TI subgroups and depth 3 subgroups in simple Suzuki groups

In this paper, we determine the TI subgroups of the simple Suzuki groups Sz.q/. More generally, we determine those nontrivial subgroups that are disjoint from some of their conjugates. It turns out that the latter are exactly those subgroups that have ordinary depth 3. The Sylow 2-subgroups of simple Suzuki groups belong to the class of so-called Suzuki 2-groups, which have been studied extensively by Higman. These results were extended later by Goldschmidt, Shaw, Shult, Gross, Wilkens and Bryukhanova. As a corollary of our investigations, we get some interesting results for the Sylow 2-subgroups of Suzuki groups, as well. We relate this to an open problem on Suzuki 2-groups, and we ask a question concerning that. We also give some characterization of Suzuki groups. 1 Motivation and results In an earlier paper [14], we determined the ordinary and combinatorial depth of several subgroups of the simple Suzuki groups Sz.q/. For the definitions of these notions, see Definition 2.5 and Definition 2.8 in Section 2. There are several ways to define ordinary depth; a good summary of these can be found in [18]; see also [6, Definition 3.5] and the introduction of [8]. The notion of combinatorial depth was introduced in [2, Definition 3.2]. We use [2, Theorem 3.9] as an equivalent definition. A subgroup L G is called TI if, for every x 2 G, from L \ L ¤ 11o, it follows that x 2 NG.L/. In an arbitrary group, non-normal TI subgroups are always of combinatorial depth three, hence also of ordinary depth three. Moreover, nontrivial subgroups having a disjoint conjugate are always of ordinary depth three in every group. However, in general, the converse is not true, e.g. L D A5 has in G D A7 ordinary depth 3, but there is no element x 2 G such that L \ L D 11o. (A similar example is A6 in A10.) The first author was supported by the Stipendium Hungaricum PhD fellowship at the Budapest University of Technology and Economics. The second and the third author were supported by the NKFI Grants No. 115288 and 115799. Open Access. © 2021 Janabi, Héthelyi and Horváth, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 International License. 602 H. A. Janabi, L. Héthelyi and E. Horváth It is an open problem how to characterize subgroups of ordinary depth 3 in a group-theoretical way in an arbitrary group. We will show in Theorem 1.3 that, in the case of simple Suzuki groups, subgroups of ordinary depth 3 are exactly those nontrivial subgroups that have a disjoint conjugate. In Theorem 1.1, we also characterize those nontrivial TI subgroups of Sz.q/ which are non-cyclic elementary abelian 2-subgroups. These are exactly those subgroups that are conjugate to the center of a Sylow 2-subgroup of a smaller Suzuki subgroup Sz.s/ Sz.q/. This property can help the recognition of those subgroups of Sz.q/ which are isomorphic to a Suzuki subgroup Sz.s/ Sz.q/. Recognition of Suzuki groups in GL.4; q/ is considered in some recent papers; see [1,4]. In another paper [23], Suzuki groups were used to construct some block designs. So results about intersections in Suzuki groups might also be helpful in combinatorial investigations. In this paper, we will use the following notation. Let G be the simple Suzuki group Sz.q/, where q D 2. Let F be a fixed Sylow 2-subgroup of G. Then NG.F / D FH , where jH j D q 1. Let A1 and A2 be Hall subgroups of G of orders q C 2r C 1 and q 2r C 1, respectively, where r D 2; see Theorem 2.1 and also [17, Theorem 3.10, Chapter XI] and [22, Theorem 4.12]. We will denote by K2n an elementary abelian subgroup of order 2. The center Z.F / of the Sylow 2-subgroup F of the Suzuki group Sz.q/ will be of order q D 2, and we will denote 2mC 1 by f . We also suppose that m > 0. The main results of this paper are the following theorems. Theorem 1.1. If G D Sz.q/ is a simple