasP in its kernel, we can viewθ∗ ∈ Irr(X).Also, theQ-Glauberman correspondent ofθ∗ is what we writeθ . Now, we also have thcharacterη0 ∈ Irr(Y ) (uniquely determined byη0). Furthermore, we have thatθ lies aboveη if and only if θ∗ lies aboveη0. By Theorem (13.29) of [4], we have thatθ∗ ∈ Irr(NN(P ))

lies aboveη0 if and only if θ lies aboveη, and this proves the theorem.✷Suppose again that we are in the hypothesis of Theorem 3.2. Since the bijec

Theorem 3.2 is the composition of Glauberman and Isaacs correspondences, w


.r, the

d

p.

t

cters

that if A acts onG stabilizingQ andN , thenA commutes with the -correspondenceThis follows from the comments in the last paragraph of Section 2. In particula˜ -correspondence commutes with the action ofNG(Q) onN .

4. Reviewing vertices

Suppose thatG is p-solvable. Ifα ∈ Irr(G) is p-special andβ ∈ Irr(G) is p′-special,we know that the productχ = αβ is irreducible and that the factorsα andβ are unique(see Proposition (7.1) of [1]). The irreducible characters ofG obtained this way are callefactorable. (We sayp-factorable if we wish to mention the primep.) We usually writeχp = α andχp′ = β . We will also use that thep-special characters ofG remain irreduciblewhen restricted to the Sylowp-subgroups ofG. In fact, this restriction is a one to one ma(See Proposition (6.1) of [1].)

Suppose now thatχ ∈ Irr(G). In [8], we associate toχ a unique pair(W,γ ), whereW ⊆ G, γ ∈ Irr(W) is p-factorable andγG = χ , uniquely determined byχ up to G-conjugacy. This association is made by induction on|G|. Givenχ , we consider the seP(χ) = {(N, θ), whereN � G andθ ∈ Irr(N) is factorable lying belowχ}. We order thisset of pairs by setting(N, θ) � (M,η) if N ⊆ M andθ lies belowη. If χ is p-factorable,then we let(W,γ ) = (G,χ). If χ is not factorable, we choose(N, θ) ∈ P(χ) be maximalbelow (G,χ). By Theorem (2.2) of [8],(N, θ) is unique up toG-conjugacy. Also, thestabilizerT of θ in G is proper inG (by Corollary (2.4) of [8]). Hence, ifψ ∈ Irr(T |θ) isthe Clifford correspondent ofχ aboveθ , we have thatψ is uniquely determined byχ upto G-conjugacy. SinceT is proper inG, by induction, we have associated toψ a uniquepair(W,γ ) up toT -conjugacy. Then we associate toχ all G-conjugates of the pair(W,γ )

and call each of these pairs anucleus(via normal pairs) forχ . Now, if Q is ap-subgroupof G andδ ∈ Irr(Q), we say that(Q, δ) is avertexof χ if there is a nucleus(W,γ ) of χ

with Q ∈ Sylp(W) and(γp)Q = δ. We denote by

Irr(G|Q,δ)

the set of irreducible characters ofχ with vertex(Q, δ).

5. Main results

In our Theorem A, we will construct a natural injection between irreducible charaof G and defect zero characters of a certain factor of a local subgroup ofG. In fact, it isno loss to work withrelative defect zero characters. If N � G andθ ∈ Irr(N), the set ofrelative defect zerocharacters ofG with respect toθ is

rdz(G|θ) = {χ ∈ Irr(G|θ) such that

(χ(1)/θ(1)

)p

= |G/N |p}.

Recall that we are writing Irr0(G) for the set of defect zero characters ofG. Of course,Irr0(G) = rdz(G|1).


s

Theorem 5.1. Suppose thatQ is a normalp-subgroup ofG and thatδ ∈ Irr(Q) is G-invariant. IfG is p-solvable, then there is a natural bijection

rdz(G|δ) → Irr0(G/Q).

Proof. This follows from Theorem (3.6) of [9], by lettingN = Q, α = 1N andβ = δ. ✷We need the following.

Lemma 5.2. Suppose thatG = NM, whereN,M are normal subgroups ofG. LetP be aSylowp-subgroup ofG. ThenNG(P) = NN(P )NM(P).

Proof. First we do the case whereG = N ×M. We have thatP = (P ∩N)× (P ∩M) andNG(P) = NN(P ∩N)×NM(P ∩M). However,NN(P ∩N) = NN(P ) andNM(P ∩M) =NM(P). Now, let �G = G/N ∩ M. By the first part, we have that

N�G(�P) = N �M(�P )N�N(�P ).

