Character degrees in 𝜋-separable groups

Theorem A.

Let G be a π-separable group, and let p be any prime. Then G has a normal abelian Sylow p-subgroup if and only if p does not divide the degree of any character in Bπ⁡(G)∪Bπ′⁡(G).

Corollary B.

Let G be a π-separable group, and let p be any prime. Then p divides the degree of some characters in Irr⁡(G) if and only if it divides the degree of some characters in Bπ⁡(G)∪Bπ′⁡(G).

Theorem C.

Let G be a π-separable group. Then Irrπ′⁡(G)=Lin⁡(G) if and only if Irrπ′⁡(G)∩(Bπ⁡(G)∪Bπ′⁡(G))⊆Lin⁡(G).

Theorem D.

Let G be a π-separable group, let H be a Hall π-subgroup for G, and let N=NG⁡(H). Then

1. Irrπ′⁡(G)∩Bπ⁡(G)⊆Lin⁡(G) if and only if G′∩H=H′;

2. Irrπ′⁡(G)∩Bπ′⁡(G)⊆Lin⁡(G) if and only if G′∩N≤H.

Definition ([2]).

Let χ∈Irr⁡(G); then χ is said to be a π-special character if, for any M≤G subnormal in G, every irreducible constituent of χM has order and degree which are both π-numbers.

Theorem 2.1 ([5]).

Let G be a π-separable group, and let χ∈Irr⁡(G). Then there exist a subgroup W≤G, canonically defined up to conjugation, α∈Xπ⁡(W) and β∈Xπ′⁡(W) such that χ=(α⁢β)G.

Theorem 2.2 ([5, Lemma 5.4]).

Let χ∈Bπ⁡(G); then χ is π-special if and only if χ⁢(1) is a π-number.

Theorem 2.3 ([9, Theorem 3.6]).

Let G be a π-separable group, H∈Hallπ⁡(G), and let χ∈Irrπ′⁡(G). Then there exist a subgroup H≤W≤G, α∈Xπ⁡(W) linear and β∈Xπ′⁡(W) such that χ=(α⁢β)G. Moreover, α and β are unique in Irr⁡(W) with this property. Finally, W can be chosen as the (unique) maximal subgroup of G such that αH extends to W.

Theorem 2.4.

Let G be a π-separable group, and let H≤G such that |G:H| is a π′-number. Let ψ∈Irr⁡(H) be a π-special character, and suppose ψ extends to G. Then ψ has an extension to G which is also a π-special character.

Proof.

This is a direct consequence of the more general result [7, Theorem F]. ∎

Theorem 2.5.

Let G be π-separable, let M⊴G, and let χ∈Bπ⁡(G); then every irreducible constituent of χM belongs to Bπ⁡(M).

On the other hand, let ψ∈Bπ⁡(M); if |G:M| is a π-number, then every irreducible constituent of ψG is in Bπ⁡(G) while, if |G:M| is a π′-number, then there exists a unique irreducible constituent of ψG which belongs to Bπ⁡(G).

In particular, if ψ∈Bπ⁡(M), then there always exists at least one character χ∈Irr⁡(G∣ψ) which belongs to Bπ⁡(G).

Proof.

This is a direct consequence of [5, Theorem 6.2] and [5, Theorem 7.1] ∎

Theorem 2.6 ([5, Theorem 8.1]).

Let χ∈Bπ⁡(G), and let H∈Hallπ⁡(G); then there exists an irreducible constituent φ of χH such that φ⁢(1)=χ⁢(1)π. Moreover, for any irreducible constituent φ of χH such that φ⁢(1)=χ⁢(1)π, the multiplicity of φ in χH is 1 and φ does not appear as an irreducible constituent of the restriction to H of any other character in Bπ⁡(G).

Theorem 2.7 ([6, Corollary 6.1] or [8, Theorem 5.13]).

Let H be a Hall π-subgroup of a π-separable group G, and let φ∈Irr⁡(H). If φ is primitive, then it is a Fong character associated with some character in Bπ⁡(G). If φ1 is another primitive irreducible character of H, then φ and φ1 are associated with the same character in Bπ⁡(G) if and only if they are NG⁡(H)-conjugated.

Example 2.8.

