The construction of nuclei for normal constituents of B_π-characters

Theorem A

Let N ◃ G , where G is π -separable. Let χ ∈ B π ( G ) , and let ( W , γ ) be a nucleus for χ . Write V = N ∩ W , and let τ ∈ Irr ( V ) lie under γ .

1. If G / N is a π ′ -group, then ( V , τ ) is a nucleus for some irreducible constituent of χ N .

2. If G / N is a π -group, and assume further that some Fong character associated with χ is quasi-primitive. Then, ( V , τ ) is a nucleus for some irreducible constituent of χ N .

Corollary B

Let N ◃ G , where G is π -separable. Let χ ∈ B π ( G ) have π ′ -degree, and let ( W , γ ) be a nucleus for χ . Write V = N ∩ W , and let τ ∈ Irr ( V ) lie under γ . Then, ( V , τ ) is a nucleus for some irreducible constituent of χ N .

Lemma 2.1

Let ( S , φ ) ∈ S ∗ ( G ) , where G is π -separable, and let N ◃ G , where G / N is either a π -group or a π ′ -group. Write D = S ∩ N , and let δ ∈ Irr ( D ) lie under φ . Then, ( D , δ ) ∈ S ∗ ( N ) .

Proof

This is precisely Lemma 4.7 of [8].□

Lemma 2.2

Let N ◃ G , where G is π -separable, and let θ ∈ Irr ( N ) be π -special.

1. If G / N is a π -group, then every member of Irr ( G ∣ θ ) is π -special.

2. If G / N is a π ′ -group and θ is invariant in G , then θ has a canonical extension χ ∈ Irr ( G ) and χ is the unique π -special character in Irr ( G ∣ θ ) . If θ is not invariant in G , then no member of Irr ( G ∣ θ ) is π -special.

Proof

See Theorem 2.4 of [8].□

Lemma 2.3

Let ( S , φ ) ∈ S ∗ ( G ) , where G is π -separable. Suppose that ( D , δ ) ≤ ( S , φ ) , where D ◃ S and S / D is either a π -group or a π ′ -group. Then, G δ ≤ N G ( S ) .

Proof

This is precisely Lemma 4.16 of [8].□

Lemma 2.4

Let G be π -separable, and suppose ( S , φ ) ∈ S ∗ ( G ) . If G φ = G , then S = G .

Proof

This is precisely Lemma 4.10 of [8].□

Lemma 2.5

Let ( S , φ ) ∈ S ∗ ( G ) , where G is π -separable. Then, induction defines a bijection Irr ( G φ ∣ φ ) → Irr ( G ∣ φ ) .

Proof

This is precisely Theorem 4.9 of [8].□

Lemma 2.6

Let α ∈ Irr ( H ) be quasi-primitive, where H is a Hall π -subgroup of a π -separable group G, and let H ≤ W ≤ G , where W is the largest subgroup to which α extends. Let γ ∈ Irr ( W ) be the unique π -special extension of α to W , and write χ = γ G . Then, χ is irreducible, and in fact, χ ∈ B π ( G ) . In addition, ( W , γ ) is a nucleus for χ .

Proof

See Theorem 4.30 of [8].□

Lemma 2.7

Suppose that G is π -separable, and let H ≤ G , where ∣ G : H ∣ is a π ′ -number. Then, restriction defines an injection map from X π ( G ) to X π ( H ) . In particular, this holds if H is a Hall π -subgroup of G .

Proof

This is precisely Theorem 2.10 of [8].□

Lemma 2.8

Let N ◃ G , where G is π -separable and G / N is a π -group, and let ( S , φ ) ∈ S ∗ ( G ) . Let D = S ∩ N , and let δ ∈ Irr ( D ) lie under φ . Then,

1. G φ ∩ N δ ◃ G φ ;

2. G φ / ( G φ ∩ N δ ) is a π -group;

3. G φ ∩ N δ ◃ N δ ;

4. N δ / ( G φ ∩ N δ ) is a π -group.

Proof

See Lemma 4.17 of [8].□

Lemma 2.9

Let N ◃ G , where G is π -separable and G / N is a π ′ -group. Let χ ∈ B π ( G ) , and suppose that N ≤ K ≤ G . Then, every irreducible constituent of χ K lies in B π ( K ) .

Proof

This is precisely Theorem 4.25 of [8].□

Lemma 2.10

Let N ◃ G , where G is π -separable and G / N is a π ′ -group, and let ψ ∈ Irr ( N ) .

1. If χ ∈ B π ( G ) lies over ψ , then ψ ∈ B π ( N ) .

2. If ψ ∈ B π ( N ) , then there exists a unique character χ ∈ B π ( G ) that lies over ψ . In addition, [ χ N , ψ ] = 1 .

3. Suppose that ψ and χ are as in (b). Let ( V , τ ) be a nucleus for ψ , and write W = G τ . Then, W / V is a π ′ -group, and ( W , γ ) is a nucleus for χ , where γ is the canonical extension of τ to W .