Hence, we deduce that

NG(P)(N ∩ M) = NM(P)NN(P )(N ∩ M).

Therefore

NG(P) = NG(P) ∩ NM(P)NN(P )(N ∩ M)

= NM(P)NN(P )(NG(P) ∩ N ∩ M

)= NM(P)NN(P ),

as wanted. ✷Lemma 5.3. LetQ be ap-subgroup ofG and letN,M � G. Suppose thatQ ∈ Sylp(QN)

andQ ∈ Sylp(QM). Then

NNM(Q) = NN(Q)NM(Q).

Proof. We argue by induction on|G|. Of course, we have thatNN(Q)NM(Q) ⊆ NNM(Q).It is clear that we may assume thatG = NMQ. Write L = NM. SinceG = (QN)(QM)

it is clear thatQ is a Sylowp-subgroup ofG. Let R = Q ∩ NM ∈ Sylp(NM). By theFrattini argument, we have thatG = LH , whereH = NG(R). Notice thatNG(Q) ⊆ H .Suppose thatH is proper inG. We have thatN ∩ H andM ∩ H are normal subgroupof H , and by induction we will have that

N(H∩N)(H∩M)(Q) = NH∩N(Q)NH∩M(Q) = NN(Q)NM(Q).

Now, by Lemma 5.2, we have that(H ∩ N)(H ∩ M) = H ∩ NM. Hence,

N(H∩N)(H∩M)(Q) = NH∩NM(Q) = NG(Q) ∩ H ∩ NM = NG(Q) ∩ NM = NNM(Q).


toione pair

ique

The lemma follows in this case. Hence, we may assume thatR � G. Now, by coprimeaction, we have that

NNM(Q)/R = NNM/R(Q/R) = CNM/R(Q) = C(NR/R)(MR/R)(Q)

= CNR/R(Q)CMR/R(Q) ⊆ NNR/R(Q/R)NMR/R(Q/R)

⊆ NN(Q)NM(Q)R/R.

Hence, we deduce thatNNM(Q) ⊆ NN(Q)NM(Q)R. HoweverR = (Q∩N)(Q∩M), andthereforeNN(Q)NM(Q)R = NN(Q)NM(Q). ✷

This is our main result.

Theorem 5.4. Suppose thatG is a group of odd order. LetQ be ap-subgroup ofG, andlet δ ∈ Irr(Q). There is a natural injection

Irr(G|Q,δ) → rdz(NG(Q,δ)|δ).

Proof. Suppose thatχ ∈ Irr(G|Q,δ). Let (W,γ ) be a nucleus ofχ such thatQ ∈ Sylp(W)

and(γp)Q = δ. Since(γp)Q = δ, notice thatNW(Q,δ) = NW(Q). By Theorem 3.1, wehave a uniquely defined

γ ∗p′ ∈ Irr

(NW(Q)/Q

).

Also, since(γp)Q is irreducible, we have that

(γp)NW (Q)γ∗p′ ∈ Irr

(NW(Q)

)by Corollary (6.17) of [4]. Also, notice that this character lies aboveδ. We prove byinduction on|G|, that (

(γp)NW (Q)γ∗p′)NG(Q,δ) ∈ Irr

(NG(Q,δ)

).

If this is true, it is easy to check that this a relative defect zero character ofNG(Q,δ) aboveδ sinceNW(Q)/Q is ap′-group. (The reader might want to see Theorem (3.5) of [9].)

If χ is factorable,(W,γ ) = (G,χ), Q ∈ Sylp(G) and in this case what we wishprove is a triviality. Suppose then thatχ is not factorable. In this case, by the constructof the normal nucleus (see Section 4), we can find a maximal normal factorabl(N, θ) of G below (W,γ ) such that ifT is the stabilizer ofθ in G, thenW ⊆ T < G,ψ = γ T ∈ Irr(T |Q,δ) and (W,γ ) is a nucleus forψ . Since W ⊆ T and Q ⊆ W ,notice thatθ is Q-invariant. In particular,θp′ is Q-invariant. Also,Q ∩ N ∈ Sylp(N)

(becauseQ ∈ Sylp(W) andN � W ), and then, by Theorem 3.2, we have defined a uncharacter

θp′ ∈ Irr(NN(Q)/Q ∩ N

).


ows

e

t

e

By the same argument as in the previous paragraph, we have that

η = (θp)NN(Q)θp′ ∈ Irr(NN(Q)

).

