Lemma 1.

If x is a non-vanishing element of G and θ∈Irr⁡(N) for a normal subgroup N of G, some G-conjugate of θ is fixed by x.

Proof.

This is [4, Lemma 2.3]. Or note that, via Clifford theory, each χ∈Irr⁡(G|θ) vanishes off ⋃g∈GIG⁢(θg). ∎

Proposition 2.

If an abelian group A acts faithfully and completely reducibly on a finite-dimensional vector space V (possibly of mixed characteristic), then A has a regular orbit on V.

Proof.

We argue by induction on dim⁡(V). If dim⁡(V)=0, then A=1 and the zero vector is in a regular orbit of A. If V is the direct sum of proper A-invariant subspaces U and W, then the inductive hypothesis implies the existence of u∈U and w∈W such that CA⁢(u)=CA⁢(U) and CA⁢(w)=CA⁢(W). Then

and u+w is in a regular A-orbit. Finally, if V≠{0} is an irreducible A-module, let x be a non-zero vector of V and set B=CA⁢(x). Then CV⁢(B)≠{0} and CV⁢(B) is A-invariant because B is normal in A. By irreducibility, CV⁢(B)=V. Since A acts faithfully on V, B=1. Now x is in a regular A-orbit of V. ∎

Lemma 3.

The following statements hold.

1. If M and N are normal subgroups of G with N⊆M and if there exists θ∈Irr⁡(M) that vanishes on M-N, then every non-vanishing element of G lying in M indeed lies in N.

2. Suppose that G is solvable and Fi⁢(G)⊆M⊆Fi+1⁢(G) with i≥1 and M normal in G. If M/Fi⁢(G) is abelian, then every non-vanishing element of G lying in M lies in Fi⁢(G).

3. If G is supersolvable, then every non-vanishing element of G lies in Z⁢(F⁢(G)).

Proof.

(a) Every G-conjugate of θ vanishes on M-N. Choose χ∈Irr⁡(G|θ). By Clifford theory, χM is a sum of G conjugates of θ, and thus χ vanishes on M-N, proving (i).

(b) Let J/Fi-1⁢(G)=Φ⁢(G/Fi-1⁢(G)). By a theorem of Gaschutz [2, Sätze, III, 4.2, 4.3 and 4.5], Fi⁢(G)/J is a completely reducible and faithful G/Fi⁢(G)-module (possibly of mixed characteristic). Then V=Irr(Fi(G)/J)) is a completely reducible and faithful G/Fi⁢(G)-module. Since M/Fi is abelian and acts completely reducibly and faithfully on V, then M/Fi has a regular orbit on V by Proposition 2, i.e. there exists α∈Irr⁡(Fi⁢(G)) whose inertia group in M is Fi. Then α induces to an irreducible character θ of M. Since θ vanishes on M-Fi⁢(G), part (i) implies that every non-vanishing element of G lying in M lies in Fi⁢(G).

(c) This is [4, Theorem B]. ∎

Theorem 4.

Suppose that V is a faithful irreducible G-module and that there exists 1≠x∈F⁢(G) such that each v∈V is centralized by a G-conjugate of x. Then there exists a faithful irreducible H-module W and a transitive permutation group S of degree n such that G is isomorphic to a subgroup of H wreath S (i.e. H≀S) in its action on V=Wn. Here H is a factor group of a subgroup of G. Furthermore, we have that

1. |W|=q2 for a Mersenne prime q, H=F⁢(H)⊆Γ⁢(W) and the Sylow-2-subgroup of H is semi-dihedral of order 4⁢(q+1),

2. |W|=52, F⁢(H)≅Q8⁢Y⁢Z4 and H/F⁢(H)≅Z3 or S3, or

3. |W|=34, F⁢(H)≅Q8⁢Y⁢D8, with H/F⁢(H)≅Z5, D10, F20, A5 or S5.

Furthermore, if V is a quasi-primitive module, then V=W and G=H.

Proof.

See [7, Corollary 2.3 and Theorem 2.1]. ∎

Corollary 5.

Suppose that V is a faithful quasi-primitive irreducible G-module and that there exists 1≠x∈F⁢(G) such that each v∈V is centralized by a G-conjugate of x. Then there exists χ∈Irr⁡(G) such that χ⁢(x)=0.

Proof.

Note V and G satisfy Theorem 4 with G=H and V=W. If (b) or (c) holds, set U=Z⁢(F). If (a) holds, then G=F⁢(G) has a self-centralizing cyclic subgroup U of index two. Since U is abelian, U has a regular orbit on V, and so x∈F⁢(G)-U. In all cases, there exists θ∈Irr⁡(F⁢(G)) that vanishes on F⁢(G)-U. By Lemma 3 (a), there exists χ∈Irr⁡(G) such that χ⁢(x)=0. ∎

Lemma 6.