Proof

See Theorem 4.19 of [8].□

Proof of Theorem A

(1) Choose ( S , φ ) ∈ S ∗ ( G ) with ( S , φ ) ≤ ( W , γ ) and W ≤ G φ , and let S 1 = N ∩ S . It is clear that S 1 ≤ V . Let φ 1 ∈ Irr ( S 1 ) lie under φ , and observe that ( S 1 , φ 1 ) ∈ S ∗ ( N ) by Lemma 2.1. Since χ ∈ B π ( G ) , we have that both γ and φ are π -special. Note that since W / V is a π ′ -group, it follows that γ V is irreducible, and thus τ = γ V is π -special. This means that τ is W-invariant, and γ is a π-special extension of τ to W. By Lemma 2.2(b), we have γ is the unique π-special character in Irr(W|τ). Also, since S/S ₁ is a π′-group and φ is π-special, it follows that φ S 1 is irreducible, and thus φ S 1 = φ 1 is π-special. This means that φ ₁ is S-invariant, and φ is a π-special extension of φ ₁ to S. By Lemma 2.2(b), we have φ is the unique π-special character in Irr(S|φ ₁). Since φ 1 lies under γ , and γ V = τ is irreducible, it follows that φ 1 lies under τ .

We claim that G φ 1 = G φ . Since G φ normalizes S ∩ N = S 1 and stabilizes φ , we see that it also stabilizes φ 1 because φ 1 = φ S 1 , and thus G φ ≤ G φ 1 . On the other hand, by Lemma 2.3, we know that G φ 1 normalizes S . Since G φ 1 stabilizes S 1 and φ 1 , by the uniqueness of φ , we know that G φ 1 stabilizes φ , and thus G φ 1 ≤ G φ , as claimed. So we conclude that N φ 1 = G φ 1 ∩ N = G φ ∩ N , and thus N φ 1 ◃ G φ and G φ / N φ 1 is a π ′ -group.

We proceed by induction on ∣ G ∣ . Suppose first that G φ = G . Then, S = G by Lemma 2.4, and thus V = N , and the π -special character τ is an irreducible constituent of χ N . Then, ( V , τ ) is a nucleus for τ . We can assume now that G φ < G . By Lemma 2.5, there exists a unique irreducible character ξ ∈ Irr ( G φ ∣ φ ) such that ξ G = χ . Note that ξ ∈ B π ( G φ ) , and ( W , γ ) is also a nucleus for ξ . It follows by the inductive hypothesis applied in G φ that ( V , τ ) is a nucleus for some irreducible constituent of ξ N φ 1 . Let η = τ N φ 1 . We have that ( V , τ ) is a nucleus for η . Now η lies under ξ and over τ , and thus under χ and over φ 1 . By Lemma 2.5, we conclude that ψ = η N is irreducible. It follows that ψ lies under χ , and ( V , τ ) is a nucleus for ψ . This proves (1).

(2) Let α ∈ Irr ( H ) be a quasi-primitive Fong character associated with χ , where H is a Hall π -subgroup of G . By Lemma 2.6, it is no loss to assume that ( H , α ) ≤ ( W , γ ) because the nuclei for χ are conjugate in G , and note that W is precisely the largest subgroup of G to which α extends, and γ is the unique π -special extension of α to W .

Since G / N is a π -group, it follows that G = N H . Write D = H ∩ N . Then, D ◃ H and D = H ∩ V , and thus D is a Hall π -subgroup of V . Since α is quasi-primitive, it follows that there exists a unique β ∈ Irr ( D ) that lies under α , and thus β is the unique irreducible constituent of γ D .

We claim that γ is also quasi-primitive. Let M ◃ W . Then, M ∩ H ◃ H , and M ∩ H is a Hall π -subgroup of M . Since α is quasi-primitive, then there exists a unique ζ ∈ Irr ( M ∩ H ) that lies under α . It follows that ζ is the unique irreducible constituent of γ M ∩ H . Since M ◃ W and restriction defines an injective map from X π ( M ) into X π ( M ∩ H ) by Lemma 2.7, we have that γ M has the unique constituent ρ , and ρ M ∩ H = ζ , as desired.

We choose ( S , φ ) ∈ S ∗ ( G ) that lies under ( W , γ ) . Then, W ≤ G φ , and thus φ is the unique irreducible constituent of γ S . Write S 1 = S ∩ N , and let φ 1 ∈ Irr ( S 1 ) lie under φ , and thus under γ . It follows by Lemma 2.1 that ( S 1 , φ 1 ) ∈ S ∗ ( N ) . Note that V ◃ W and S ◃ W . Then, S 1 ◃ W because S 1 = S ∩ N = S ∩ W ∩ N = S ∩ V . Since γ is quasi-primitive, we see that φ 1 is the unique irreducible constituent of γ S 1 and thus φ 1 is the unique irreducible constituent of φ S 1 . Furthermore, we have that τ is the unique irreducible constituent of γ V because V ◃ W . This forces that φ 1 is the unique irreducible constituent of τ S 1 . This means that φ 1 is V -invariant and S -invariant. Hence, V ≤ N φ 1 . Since γ is π -special, it follows that τ is π -special, and thus τ D = β because D is a Hall π -subgroup of V .

