Class-preserving Coleman automorphisms of some classes of finite groups

Lemma 2.1

[17] Let G be a finite group with a nilpotent normal subgroup N. Assume that P is an arbitrary Sylow subgroup of N and G / N acts faithfully on Z ( P ) . Then C G ( P ) ≤ N . In particular, C G ( N ) ≤ N .

Lemma 2.2

[19] Let P be a normal p-subgroup of a finite group G . If C G ( P ) ≤ P , then G has no noninner p-central automorphisms. In particular, Out Col ( G ) = 1 .

Lemma 2.3

Let G be a finite group, H be a subgroup of G, and let σ be an automorphism of G of p -power order, where p is a prime. If there is x ∈ G such that σ ∣ H = conj ( x ) ∣ H , then there exists some γ ∈ Inn ( G ) such that σ γ ∣ H = i d ∣ H and σ γ is still of p-power order.

Proof

Set o ( σ ) = p i , where i ∈ N . Write β ≔ conj ( x ) . Then σ ∣ H = β ∣ H , i.e., σ β − 1 ∣ H = i d ∣ H . Let n ∈ N such that ( σ β − 1 ) n be the p -part of σ β − 1 with ( n , p ) = 1 . Then there exists s , t ∈ Z such that s n + t p i = 1 . Obviously, ( σ β − 1 ) s n is of p -power order and ( σ β − 1 ) s n ∣ H = i d ∣ H . Note that Inn ( G ) ⊴ Aut ( G ) , so there exists some γ ∈ Inn ( G ) such that ( σ β − 1 ) s n = σ s n γ = σ 1 − t p i γ = σ γ . Hence, γ is the desired inner automorphism.□

Lemma 2.4

[1] Let p be a prime, and σ an automorphism of G of p-power order. Assume further that there is N ⊴ G such that σ fixes all elements of N, and that σ induces the identity on G / N . Then σ induces the identity on G / O p ( Z ( N ) ) . If σ fixes in addition a Sylow p-subgroup of G element-wise, then σ is an inner automorphism.

Lemma 2.5

[1] Let N ⊴ G and let p be a prime which does not divide the order of G / N . Then the following hold.

1. If σ ∈ Aut ( G ) is a class-preserving or Coleman automorphism of G of p -power order, then σ induces a class-preserving or a Coleman automorphism of N , respectively;

2. If Out c ( N ) or Out Col ( N ) is a p ′ -group, then so is Out c ( G ) or Out Col ( G ) . If Out c ( N ) ∩ Out Col ( N ) is a p ′ -group, then so is Out c ( G ) ∩ Out Col ( G ) .

Lemma 2.6

Let G be a finite group and let N be a subgroup of G . Let σ be an automorphism of G of p -power order with p a prime. Suppose that σ fixes N and σ ∣ N = conj ( x ) ∣ N for some x ∈ G . Then there exists a p -element y ∈ G such that σ ∣ N = conj ( y ) ∣ N .

Proof

Let o ( σ ) = p i , o ( x ) = p j t , where i , j , t ∈ N and ( p , t ) = 1 . Set k = max { i , j } . Since ( p k , t ) = 1 , it follows that there exists u , v ∈ Z such that u p k + v t = 1 . Write y = x v t . Then it is obvious that y is a p -element. For any z ∈ N , since z = z σ u p k = z x u p k , it follows that z σ = z x = z x u p k + v t = ( z x u p k ) x v t = z x v t = z y , namely, σ ∣ N = conj ( y ) ∣ N .□

Lemma 2.7

[20] Let A be an abelian p ′ -group and let N be a p-group on which A acts, where p is a prime. Then C A ( N ) = C A ( x ) = C A ( N ¯ ) = C A ( x ¯ ) for some x ∈ N , where N ¯ = N / Φ ( N ) with Φ ( N ) being the Frattini subgroup of N .

Theorem 3.1

Let G be a finite group with a nontrivial nilpotent normal subgroup N. Assume that G / N acts faithfully on the center of each Sylow subgroup of N. Then Out c ( G ) ∩ Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

Proof

Let q ∈ π ( G ) and let σ ∈ Aut c ( G ) ∩ Aut Col ( G ) be of q -power order. We have to show that σ ∈ Inn ( G ) . If ∣ π ( N ) ∣ = 1 , then by Lemma 2.1, C G ( N ) ≤ N . Furthermore, by Lemma 2.2, this implies that Out Col ( G ) = 1 . In particular, Out c ( G ) ∩ Out Col ( G ) = 1 . Hereafter, we assume that ∣ π ( N ) ∣ = r > 1 .

Claim 1. σ ∣ N = i d ∣ N .