Suppose that V is a (finite) faithful completely reducible G-module for a solvable group G. Assume that dim⁡(V)≤2, that |V|=2j with j≤8, that |V|=3k with k≤5, or that V is irreducible with prime dimension. If x∈G is non-vanishing and x fixes an element in each G-orbit on V, then x=1.

Proof.

We argue by induction on |V|⁢|G|. Suppose that V=V1⊕V2 for G-modules Vi, and let Ci=CG⁢(Vi). Now Ci⁢x is a non-vanishing element of G/Ci, and Ci⁢x fixes an element of each G/Ci-orbit on Vi. If both Vi are non-trivial, then induction shows x∈C1∩C2=1. So we may assume that V is irreducible. We let F=F⁢(G).

If x lies in a normal subgroup U of G such that U has a regular orbit O in V, then x=1 (see discussion two paragraphs before Lemma 1). In particular, if x lies in an abelian normal subgroup of G, then x=1 as desired by Proposition 2.

If G is supersolvable, then every non-vanishing element of G lies in Z⁢(F⁢(G)) by Lemma 3 (c), and so x=1 by the last paragraph. In particular, the result holds if dim⁡(V)=1 of if |V|=22 or if G⊆Γ⁢(V). If |V|=2p for a prime p, then G⊆Γ⁢(V) by [5, Corollary 2.13] and so x=1 as desired.

Suppose that x∈F. Then, by Theorem 4, either x=1 or both char⁡(V) is odd and dim⁡(V) is even. Given our hypotheses, we see that V is quasi-primitive, that G is a subgroup of the wreath product Γ⁢(32)≀Z2, or that x=1. If V is quasi-primitive, then x=1 by Corollary 5. If G⊆Γ⁢(32)≀Z2, G is a 2-group and x=1 by the last paragraph. In all cases, x=1 if x∈F⁢(G).

Theorem 2.4 of [4] shows that if a solvable group H has Fitting length f>1, then every non-vanishing element of H lies in Ff-1⁢(H). It follows from the last paragraph and Lemma 3 (b) that x=1 if the Fitting length fl⁡(G) is at most 2 or if fl⁡(G)=3 and F2⁢(G)/F⁢(G) is abelian.

Suppose M is a normal subgroup of G with F/M a faithful completely reducible G/F-module (possibly of mixed characteristic). This does occur when M=Φ⁢(G) by Gaschutz’s theorem. Write F/M=M1⊕…⊕Mm for irreducible G-modules Mi. Let Di=CG⁢(Mi). Then W=Irr⁡(F/M) is a direct sum of irreducible G-modules W1⊕…⊕Wm such that |Wi|=|Mi| and Di=CG⁢(Wi). Since x is a non-vanishing element of G, then x fixes an element of each G/Di-orbit on Wi and also Di⁢x is a non-vanishing element of G/Di. Assume for each i, one of the following occurs: dim⁡(Mi) is 1 or prime, |Mi|=2j with j≤8, or |Mi|=3k with k≤5. Then the induction argument shows that Di⁢x=1 for all i. Then x∈D1∩…∩Dm=F. By the next to last paragraph, x=1.

Assume dim⁡(V)=p, a prime. We apply [5, Theorem 2.12] to determine the structure of G. If V is quasi-primitive, then G is isomorphic to a subgroup of Γ⁢(V) or F/Z⁢(G) is a faithful irreducible module G/F-module of order p2. Then x=1 by the third paragraph or the last paragraph, respectively. If V is not a quasi-primitive module, then G has a normal subgroup N such that VN is a direct sum of p one-dimensional N-modules permuted transitively by G/N. Then N is abelian and G/N is a subgroup of the Frobenius group of order p⁢(p-1). Then fl⁡(G)≤3 and F2⁢(G)/F⁢(G) is abelian. By the next to last paragraph, x=1 in this case. So x=1 if dim⁡(V) is prime.

Suppose that |V|=24 or 28. Since char⁡(V)=2 and dim⁡(V) is a power of 2, we see that F must be an abelian group of odd order by [5, Corollary 2.5]. If V is quasi-primitive, then G⊆Γ⁢(V) by [5, Corollary 2.3], and so x=1 by the third paragraph. So we assume V is not quasi-primitive. If |V|=24, then G is isomorphic to a subgroup of the wreath product S3≀Z2 which has Fitting length 2, and so x=1 by the fifth paragraph. We can assume that |V|=28. Since V is not quasi-primitive, G is isomorphic to a subgroup of Γ⁢(24)≀Z2 or a subgroup of S3≀S4. Thus |F| divides 52⋅32 or 35 (respectively). Since F/Φ⁢(G) is a completely reducible and faithful G/F-module, it follows from the sixth paragraph that x=1.