Suzuki group, then G has the following TI subgroups. (i) Cyclic subgroups of prime order and the trivial subgroup. (ii) Subgroups F;H;A1; A2, their characteristic subgroups and the conjugates of these. (iii) An elementary abelian subgroup K2n of order 2 > 2 is a TI-subgroup if and only if it is the center of a Sylow 2-subgroup of a simple Suzuki subgroup G1 G, or a conjugate to it. This holds if and only if n > 1 and n j f . (Remember jZ.F /j D 2 .) These subgroups are exactly those non-cyclic elementary abelian 2-subgroups of G that have combinatorial depth 3. All other nontrivial subgroups are not TI. Remark 1.2. This classification of TI subgroups shows that [1, Theorem 2.1 (7)] is wrong; the centralizer of an element of order 4 cannot be TI. TI subgroups and depth 3 subgroups in simple Suzuki groups 603 Theorem 1.3. Subgroups of ordinary depth 3 of a simple Suzuki groupG D Sz.q/ are the following. (i) Every nontrivial subgroup contained in a maximal subgroup different from a conjugate of NG.F /. (ii) F and all its nontrivial subgroups, and the conjugates of these. (iii) All nontrivial subgroups U D F1K of NG.F /, where F1 < F andK H , and the conjugates of these subgroups. Moreover, a nontrivial subgroupL G D Sz.q/ is of ordinary depth 3 if and only if there exists an element x 2 G with L \ L D 11o. As a corollary, we get a characterization of Suzuki groups. Corollary 1.4. LetG be a simple Zassenhaus group acting on q C 1 points, where q is a 2-power. Then the following are equivalent.


Motivation and results
In an earlier paper [14], we determined the ordinary and combinatorial depth of several subgroups of the simple Suzuki groups Sz.q/. For the definitions of these notions, see Definition 2.5 and Definition 2.8 in Section 2. There are several ways to define ordinary depth; a good summary of these can be found in [18]; see also [6,Definition 3.5] and the introduction of [8]. The notion of combinatorial depth was introduced in [2, Definition 3.2]. We use [2,Theorem 3.9] as an equivalent definition. A subgroup L Ä G is called TI if, for every x 2 G, from L x \ L ¤ ¹1º, it follows that x 2 N G .L/. In an arbitrary group, non-normal TI subgroups are always of combinatorial depth three, hence also of ordinary depth three. Moreover, nontrivial subgroups having a disjoint conjugate are always of ordinary depth three in every group. However, in general, the converse is not true, e.g. L D A 5 has in G D A 7 ordinary depth 3, but there is no element x 2 G such that L \ L x D ¹1º. (A similar example is A 6 in A 10 .) It is an open problem how to characterize subgroups of ordinary depth 3 in a group-theoretical way in an arbitrary group.
We will show in Theorem 1.3 that, in the case of simple Suzuki groups, subgroups of ordinary depth 3 are exactly those nontrivial subgroups that have a disjoint conjugate.
In Theorem 1.1, we also characterize those nontrivial TI subgroups of Sz.q/ which are non-cyclic elementary abelian 2-subgroups. These are exactly those subgroups that are conjugate to the center of a Sylow 2-subgroup of a smaller Suzuki subgroup Sz.s/ Ä Sz.q/. This property can help the recognition of those subgroups of Sz.q/ which are isomorphic to a Suzuki subgroup Sz.s/ Ä Sz.q/. Recognition of Suzuki groups in GL.4; q/ is considered in some recent papers; see [1,4]. In another paper [23], Suzuki groups were used to construct some block designs. So results about intersections in Suzuki groups might also be helpful in combinatorial investigations.
In this paper, we will use the following notation. Let G be the simple Suzuki group Sz.q/, where q D 2 2mC1 . Let F be a fixed Sylow 2-subgroup of G. Then N G .F / D FH , where jH j D q 1. Let A 1 and A 2 be Hall subgroups of G of orders q C 2r C 1 and q 2r C 1, respectively, where r D 2 m ; see Theorem 2.1 and also [17,Theorem 3.10, Chapter XI] and [22,Theorem 4.12].