Let π ( N ) = { p 1 , p 2 , … , p r } and let P i ∈ Syl p i ( N ) , where i = 1 , 2 , … , r . Then N = P 1 × P 2 × ⋯ × P r . Since σ ∈ Aut Col ( G ) , there exists some h i ∈ G such that

For any z i ∈ Z ( P i ) , by equation (1), we have

For any z i ∈ Z ( P i ) and z j ∈ Z ( P j ) with i ≠ j , by equation (2),

On the other hand, since σ ∈ Aut c ( G ) , there exists h ∈ G such that

Combining equation (3) with (4), we obtain

Since N is nilpotent, by equation (5), we have h i − 1 z i h i = h − 1 z i h and h j − 1 z j h j = h − 1 z j h . That is,

Since ⟨ h ¯ i h ¯ i − 1 ⟩ is cyclic, there exists some element in Z ( P i ) , say z i , such that

So by equations (6) and (7), h N = h i N = h j N . As i , j are arbitrary, we have h N = h 1 N = h 2 N = ⋯ = h r N . Set h i = h n i with n i ∈ N , i = 1 , 2 , … , r . For any x i ∈ P i , by equation (1),

As N is nilpotent, we may assume n i ∈ P i in equation (8). Write n ≔ n 1 n 2 ⋯ n r . Then, by equation (8), for any x = x 1 x 2 ⋯ x r ∈ N with x i ∈ P i ,

This shows that σ ∣ N = conj ( h n ) ∣ N . By Lemma 2.3, we may assume that σ ∣ N = i d ∣ N , as claimed.

Claim 2. σ ∣ G / N = i d ∣ G / N .

For any g ∈ G and n ∈ N , by Claim 1, n g = ( n g ) σ = n g σ , implying g σ g − 1 ∈ C G ( N ) . Recall that C G ( N ) ≤ N . So the preceding equality implies that σ ∣ G / N = i d ∣ G / N , as claimed.

Claim 3. σ ∈ Inn ( G ) .

By Lemma 2.4, Claims 1 and 2 yield that σ ∣ G / O q ( Z ( N ) ) = i d ∣ G / O q ( Z ( N ) ) . If q ∉ π ( N ) , then the preceding equation implies that σ = i d . It remains to consider the case q ∈ π ( N ) . Let Q be a Sylow q -subgroup of G fixed by σ . Then Q 1 ≔ Q ∩ N is the Sylow q -subgroup of N . Since σ ∈ Aut Col ( G ) , there exists some q -element g ∈ G such that σ ∣ Q = conj ( g ) ∣ Q . In particular, σ ∣ Z ( Q 1 ) = conj ( g ) ∣ Z ( Q 1 ) . On the other hand, By Claim 1, σ ∣ Z ( Q 1 ) = i d ∣ Z ( Q 1 ) . Consequently, g ∈ C G ( Z ( Q 1 ) ) . It follows that g N ∈ C G / N ( Z ( Q 1 ) ) . From this we deduce that g N = N since the action of G / N on Z ( Q 1 ) is faithful. So g ∈ N and hence g ∈ Q 1 . Note that σ ∣ Q 1 = conj ( g ) ∣ Q 1 = i d ∣ Q 1 . So g ∈ Z ( Q 1 ) ≤ Z ( N ) . It follows that σ conj ( g − 1 ) ∣ N = i d ∣ N , σ conj ( g − 1 ) ∣ G / N = i d ∣ G / N , and σ conj ( g − 1 ) ∣ Q = i d Q . So by Lemma 2.4 σ ∈ Inn ( G ) . We are done.□

Corollary 3.2

Let G = N w r K be the standard wreath product of N by K, where N is a nontrivial nilpotent group and K is an arbitrary group. Then Out c ( G ) ∩ Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

Proof

Let ∣ K ∣ = r . Then G = N w r K = N r ⋊ K , where N r is the direct product of r copies of N . Let p ∈ π ( N ) and let P ∈ Syl p ( N r ) . We will show that K acts faithfully on Z ( P ) . Since N is a nilpotent group, thus N r is also a nilpotent group. Obviously, K acts on Z ( P ) . For any y ∈ Z ( P ) , if y h = y , where h ∈ K . Since the intersection of Z ( P ) with each component of N r is nontrivial, i.e., Z ( P ) is extensive in N r , we deduce that h = 1 , this shows that K acts faithfully on Z ( P ) . Thus, the assertion follows from Theorem 3.1.□

Corollary 3.3

Let G = N w r S m , where N is a finite nilpotent group and S m is the group of all permutations on m letters. Then the normalizer property holds for G.

Corollary 3.4

Let G be the holomorph of an arbitrary nilpotent group N , i.e., G = N ⋊ Aut ( N ) . Then Out c ( G ) ∩ Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

Theorem 3.5

Let G = N A , where N is a nilpotent normal subgroup and A is an abelian subgroup. Then Out c ( G ) ∩ Out Col ( G ) is a p ′ -group for each p ∈ π ( G ) ⧹ π ( N ) ∩ π ( A ) .

Proof

For any p ∈ π ( G ) ⧹ π ( N ) ∩ π ( A ) , let ρ ∈ Aut c ( G ) ∩ Aut Col ( G ) be of p -power order. We will show that ρ is an inner automorphism.