Now we prove G φ ∩ N = N φ 1 . By Lemma 2.8, we conclude that N φ 1 / ( G φ ∩ N φ 1 ) is a π -group. Since ∣ N φ 1 : N φ 1 ∩ G φ ∣ divides ∣ N : D ∣ , and ∣ N : D ∣ = ∣ G : H ∣ is a π ′ -number, we see that ∣ N φ 1 : N φ 1 ∩ G φ ∣ = 1 . Hence, N φ 1 ≤ G φ , and thus N φ 1 ≤ G φ ∩ N . In order to prove G φ ∩ N ≤ N φ 1 , it suffices to show that G φ ∩ N stabilizes ( S 1 , φ 1 ) . Since S 1 = S ∩ N and N ◃ G , it follows that G φ ∩ N stabilizes S , φ , and S 1 . Recall that φ 1 is the unique irreducible constituent of φ S 1 ; it follows that G φ ∩ N stabilizes φ 1 , as wanted.

Work by induction on ∣ G ∣ . If G φ = G , by Lemma 2.4, we conclude that S = G . Then, V = N and the π -special character τ is an irreducible constituent of χ N , and thus ( V , τ ) is a nucleus for itself. Now we assume G φ < G . By Lemma 2.5, there exists a unique irreducible character ξ ∈ Irr ( G φ ∣ φ ) such that ξ G = χ . By the inductive hypothesis applied in the group G φ with respect to the normal group N φ 1 and the character ξ ∈ Irr ( G φ ) , we know that ( V , τ ) is a nucleus for some irreducible constituent of ξ N φ 1 . Let η = τ N φ 1 . Then, we have that ( V , τ ) is a nucleus for η . Now η lies under ξ and over τ , and thus under χ and over φ 1 . By Lemma 2.5, it follows that η N = ψ is irreducible. We see that ψ lies under χ and ( V , τ ) is a nucleus for ψ , and the result follows.□

Corollary 3.1

Let N ◃ G , where G is π -separable and G / N is a π ′ -group and let N ≤ K ≤ G . Let χ ∈ B π ( G ) and suppose that ξ is an arbitrary irreducible constituent of χ K . Then, we can choose some nucleus ( W , γ ) for χ such that ( W ∩ K , τ W ∩ K ) is a nucleus for ξ .

Proof

Let ψ be an arbitrary irreducible constituent of ξ N . Then, ξ ∈ B π ( K ) and ψ ∈ B π ( N ) by Lemma 2.9. Since N ◃ G , it follows that all of the irreducible constituents of χ N form an orbit under the conjugation action of G on Irr ( N ) . By Theorem A(1), we can choose a nucleus ( W , γ ) for χ such that ( W ∩ N , γ W ∩ N ) is a nucleus for ψ and note that W is the stabilizer of ( W ∩ N , γ W ∩ N ) in G and γ is the canonical extension of γ W ∩ N to W . Hence, γ W ∩ K is the canonical extension of γ W ∩ N to W ∩ K and W ∩ K is the stabilizer of ( W ∩ N , γ W ∩ N ) in K . By Lemma 2.10, we have that ( W ∩ K , γ W ∩ K ) is a nucleus for ξ .□

Proof of Corollary B

Since χ = γ G , it follows that χ ( 1 ) = ∣ G : W ∣ γ ( 1 ) . We know that χ ( 1 ) is a π ′ -number. Hence, γ ( 1 ) = 1 , and ∣ G : W ∣ is a π ′ -number. Then, there exists a Hall π -subgroup H of G with H ≤ W . Writing λ = γ H , we conclude that λ ∈ Irr ( H ) is linear and is quasi-primitive certainly.

If ∣ G : N ∣ = 1 , then ( V , τ ) = ( W , γ ) , and the result is trivial. We can therefore assume that N < G , and we proceed by induction on ∣ G : N ∣ . Let M / N be a chief factor of G . Then, we have that M / N is a π ′ -group or a π -group. Observing that ∣ G : M ∣ < ∣ G : N ∣ and M ◃ G , by the inductive hypothesis, we conclude that ( M ∩ W , γ M ∩ W ) is a nucleus for an irreducible constituent ψ of χ M .

If M / N is a π ′ -group, by Theorem A(1), we know that ( N ∩ M ∩ W , ( γ M ∩ W ) ( N ∩ M ∩ W ) ) = ( N ∩ W , γ N ∩ W ) = ( V , τ ) is a nucleus for an irreducible constituent θ of ψ N . Obviously, θ is an irreducible constituent of χ N . If M / N is a π -group and note that λ M ∩ H is quasi-primitive, where M ∩ H is a Hall π -subgroup of M , by Theorem A(2), we also conclude that ( V , τ ) is a nucleus for an irreducible constituent of χ , and the proof is complete.□