Now, since restriction ofp-special characters to a Sylow subgroup is one to one, it follthat x ∈ NG(Q) fixes θp if and only if x fixes (θp)NN(Q). By the remark following theproof of Theorem 3.2, we deduce thatNT (Q) is the stabilizer ofη in NG(Q). In particular,NT (Q, δ) is the stabilizer ofη in NG(Q,δ). Now, T < G, and by induction, we havthat (

(γp)NW (Q)γ∗p′)NT (Q,δ) ∈ Irr

(NT (Q, δ)

).

By the Clifford correspondence (Theorem (6.11) of [4]), it suffices to show that

((γp)NW (Q)γ

∗p′)NT (Q,δ)

lies aboveη. Now, we have thatγ lies aboveθ . (In fact, γN = eθ .) Hence,γp liesaboveθp andγp′ lies aboveθp′ . By Theorem 3.2 applied inW , we deduce thatγ ∗

p′ lies

aboveθp′ . Now, it is obvious that(γp)NW (Q) lies above(θp)NN(Q) and we deduce tha(γp)NW (Q)γ

∗p′ lies aboveη. Hence((γp)NW (Q)γ

∗p′)NT (Q,δ) lies aboveη, and we deduce

that ((γp)NW (Q)γ

∗p′)NG(Q,δ) ∈ Irr

(NG(Q,δ)

)by the Clifford correspondence.

Now, givenχ ∈ Irr(G|Q,δ), we could have chosen a different nucleus(W0, γ0) for χ ,whereQ ∈ Sylp(W0) and((γ0)p)Q = δ. In this case, since nuclei forχ areG-conjugate,we will have that(W0, γ0) = (Wx, γ x) for somex ∈ G. Now, Q,Qx are Sylowp-subgroups ofWx and we deduce thatQwx = Q for somew ∈ W (by Sylow theory). Also,

δ = ((γ0)p

)Q

= (γ wxp

)Q

= ((γp)Q

)wx = δwx.

Hence, by replacingx by wx, it is no loss to assume thatx ∈ NG(Q,δ). Therefore, we sethat the character (

(γp)NW (Q)γ∗p′)NG(Q,δ) ∈ Irr

(NG(Q,δ)

)only depends onχ (once we have fixed(Q, δ)). Hence, we see that

χ �→ ((γp)NW(Q)γ

∗p′)NG(Q,δ) ∈ Irr

(NG(Q,δ)

)is a natural correspondence Irr(G|Q,δ) → rdz(NG(Q,δ)|δ).


nce of

an

t,

ow

In order to finish the proof of this theorem, we need to show that our correspondecharacters is one to one. Suppose now thatµ ∈ Irr(G|Q,δ) has nucleus(U,ρ) such thatQ ∈ Sylp(U) and(ρp)Q = δ, and assume that(

(ρp)NU (Q)ρ∗p′)NG(Q,δ) = (

(γp)NW (Q)γ∗p′)NG(Q,δ) = ν.

We prove by induction on|G| thatχ = µ. Arguing as in the previous paragraph, we cfind a normal maximal factorable pair(M, τ) below (U,ρ) such that ifI is the stabilizerof τ in G, thenU ⊆ I , ϕ = ρI ∈ Irr(I |Q,δ) and with(U,ρ) being a nucleus ofϕ.

By the first paragraph of the proof, we know thatν lies above(θp)NN(Q)θp′ . By thesame reason, it also lies above(τp)NM(Q)τp′ . Now, letε ∈ Irr(NN(Q)NM(Q)) be belowν

and above(θp)NN(Q)θp′ . Also, let ξ ∈ Irr(NM(Q)) be belowε. Hence,ξ is belowν andwe deduce thatξ is NG(Q,δ)-conjugate to(τp)NM(Q)τp′ . By replacing(U,ρ) by someNG(Q,δ)-conjugate (as we can), it is no loss to assume thatε lies above(τp)NM(Q)τp′

and above(θp)NN(Q)θp′ . Now, by Theorem (2.2) of [8], we have thatε is factorable. Inparticular,εp′ lies aboveθp′ and aboveτp′ . Now, we have thatQ ∩ N ∈ Sylp(N) andQ ∩ M ∈ Sylp(M). ThusQ ∈ Sylp(NQ) andQ ∈ Sylp(MQ). ThusQ ∈ Sylp(NMQ)

and alsoQ ∩ NM ∈ Sylp(NM). By Lemma 5.3, we have that

NNM(Q) = NM(Q)NN(Q).