We will denote by K 2 n an elementary abelian subgroup of order 2 n . The center Z.F / of the Sylow 2-subgroup F of the Suzuki group Sz.q/ will be of order q D 2 2mC1 , and we will denote 2m C 1 by f . We also suppose that m > 0. The main results of this paper are the following theorems. Theorem 1.1. If G D Sz.q/ is a simple Suzuki group, then G has the following TI subgroups.
(i) Cyclic subgroups of prime order and the trivial subgroup.
(ii) Subgroups F; H; A 1 ; A 2 , their characteristic subgroups and the conjugates of these.
(iii) An elementary abelian subgroup K 2 n of order 2 n > 2 is a TI-subgroup if and only if it is the center of a Sylow 2-subgroup of a simple Suzuki subgroup (i) Every nontrivial subgroup contained in a maximal subgroup different from a conjugate of N G .F /.
(ii) F and all its nontrivial subgroups, and the conjugates of these.
(iii) All nontrivial subgroups U D F 1 K of N G .F /, where F 1 < F and K Ä H , and the conjugates of these subgroups.
Moreover, a nontrivial subgroup L Ä G D Sz.q/ is of ordinary depth 3 if and only if there exists an element x 2 G with L \ L x D ¹1º.
As a corollary, we get a characterization of Suzuki groups.
Corollary 1.4. Let G be a simple Zassenhaus group acting on q C 1 points, where q is a 2-power. Then the following are equivalent.
(ii) If F 2 Syl 2 .G/, then every subgroup not containing a conjugate of the subgroup F has a disjoint conjugate.
Proof. By Theorem 1.3 and by the description of subgroups of Sz.q/ in Theorem 2.1, we have that (i) implies (ii). For the other direction, observe that the only simple Zassenhaus groups with q even are Sz.q/ and PSL.2; q/. In PSL.2; q/, there are dihedral subgroups of order 2.q C 1/, that are not disjoint from their conjugates; see [7].
We also have a wider class of groups where ordinary depth 3 is equivalent to having a disjoint conjugate. Corollary 1.5. Let G be a simple Zassenhaus group acting on q C 1 points, where q is an odd power of 2. Then the following are equivalent for a nontrivial subgroup L Ä G.
(i) The subgroup L Ä G has ordinary depth 3.
(ii) The subgroup L Ä G has a disjoint conjugate.
Proof. Since these Zassenhaus groups are Sz.q/ and PSL.2; q/, we have to prove the equivalence for these groups. In Theorem 1.3, we proved the equivalence for Sz.q/. In [7], this is proved for PSL.2; 2 2nC1 /.
We also get some information about Sylow 2-subgroups of Suzuki groups. Corollary 1.6. Every Sylow 2-subgroup of G D Sz.q/ is the union of conjugates of the Sylow 2-subgroups of a smallest simple Suzuki subgroup G 1 contained in G. If F 2 Syl 2 .G/ and F 1 2 Syl 2 .G 1 / are contained in F , then F D S x2N G .F / F 1 x . Every element of order 4 of F is in exactly one conjugate of F 1 , and any two of the conjugates of F 1 in this union either have trivial intersection or their intersection is their center, which is a conjugate of 1 .F 1 /.
We give a proof of this in Section 3.
, and in this union, any two conjugates of F 2 either have trivial intersection, or the intersection is a conjugate of 1 .F 2 /. This is because there are two conjugacy classes of order 4 elements in G. Let they be K G .a/ and K G .a 1 /, where a 2 F . Consider S x2N G .F / hai x . This will contain all elements of order 4 in F , i.e. all elements of F n Z.F /. However, each involution in F is a square of an element of order 4, since this is true for at least 2 involutions and the other involutions are conjugate by elements of H Ä N G .F /. Hence we are done.