Let G=H⋉M, with H=SL⁡(2,3) acting canonically on the vector space M=(ℤ3)2. Computing the character table of G, we can see that cd⁡(G)={1,2,3,8} and, with a little more work, it is not hard to prove that cdB3⁡(G)={1,8} and cdB2⁡(G)={1,2,3}.

Now, let Γ=G≀C2, let θ∈B3⁡(G) of degree 8, and let η∈B2⁡(G) of degree 3. The character θ×η∈Irr⁡(G×G) induces irreducibly to Γ and χ⁢(1)=48.

Suppose that there exists ψ∈B2⁡(Γ)∪B3⁡(Γ) such that ψ⁢(1)=χ⁢(1), and let λ1×λ2 be an irreducible constituent of ψG×G. Then λ1 and λ2 are either both in B2⁡(G) or they are both in B3⁡(G). Moreover, since ψ⁢(1)=48, then we have λ1⁢(1)⁢λ2⁢(1)∈{24,48}. However, neither 24 nor 48 can be written as a product of two numbers in cdB2⁡(G) or as a product of two numbers in cdB3⁡(G). It follows that 48∈cd⁡(Γ) but 48∉cdB2⁡(Γ)∪cdB3⁡(Γ).

Lemma 3.1.

Let G be a group, let N be a normal minimal π′-subgroup, and let M⊴G such that M/N is an abelian π-group. Furthermore, suppose that Oπ⁡(M)=1. Then there exists a character χ∈Bπ′⁡(G) such that χ⁢(1) is divided by |M:N|.

Proof.

Let A be a complement for N in M, which exists by the Schur–Zassenhaus theorem. Since CA⁡(N)≤Z⁡(M) and A is a π-group,

Thus, A acts faithfully on N. By [1, Lemma 2.8], there exists some character τ∈Irr⁡(N) such that η=τM∈Irr⁡(M). In particular, |M:N| divides η⁢(1). Since τ∈Bπ′⁡(N)=Irr⁡(N) and N is normal in M, by Theorem 2.5, the character η is in Bπ′⁡(M), too. It follows that Bπ′⁡(G∣η) is nonempty and |M:N| divides the degree of every character in Bπ′⁡(G∣η). ∎

Theorem 3.2 ([8, Theorem 3.17]).

Let G be π-separable; then Bπ⁡(G)=Xπ⁡(G) if and only if G has a normal π-complement.

Proof.

Note at first that, if G has a normal π-complement H, then it follows that Bπ⁡(G)=Xπ⁡(G)=Irr⁡(G/H). Thus, there is only one implication to be proved.

Let us assume that Bπ⁡(G)=Xπ⁡(G) and prove the thesis by induction on |G|. At first, let us assume that Oπ′⁡(G)=1 since, otherwise, the thesis would follow by induction.

Let N be a normal minimal subgroup of G, and suppose it to be a π-group. Since the hypotheses are preserved by factor groups, if H is a Hall π′-subgroup of G, then, by induction on |G|, we have that HN is normal in G. In particular, it follows that there exists K⊲G such that K/N is a π′-chief factor of G. Since |N| and |K/N| are coprime, at least one of them is odd and, thus, since an odd group is solvable and both N and K/N are normal minimal in G, at least one of them is abelian.

Suppose K/N is abelian. Since we have assumed Oπ′⁡(G)=1, it follows by Lemma 3.1 that there exists a character in Bπ⁡(G) whose degree is divided by the π′-number |K:N|, contradicting the hypothesis.

Suppose that K/N is not abelian, so N has to be, and let λ∈Irr⁡(N)=Bπ⁡(N). If λ is not K-invariant, then the degree of some θ∈Bπ⁡(K∣λ) is divided by some primes in π′ and, therefore, so is the degree of some character χ∈Bπ⁡(G∣θ), in contradiction with the hypothesis. It follows that K fixes every character of N and, thus, it also centralizes N since N is abelian. If B is a complement of N in K, then it is normal in K. In particular, 1<B=Oπ′⁡(K)≤Oπ′⁡(G).

Therefore, Oπ′⁡(G)≠1, and the thesis follows by induction on |G|. ∎

Proof of Theorem A.

It can be observed that there is little to prove in one direction, it being a consequence of the Ito–Michler theorem. Thus, we assume that p does not divide the degree of any character in Bπ⁡(G)∪Bπ′⁡(G), and we first prove that G has a normal Sylow p-subgroup. We argue by induction on |G|.

