Theorem.

Let G be a soluble homomorphic image of a periodic linear group, and let N be a locally nilpotent normal subgroup of G such that no G-chief factor of N is G-isomorphic to any G-chief factor of G / N . If A and B are subgroups of G with G = A ⁢ N = B ⁢ N and A ∩ N = B ∩ N , then A g = B for some g ∈ G .

Proof.

First, we show that it is possible to assume G = 〈 A , B 〉 . Let X = 〈 A , B 〉 . Since G / N = X ⁢ N / N ≃ X X / X ∩ N , the X-chief factors of G / N coincide with the X-chief factors of X / X ∩ N . Now, let U / V be an X-chief factor of X ∩ N . Since N is locally nilpotent, it is also hypercentral,^[1] so there is a smallest ordinal number α such that Z α ⁢ ( N ) ⁢ V ≥ U . Suppose first α is limit. Then Z α ⁢ ( N ) = ⋃ β < α Z β ⁢ ( N ) . Let γ < α be such that Z γ ⁢ ( N ) ⁢ V ∩ ( U ∖ V ) ≠ ∅ . Then ( Z γ ⁢ ( N ) ⁢ V ∩ U ) / V is a non-trivial X-invariant subgroup of U / V , so ( Z γ ⁢ ( N ) ⁢ V ∩ U ) / V = U / V and Z γ ⁢ ( N ) ⁢ V ≥ U . This contradiction shows that α is successor. Therefore, T ⁢ V ∩ U = V and T ⁢ U / T ⁢ V ≃ X U / V , where T = Z α - 1 ⁢ ( N ) . Put Z = Z α ⁢ ( N ) . Then

It follows that U / V is X-isomorphic to an irreducible X-section U 1 / V 1 of N such that U 1 ≤ Z α ⁢ ( N ) and V 1 ≥ Z α - 1 ⁢ ( N ) . Moreover, U 1 / V 1 is a G-chief factor of N, so it is not X-isomorphic to any X-chief factor of G / N = X ⁢ N / N ≃ X X / X ∩ N . Clearly, A ∩ ( X ∩ N ) = B ∩ ( X ∩ N ) , so the hypotheses hold for X and we may assume G = X .

Now we show that it is even possible to require that A ∩ N = B ∩ N = { 1 } . Assume in fact the result holds in such a case, and put M = A ∩ N = B ∩ N . Since G = 〈 A , B 〉 , we have that M is normal in G, and consequently

satisfies the hypotheses of the statement. Since G / M ∩ A / M = G / M ∩ B / M = { 1 } , there is g ∈ G such that ( A / M ) g = ( B / M ) . Therefore A g = B and we are done.

In what follows we hence assume G = 〈 A , B 〉 and A ∩ N = B ∩ N = { 1 } . Further reductions are allowed by the following claim: if the statement holds for G / L , where L is a G-invariant subgroup of N, then we may assume N is a subgroup of L. In fact, we can find g ∈ G with ( A ⁢ L ) g = ( B ⁢ L ) . Put X = A g ⁢ L = ( A ⁢ L ) g = B ⁢ L . The argument employed in the first paragraph of the proof yields that the X-chief factors of L = X ∩ N are not X-isomorphic to any of the X-chief factors of X / L , so X satisfies the hypotheses of the statement when A is replaced by A g . Thus, may replace G by X, A by A g , and N by L; however, if we still want G = 〈 A , B 〉 , we need to make a further reduction and so we need actually to replace N by a subgroup of L. This proves the claim. It is also clear we may assume ( A ∩ B ) G = { 1 } .

We first prove the statement in case N is finite: here we use induction on the order of N. By the above paragraph we can hence assume N is a minimal normal subgroup of G, so in particular it is an elementary abelian q-group for some prime q. Let C = C G ⁢ ( N ) = C A ⁢ ( N ) × N = C B ⁢ ( N ) × N , so | G : C | is finite. Clearly, C A ⁢ ( N ) is normalized by A and N, so is normal in G; similarly, C B ⁢ ( N ) is normal in G. Since any G-chief factors of N is not G-isomorphic to any of those of G / N , it follows that C A ⁢ ( N ) ⁢ C B ⁢ ( N ) / C B ⁢ ( N ) must be trivial, so C A ⁢ ( N ) ≤ C B ⁢ ( N ) . Similarly, C B ⁢ ( N ) ≤ C A ⁢ ( N ) and hence C A ⁢ ( N ) = C B ⁢ ( N ) = { 1 } . Therefore G = 〈 A , B 〉 is finite and the statement follows from the main theorem of [8].

Write G = ℋ / ℳ , where ℋ is a soluble periodic linear group of characteristic p (see [10, Theorem 9.20]) and ℳ ⁢ ⊴ ⁢ ℋ . Now, we divide the proof in two sub-cases according to the fact that p belongs to π ⁢ ( N ) or not. Suppose first p ∉ π ⁢ ( N ) . It follows from [10, Theorem 9.24] that N is the image in G of a p ′ -subgroup 𝒬 of ℋ . Now, 𝒬 is a d-group, so it contains an abelian normal subgroup of finite index (see for instance [10, Lemma 9.17]), and hence also N is abelian-by-finite; in particular, N contains a characteristic abelian subgroup U of finite index. Since the statement is true for G / U , we can assume N is abelian. As in the case in which N is finite, we have C G ⁢ ( N ) = N , so O p ⁢ ( G ) = { 1 } . Since ℋ is triangularizable-by-finite (by the Lie–Kolchin theorem), we see that G is abelian-by-finite; let V be an abelian normal subgroup of finite index. The reductions we discussed above make it possible to assume that N ≤ N ∩ V , so N = C G ⁢ ( N ) = V , and hence that A , B are finite. This means that G is finite and again the statement follows from the main theorem of [8].

Suppose p ∈ π ⁢ ( N ) . Since N is locally nilpotent, it contains a unique Sylow p-subgroup, and, by the previous paragraph, we may reduce to the case in which N is a p-group. Let P be a Sylow p-subgroup of G; in particular, P ≥ N . Now, P is a homomorphic image of a Sylow p-subgroup of ℋ , so it is nilpotent (see [10, 2.6]) and P 1 = O p ⁢ ( G ) has finite index in P; note that the latter assertion follows from the fact that G is (p-group)-by-( p ′ -group)-by-finite by Lie–Kolchin theorem. Using induction on the smallest length of a series of P 1 -invariant subgroups of N whose factors are centralized by P 1 , we may assume N is abelian and is centralized by a G-invariant subgroup of finite index of P 1 . Since N is abelian, we get C A ⁢ ( N ) = C B ⁢ ( N ) = { 1 } , which means that N has finite index in P, and consequently that the Sylow p-subgroups of G / N are finite. Repeating (if necessary) this reduction we finally obtain N = P 1 , so A ≃ B ≃ G / N has no non-trivial normal p-subgroups. Let D A and D B be the finite residuals of A and B, respectively. Since these coincide with the finite residual of maximal p ′ -subgroups of A and B, respectively, and, in turns, these are maximal p ′ -subgroups of G, it follows that D A and D B are conjugate; in particular, there is g ∈ G such that D A g = D B . Replacing A by A g , we get that D A = D B ≤ ( A ∩ B ) G = { 1 } , so A and B are finite and such is G. A final application of the main result of [8] completes the proof. ∎

Remark.

The first paragraph of the proof actually works for an arbitrary group G having a hypercentral normal subgroup N. It basically says that if X is any subgroup of G with X ⁢ N = G , then the X-chief factors of X ∩ N can be “naturally seen” as G-chief factors of N.