Also, Q ∩ NM is the normal Sylowp-subgroup ofNNM(Q). Hence, we have thatεp′ ∈Irr(NNM(Q)/Q∩NM). By Theorem 3.2, we can writeεp′ = α for someα ∈ Xp′,Q(NM).Now, sinceεp′ lies aboveθp′ by Theorem 3.2, we have thatα lies aboveθp′ . By the sameargument,α lies aboveτp′ . Now, sinceδ extends toW andQN ⊆ W , we have thatδ ex-tends toQN . By the same reason,δ extends toQM. By Theorem A of [2], we have thatδextends toQ(NM). By Theorem F of [3], letδ be thep-special extension ofδ toQNM. Bythe uniqueness of thep-special extensions (Proposition (6.1) of [1]), we have thatδQN =(γp)QN . Sinceγp lies aboveθp, we deduce thatδ lies aboveθp . By the same argumenit lies aboveτp. SinceN ⊆ NM � QNM, we may find ap-special characterβ1 of NM

lying aboveθp and belowδ. Now,αβ1 lies aboveθ . However,(N, θ) was maximal amongfactorable normal pairs ofG. Therefore, we deduce that(N, θ) = (NM,αβ1). By the sameargument, we may find ap-special characterβ2 of NM lying aboveτp and belowδ. There-fore, we deduce that(M, τ) = (NM,αβ2). Therefore,N = M andθp′ = τp′ . Also, θp andτp lies belowδ ∈ Irr(QN), and thereforeθp andτp areQ-conjugate. Butθp is Q-invariant(recall thatQ ⊆ W ⊆ T ). So we have thatθp = τp and(N, θ) = (M, τ). Hence,I = T .

If χ is factorable, then(N, θ) = (G,χ), (M, τ) = (G,µ) and we deduce thatχ = µ. Ifχ is not factorable, thenI = T < G. By the construction of our correspondence, we knthat (

(γp)NW (Q)γ∗p′)NT (Q,δ) and

((ρp)NU (Q)ρ

∗p′)NT (Q,δ)

are the Clifford correspondents ofν above

η = (θp)NN(Q)θp′ = (τp)NN(Q)τp′ .


9)

gh

5)

Z. 212

73.1–

86–195.

By uniqueness of Clifford correspondents, we deduce that((γp)NW (Q)γ

∗p′)NT (Q,δ) = (

(ρp)NU(Q)ρ∗p′)NT (Q,δ)

.

By the inductive hypothesis, we haveγ T = ρT . Hence

χ = γG = ρG = µ,

and the proof of the theorem is complete.✷Next is Theorem A of the introduction.

Theorem 5.5. Suppose thatG is a group of odd order. LetQ be ap-subgroup ofG and letδ ∈ Irr(Q). Then there is a natural injection


).

Proof. This is a consequence of Theorems 5.1 and 5.4.✷

References

[1] D. Gajendragadkar, A characteristic class of characters of finiteπ -separable groups, J. Algebra 59 (197237–259.

[2] I.M. Isaacs, Extensions of characters from Hallπ -subgroups ofπ -separable groups, in: Proc. EdinburMath. Soc., Vol. 38, 1985, pp. 313–317.

[3] I.M. Isaacs, Induction and restriction ofπ -special characters, Can. J. Math. 38 (1986) 576–604.[4] I.M. Isaacs, Character Theory of Finite Groups, Dover, New York, 1994.[5] I.M. Isaacs, G. Navarro, Weights and vertices for characters ofπ -separable groups, J. Algebra 177 (199

339–366.[6] I.M. Isaacs, G. Navarro, Characters ofp′-degree ofp-solvable groups, J. Algebra 246 (2001) 394–413.[7] G. Navarro, Weights, vertices and a correspondence of characters in groups of odd order, Math.

(1993) 535–544.[8] G. Navarro, Vertices for characters ofp-solvable groups, Trans. Amer. Math. Soc. 354 (2002) 2759–27[9] G. Navarro, Defect of characters and local subgroups inp-solvable groups, J. Group Theory 5 (2002) 24

268.[10] L. Sanus, Induction and character correspondences in groups of odd order, J. Algebra 249 (2002) 1[11] T. Wolf, Character correspondences in solvable groups, Ill. J. Math. 22 (1978) 327–340.

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A new character correspondence in groups of odd order