About the structure of the paper: In Section 2, we give some properties of the Suzuki groups that are needed in the proofs; we also give the definitions of combinatorial and ordinary depth of a subgroup. In Section 3, we give the proof of Theorem 1.1 and of Corollary 1.6. In Section 4, we prove Theorem 1.3. At the end of the paper, we formulate two questions.

Preliminaries
Suzuki groups Sz.q/ are twisted groups of Lie type 2 B 2 .q/, where q´2 2mC1 . If m > 0, then they are simple. Suzuki groups are also doubly transitive permutation groups on q 2 C 1 points; they belong to the class of Zassenhaus groups. Suzuki groups also can be defined as subgroups of GL.4; q/. The order of Sz.q/ is .q 2 C 1/.q 1/q 2 . The order of Sz.q/ is not divisible by 3; however, it is always divisible by 5. For further information, see [14].
We will use the following results on the Suzuki groups; see [17, Theorem 3.10, Chapter XI] and [22,Theorem 4.12].  where F 2 Syl 2 .G/ and H is cyclic of order q 1.
(3) The cyclic Hall subgroups A 1 ; A 2 of orders q C 2r C 1, q 2r C 1, respectively, where r D 2 m and jA 1 jjA 2 j D q 2 C 1.
(5) Subgroups of the form Sz.s/, where s is an odd power of 2, s 8, and q D s n for some positive integer n. Moreover, for every odd 2-power s, where s n D q for some positive integer n, there exists a subgroup isomorphic to Sz.s/.
(6) Subgroups (and the conjugates of the subgroups) of the above groups. (a) Let i 2 ¹1; 2º, and let (b) Let F; H; A 1 ; A 2 be as in Theorem 2.1. Then the conjugates of F; H; A 1 ; A 2 form a partition of G. In particular, F; H; A 1 ; A 2 , their conjugates and the conjugates of their characteristic subgroups are TI sets in G.
The Sylow 2-subgroup F of G is a Suzuki 2-group. This means that it is a nonabelian 2-group, having more than one involution, and having a solvable group of automorphisms which permutes the set of involutions of F transitively. See [16, p. 299] for details.
The group F is a class 2 group of order q 2 and exponent 4. Moreover, its center Z.F / D F 0 Dˆ.F / is of order q. The involutions in F together with the identity element constitute Z.F /, and F does not contain any quaternion subgroups. A nontrivial element of F is real in G if and only if it is an involution. (An element of a group G is called a real element in G if it is conjugate in G to its inverse; see [10, p. 303]. ) The subgroup H acts sharply 1-transitively on the involutions of F , and on the nontrivial elements of F=Z.F /. The centralizer in G of every nontrivial element of F is a subgroup of F . For m > 0, Suzuki groups are simple and their orders are congruent to 2 mod 3.
Zassenhaus groups are doubly transitive permutation groups without any regular normal subgroup, where any non-identity element has at most two fixed points. Zassenhaus groups are always of degree q C 1, where q is a prime power. If q is even, then there are two series of Zassenhaus groups PSL.2; 2 n / for n > 1 and Sz.2 2mC1 / for m > 0. If q D p f is odd and f odd, then Zassenhaus groups are PGL.2; p f / and PSL The notion of ordinary depth was originally defined for von-Neumann algebras; see [9]. Later, it was also defined for Hopf algebras; see [21]. For some recent results in this direction, see [12,13,19]. In [20] and later in [6], the depth of semisimple algebra inclusions was studied. The ordinary depth of a group inclusion L Ä G (denoted by d.L; G/) is defined as the minimal depth of the group algebra inclusion CL Â CG, studied in [2]. We will use an equivalent definition which is established in [6]. For the case of depth one and two, see [3,20].
Let G be a finite group, L Ä G a subgroup. We introduce the ordinary depth as follows.