Let N be a minimal normal subgroup of G. Without loss of generality, we can assume N to be a π-group. Suppose p∈π. By induction, let K/N be a normal Sylow p-subgroup of G/N; then K is a normal π-subgroup of G which contains a Sylow p-subgroup P∈Sylp⁡(G). If P is normal in K, then it is also normal in G. Otherwise, there exists θ∈Irr⁡(K)=Bπ⁡(K) such that p∣θ(1) and, by Theorem 2.5, there exists χ∈Bπ⁡(G) lying over θ. As a consequence, p∣χ(1), in contradiction with the hypothesis.

Therefore, we can assume p∉π and, in particular, p does not divide |N|. Since N is arbitrarily chosen, we can assume that Op⁡(G)=1. As in the previous paragraph, let K/N be a normal Sylow p-subgroup of G/N, which is nontrivial because p divides |G:N|, and let C/N=Z⁡(K/N). Note that N<C⊴G and C/N is an abelian π′-group. Then, by Lemma 3.1, there exists a character χ in Bπ⁡(G) such that |C:N| divides χ⁢(1). However, since |C:N| is a power of p, this would contradict the hypothesis.

Finally, if P is a normal Sylow p-subgroup of G and γ∈Irr⁡(P), then, by Theorem 2.5, there exists χ∈Bπ⁡(G)∪Bπ′⁡(G) lying over γ and, thus, γ(1)∣χ(1). Since p∤χ⁢(1), then γ is linear. It follows that Irr⁡(P)=Lin⁡(P) and, thus, P is abelian. ∎

Theorem 4.1 ([12, Theorem 1.15]).

Let π and ω be two sets of primes, and let G be both π-separable and ω-separable. Let H be a Hall ω-subgroup of G, and let N=NG⁡(H). Then

Corollary 4.2.

Let G be a π-separable group, let H be a Hall π-subgroup of G, and let N=NG⁡(H). Then

Lemma 4.3.

Let G be a π-separable group, and let H be a Hall π-subgroup of G; then Irrπ′⁡(G)∩Bπ⁡(G)⊆Lin⁡(G) if and only if every linear character in H extends to G.

Proof.

Let λ be a linear character in H. By Theorem 2.7 and Theorem 2.6, λ is the Fong character associated with some character χ∈Irrπ′⁡(G)∩Bπ⁡(G). It follows that, if χ is linear, then it extends λ while, on the other hand, if λ extends to G, then, by Theorem 2.4, it has a linear π-special extension, which coincides with χ by Theorem 2.6. ∎

Proposition 4.4 (Theorem C).

Let G be a π-separable group. Then we have that Irrπ′⁡(G)=Lin⁡(G) if and only if Irrπ′⁡(G)∩(Bπ⁡(G)∪Bπ′⁡(G))⊆Lin⁡(G).

Proof.

One direction is obviously true. Thus, let one assume

and suppose there exists a nonlinear character χ∈Irr⁡(G) such that χ⁢(1) is a π′-number. By Theorem 2.3, there exist W≤G, α∈Xπ⁡(W) linear and β∈Xπ′⁡(W) such that χ=(α⁢β)G, W contains a Hall π-subgroup H of G and it is the maximal subgroup of G such that αH extends to W. However, by Lemma 4.3, αH extends to G; thus W=G. It follows that β is a nonlinear π′-special character of G, negating the fact that every character in Xπ′⁡(G)=Irrπ′⁡(G)∩Bπ′⁡(G) is linear, in contradiction with the hypothesis. ∎

Corollary 4.5.

Let G be a π-separable group, and let H be a Hall π-subgroup of G. Then the following statements hold.

1. Irrπ′⁡(G)={1G} if and only if Irrπ′⁡(G)∩(Bπ⁡(G)∪Bπ′⁡(G))={1G}.

2. Irrπ′⁡(G)∩Bπ⁡(G)={1G} if and only if H=H′.

3. Xπ′⁡(G)={1G} if and only if H is self-normalizing.

4. Irrπ′⁡(G)={1G} if and only if H=H′ and H is self-normalizing.

Proof.