Suppose that |V|=26. It follows from [5, Corollary 2.15] that F is extra-special of order 33 or that G is isomorphic to a subgroup S3≀S3, a subgroup of Γ⁢(23)≀Z2, or a subgroup of Γ⁢(26). In all cases, |F| divides 34⋅72. Since F/Φ⁢(G) is a completely reducible and faithful G/F-module, it follows from the sixth paragraph that x=1.

Finally, suppose that |V|=34. If V is not quasi-primitive, then G is isomorphic to a subgroup of GL⁢(2,3)≀Z2 and so fl⁡(G)≤3 and F2⁢(G)/F is an abelian 3-group. In this case, x=1 by the fifth paragraph. We may assume that V is a quasi-primitive, but G is not a subgroup of Γ⁢(V). By [5, Theorem 1.10 and Corollary 2.5], we have that F is extra-special of order 25 or that F=E⁢T, where E is extra-special of order 23 and T has a cyclic subgroup U of index 1 or 2 and |U| divides 8. In all cases, |F| divides 27. Since F/Φ⁢(G) is a completely reducible and faithful G/F-module, it follows from the sixth paragraph that x=1. ∎

Theorem 7.

Suppose that V is a (finite) quasi-primitive G-module for a solvable group G and x is a non-vanishing element of G such that x fixes an element in each G-orbit of V. Then x=1.

Proof.

Since V is quasi-primitive, every abelian normal subgroup of G is cyclic. By [5, Corollary 1.10], there exist normal subgroups F=F⁢(G), E, T, Z, and A=CG⁢(Z) such that Z=Z⁢(E) is cyclic, such that F/T≅E/Z is a completely reducible G-module (possibly of mixed characteristic) of order e2 for an integer e, and A/F acts faithfully on E/Z. Since G has no regular orbit of V (by discussion preceding Lemma 1), a theorem of Yang [9] shows that e is 1, 2, 3, 4, 8, 9 or 16.

Now W=Irr⁡(F/T) is a completely reducible G/F-module of order e2 and A/F acts faithfully on W. Let D=CG⁢(W). Then F⊆D and A∩D=F. We claim x∈D. This is trivial if e=1. Else write W=W1⊕…⊕Wk for irreducible G-modules Wi. Set Di=CG⁢(Wi) for each i. As x is a non-vanishing element of G, indeed, Di⁢x is a non-vanishing element of G/Di. Also Di⁢x must fix an element of each G/Di-orbit on Wi. Since |Wi| is 2i with i≤8 or 3j with j≤4, it follows from Lemma 6 that x∈Di for each i. Since ⋂i=1kDi=D, then x∈D, as claimed.

Now Z is cyclic and A=CG⁢(Z). Since A∩D=F, we have F=CD⁢(Z). Let λ∈Irr⁡(Z) be faithful. Since Z is cyclic and λ is faithful, it follows that ID⁢(λ)=A∩D=F. Thus every α∈Irr⁡(D|λ) vanishes on D-F. As D and F are normal in G, it follows that every non-vanishing element of G lying in D indeed lies in F by Lemma 3. By the last paragraph, x∈F. By Corollary 5, x=1. ∎

Theorem 8.

For a solvable group G, Gaschutz’s theorem shows that F⁢(G)/Φ⁢(G) is a direct sum of irreducible G-modules Mi. If each Mi is a quasi-primitive G-module, then every non-vanishing element of G lies in F⁢(G).

Proof.

Gaschutz’s theorem [2, Sätze, III, 4.2, 4.3 and 4.5] also says G/F⁢(G) acts faithfully on F⁢(G)/Φ⁢(G). If G=F⁢(G), the result holds. So we may write

for irreducible G-modules Mi with m≥1. Let Di=CG⁢(Mi), and note that

by the first sentence. Then W=Irr⁡(F⁢(G)/Φ⁢(G)) is a direct sum of irreducible G-modules W1⊕…⊕Wm such that |Wi|=|Mi| and Di=CG⁢(Wi). Since Mi is a quasi-primitive G-module, so is Wi (a dual of Mi) a quasi-primitive G-module. Since x is a non-vanishing element of G, then x fixes an element of each G/Di-orbit on Wi and also Di⁢x is a non-vanishing element of G. By Theorem 7, x∈Di for all i. Hence x∈F⁢(G). ∎