Two irreducible characters˛;ˇ2 Irr.L/ are related (denoted by˛ Gˇ) if they are constituents of the restriction j L for some 2 Irr.G/. The distance d G .˛;ˇ/ D m is the smallest integer m such that there is a chain of irreducible characters of L such that˛D 0 G 1 G G m Dˇ. If there is no such chain, then d G .˛;ˇ/ D 1, and if˛Dˇ, then the distance is zero. If X is the set of irreducible constituents of j L , then m. /´max˛2 Irr.L/ min 2X d G .˛; /.   Remark 2.7. It follows from the first part of Theorem 2.6 that if a non-normal subgroup L Ä G has a disjoint conjugate, i.e. L \ L x D ¹1º for some x 2 G, then d.L; G/ Ä 4. Since in this case Core G .L/ D ¹1º, it follows from the second part of Theorem 2.6 that d.L; G/ Ä 3. But since L is not normal in G, d.L; G/ D 3. However, the converse is not true, e.g. d.
The combinatorial depth can be defined as follows; see [2].
(ii) Let i > 1. Then d c .L; G/ Ä 2i 1 if and only if, for every x 1 ; : : : ; x i 2 G, there exist some y 1 ; : : :  are cyclic and TI, each subgroup of them is characteristic, hence TI. We will see below that not every subgroup of F is TI. (4) If we take a nontrivial subgroup Q A of A i , and a nontrivial subgroup C 1 Ä C of a cyclic complement of order 4 in the Frobenius group N G .A i /, then C 1 Q A is not TI; in particular, Sz.2/ is not TI. Let A is not TI.
(5) Let G 1´S z.s/ be a simple smaller Suzuki subgroup in G. The subgroups U of G 1 , whose order is divisible both by 2 and by some odd integer greater than 1, are not TI. Let U 2 2 Syl 2 .U /, and let U 2 Ä P 2 2 Syl 2 .G/. Then Z.P 2 / Ä N G .U 2 /; however, since Z.P 2 / -G 1 , we have that Z.P 2 / -U . If N P 2 .U 2 / Ä N G .U /, then UZ.P 2 / is a subgroup of G. It cannot be a subgroup of any maximal subgroup of type N G .A i /, N G .H / and Sz.s 1 / since the orders of these subgroups are not divisible by q. If U Ä N G .S/ for some S 2 Syl 2 .G/, then U is conjugate to a subgroup discussed in point (2); hence it is not TI. The subgroup UZ.P 2 / cannot be G since U is normal in it. So we can suppose that N P 2 .U 2 / -N G .U /. Let x 2 N P 2 .U 2 / n N G .U /. Then U 2 Ä U x \ U < U ; hence U is not TI.
(6) Let C´hci be a cyclic subgroup of F of order 4. This is not TI.
By the proof of [17, Lemma 5.9, Chapter XI], we have that C F .C / D Z.F /C is of order 2q. Since c is not real in F , we have that N F .C / D C F .C /. Thus there is an element x 2 F n N F .C /. Let u´c 2 . Then C G .u/ D F , and hence C x \ C contains u. Hence C is not TI.
(7) Let us denote an elementary abelian subgroup of order 2 n in F by K 2 n . The main results in this step are (e) and (f). This is exactly the content of (iii) in Theorem 1.1, which we will prove with the help of statements (a)-(d).
(a) Let K 2 n D Z.S 1 /, where S 1 2 Syl 2 .G 1 / and G 1 is a simple Suzuki subgroup of G. Then K 2 n is TI in G.
We may assume that S 1 Ä F . Suppose that Z.S 1 / x \ Z.S 1 / ¤ ¹1º for some x 2 G. Then there exist involutions a; b 2 Z.S 1 / with a x D b. Since F is TI, thus x 2 N G .F / D FH . Since F acts trivially on Z.S 1 /, we may suppose that x 2 H . Moreover, H acts sharply 1-transitively on the involutions of F . We also have that (b) If a non-cyclic elementary abelian subgroup of G is TI, and its order is equal to jZ.S 1 /j for a Sylow subgroup S 1 2 Syl 2 .G 1 / for some simple Suzuki subgroup G 1 of G, then this elementary abelian subgroup is the center of a Sylow 2-subgroup of a subgroup conjugate to a simple Suzuki subgroup of G.