For point (i), only one direction is needed. Suppose, thus, that

Then, by Proposition 4.4, Irrπ′⁡(G)=Lin⁡(G). It follows that every character in Irrπ′⁡(G) can be factorized as a product α⁢β, with α∈Irrπ′∩Xπ⁡(G), β∈Xπ′⁡(G); however, Irrπ′∩Xπ⁡(G) is a subset of Irrπ′∩Bπ⁡(G), while

and the two sets of characters both coincide with {1G} by hypothesis. Therefore, it follows that α=β=1G and Irrπ′⁡(G)={1G}.

Point (ii) follows directly from Lemma 4.3. In fact, if

then every character in Lin⁡(H) extends to G and, by Theorem 2.4, it has an extension in Irrπ′⁡(G)∩Bπ⁡(G), and thus Lin⁡(H)={1H}. On the other hand, if Lin⁡(H)={1H}, then there are no nonprincipal linear Fong characters of H in G, and it follows that |Irrπ′⁡(G)∩Bπ⁡(G)|=1, and the thesis follows.

Finally, point (iii) is a direct consequence of Corollary 4.2, and point (iv) follows from points (i), (ii) and (iii). ∎

Proposition 4.6.

Let G be a finite group, and let H be a Hall π-subgroup for G. Then G′∩H=H′ if and only if every character in Irr⁡(H/H′) extends to G.

Proof.

Suppose that, for every λ∈Irr⁡(H/H′), there exists χ∈Lin⁡(G) such that χH=λ. It follows that

and, therefore, G′∩H=H′.

On the other hand, suppose G′∩H=H′. Then one can write

for some K/G′∈Hallπ′⁡(G/G′) and, if we identify Irr⁡(H/H′) with Irr⁡(H⁢G′/G′), we have that every λ∈Irr⁡(H/H′) extends to λ×1K/G′∈Irr⁡(G/G′). ∎

Corollary 4.7 (Theorem D (a)).

Let G be a π-separable group, and let H be a Hall π-subgroup for G. Then Irrπ′⁡(G)∩Bπ⁡(G)⊆Lin⁡(G) if and only if G′∩H=H′.

Proof.

From Lemma 4.3, the property that every character in Irrπ′⁡(G)∩Bπ⁡(G) is linear is equivalent to the fact that every character in H/H′ extends to G. By Proposition 4.6, we deduce that it happens if and only if G′∩H=H′. ∎

Proposition 4.8 (Theorem D (b)).

Let G be a π-separable group, let H be a Hall π-subgroup for G, and let N=NG⁡(H). Then Xπ′⁡(G)⊆Lin⁡(G) if and only if G′∩N≤H.

Proof.

Assume at first that Xπ′⁡(G)⊆Lin⁡(G); therefore, if χ∈Xπ′⁡(G), then χN is linear. Suppose that, for some χ,ψ∈Xπ′⁡(G), we have χN=ψN; then

It follows that ker⁡(χ⁢ψ¯)=G, by the Frattini argument, and thus χ=ψ. Therefore, the restriction realizes an injection from Xπ′⁡(G) to Xπ′⁡(N) and, since we have |Xπ⁡(G)|=|Xπ⁡(N)|, by Corollary 4.2, it is actually a bijection. It follows that every character in Irr⁡(N/H) is the restriction of a linear character of G; thus we have that

On the other hand, suppose that G′∩N≤H. Let X be a complement for H in N, and notice that X is abelian. Moreover, notice that N⁢G′ is normal in G and it contains N; thus G=N⁢G′ for the Frattini argument. It follows that

and, thus, there is a bijection between characters in Irr⁡(X)=Irr⁡(N/H) and characters in Xπ′⁡(G/G′). However, by Corollary 4.2, we have that

and, thus, it follows that every π′-special character in G is linear. ∎

Corollary 4.9 ([11, Corollary 3]).

Let G be a π-separable group, and let H be a Hall π-subgroup for G and N=NG⁡(H). Then Irrπ′⁡(G)=Lin⁡(G) if and only if G′∩N=H′.

Proof.

If N is the normalizer in G of a Hall π-subgroup H, Proposition 4.4, Corollary 4.7 and Proposition 4.8 provide that Irrπ′⁡(G)=Lin⁡(G) if and only if both G′∩N≤H and G′∩H=H′, and the thesis follows directly from this. ∎