Case 1. p ∈ π ( N ) .

Let N p be the Sylow p -subgroup of N . Then N p char N ⊴ G . It follows that N p is a normal Sylow p -group of G . By Lemma 2.5(2) (replacing N therein with N p ), Out Col ( G ) is a p ′ -group. In particular, Out c ( G ) ∩ Out Col ( G ) is a p ′ -group.

Case 2. p ∈ π ( A ) .

Let A p be the Sylow p -subgroup of A . Then N A p is normal in G since G / N is abelian. Note that G / N A p is a p ′ -group. By Lemma 2.5, we may assume that G = N A p .

Claim 1. ρ conj ( g − 1 ) ∣ N ∈ Aut Col ( N ) for some g ∈ G .

Let π ( N ) = { p 1 , p 2 , … , p r } and let P i ∈ Syl p i ( N ) , where i = 1 , 2 , … , r . Then N = P 1 × P 2 × ⋯ × P r . Since ρ is a Coleman automorphism, by Lemma 2.6, for each P i , there exists a p -element h i ∈ G such that

By Lemma 2.7, C A p ( P i ) = C A p ( x i ) for some x i ∈ P i , where i = 1 , 2 , … , r . Write x = x 1 x 2 ⋯ x r . Then there exists g ∈ G such that x ρ = x g , i.e., ( g − 1 x 1 g ) ⋯ ( g − 1 x r g ) = ( h 1 − 1 x 1 h 1 ) ⋯ ( h r − 1 x r h r ) . From this we obtain g − 1 x i g = h i − 1 x i h i . It follows that

Since G = N ⋊ A p , we may set h i g − 1 = n a , where n ∈ N and a ∈ A p . We will show a ∈ C A p ( P i ) . Let P i ¯ ≔ P i / Φ ( P i ) , where Φ ( P i ) is the Frattini subgroup of P i . Then G acts on P i ¯ . By equation (11), we have

On the other hand,

So equations (12) and (13) imply that [ a , x ¯ i ] = 1 . Again by Lemma 2.7, a ∈ C A p ( x ¯ i ) = C A p ( P i ) . This, together with equation (10), implies that ρ conj ( g − 1 ) ∣ P i = conj ( n ) ∣ P i . This shows that ρ conj ( g − 1 ) ∣ N ∈ Aut Col ( N ) , as claimed.

Claim 2. ρ ∈ Inn ( G ) .

Since N is nilpotent, it follows that ρ conj ( g − 1 ) ∣ N = conj ( n ) ∣ N for some n ∈ N . That is, ρ ∣ N = conj ( n g ) ∣ N . With this in hand, by Lemma 2.3, we may assume that

Note that G / N is abelian. So we have

By Lemma 2.4, equations (14) and (15) yield that ρ ∣ G / O p ( Z ( N ) ) = i d ∣ G / O p ( Z ( N ) ) . Note that N is a p ′ -group. So the preceding equality is precisely ρ = i d . We are done.□

Corollary 3.6

Let G be an extension of a p-group by an abelian group. Then Out c ( G ) ∩ Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

Corollary 3.7

Let G be an extension of a nilpotent group of odd order by an abelian group. Then Out c ( G ) ∩ Out Col ( G ) is of odd order. In particular, the normalizer property holds for G.

Theorem 3.8

Let G = C A , where C is a cyclic normal subgroup and A is an abelian subgroup. Then Out c ( G ) ∩ Out Col ( G ) = 1 . In particular, the normalizer property holds for G.

Proof

Let p ∈ π ( G ) and ρ ∈ Aut c ( G ) ∩ Aut Col ( G ) be of p -power order. We will show that ρ is inner. If either p ∈ π ( C ) ⧹ π ( A ) or p ∈ π ( A ) ⧹ π ( C ) , then by Theorem 3.5 ρ ∈ Inn ( G ) . If p ∈ π ( C ) ∩ π ( A ) , then by Lemma 2.5(2) we may assume that A itself is a p -subgroup. Since C is cyclic and ρ ∈ Aut c ( G ) , it follows that there exists some g ∈ G such that ρ ∣ C = conj ( g ) ∣ C . Without loss of generality, we may set ρ ∣ C = i d ∣ C . Let P C be the Sylow p -subgroup of C . Then P = P C A is a Sylow p -subgroup of G . Without loss of generality, we may assume that ρ fixes P and ρ ∣ P = conj ( x ) ∣ P for some p -element x . Set x = a b with a ∈ P C and b ∈ A . Note that i d ∣ P C = ρ ∣ P C = conj ( a b ) ∣ P C = conj ( b ) ∣ P C . This yields that b ∈ C A ( P C ) . Since A is abelian, it follows that b ∈ Z ( P ) and thus ρ ∣ P = conj ( x ) ∣ P = conj ( a b ) ∣ P = conj ( a ) ∣ P . From this we deduce that ρ = conj ( a ) . We are done.□