Let us suppose that a non-cyclic elementary abelian subgroup K 2 n Ä F of order 2 n in G is a TI set. We suppose that 2 n D jZ.S 1 /j for S 1 2 Syl 2 .G 1 /, where G 1 is a simple Suzuki subgroup of G. The involutions of K 2 n and their H -conjugates form blocks in Z.F / since K 2 n is TI in G. Since H acts sharply 1-transitively on the involutions of Z.F /, hence Z.F / is the disjoint union of different H -conjugates of K 2 n . We claim that the normalizers in H of the elementary abelian TI subgroups Q K 2 n of order 2 n of G contained in Z.F / are the same. To see this, note that no element of H fixes any element in F ; hence the elements of the cyclic subgroup N G . Q K 2 n / \ H move the nonunit elements of Q K 2 n sharply 1transitively; hence N G . Q K 2 n / \ H has order 2 n 1. Since H is cyclic, it has only 1 subgroup of this order; thus each elementary abelian TI-subgroup of order 2 n in Z.F / has the same normalizer in H . We know that Q K 2 n Â S h2H K h 2 n . We claim that Q K 2 n is one of the conjugates of K 2 n . Suppose that a 2 Q K 2 n \ K h 1 2 n and b 2 Q K 2 n \ K h 2 2 n are two different involutions and h 1 ; h 2 2 H . Then there exists a unique h 2 H \ N G . Q However, then K h 1 2 n D .K h 1 2 n / h D K h 2 2 n . Hence Q K 2 n D K h 1 2 n . Thus all elementary abelian TI subgroups of order 2 n in F are conjugate to Z.S 1 / for a Sylow subgroup S 1 2 Syl 2 .G 1 / for some Suzuki subgroup G 1 Ä G.
(c) Let K 2 r Ä F be an elementary abelian subgroup of order 2 r in G. Then it is TI if and only if N H .K 2 r / D H 1 is of order 2 r 1.
Let K 2 r Ä F be an elementary abelian subgroup of order 2 r with the property that N H .K 2 r / D H 1 and jH 1 j D 2 r 1. Then H 1 permutes the elements of K 2 r n ¹1º sharply 1-transitively, and every h 2 H n H 1 transports each involution of K 2 r outside this group. Thus K 2 r is TI.
Conversely, let K 2 r Ä F be TI. Since by [22,Theorem 4.1 (e), (f)] H acts sharply 1-transitively on the involutions of F , if a 2 K 2 r n ¹1º, then for every Hence it follows that jN H .K 2 r /j D 2 r 1.
(d) Let r > 1, and let N H .K 2 r / D H 1 be of order 2 r 1. Then 2 r 1 divides q 1 D 2 f 1, which happens if and only if jK 2 r j D jZ.S 1 /j for the Sylow subgroup S 1 2 Syl 2 .G 1 / for a simple Suzuki subgroup G 1 Ä G. This happens if and only if r j f and r > 1.
Suppose that r > 1 and jH 1 j D 2 r 1. This divides jH j D q 1 D 2 2mC1 1. Then r j 2m C 1, and hence .2 r / k D 2 2mC1 for some positive integer k. Thus if G 1 D Sz.2 r /, then S 1 2 Syl 2 .G 1 / has center Z.S 1 / of order 2 r . Conversely, if 2 r is the size of Z.S 1 / for some S 1 2 Syl 2 .G 1 / for a simple Suzuki subgroup G 1 Ä G, then r > 1 and .2 r / k D 2 2mC1 for some positive integer k; hence 2 r 1 j 2 2mC1 1.
(e) A non-cyclic elementary abelian 2-subgroup of G is TI if and only if it is the center of a Sylow subgroup S 1 2 Syl 2 .G 1 / for some simple Suzuki subgroup G 1 Ä G or conjugate to it; in particular, Klein four subgroups of G are not TI.
One direction follows from (a); the other direction follows from (b) using (c) and (d).

(f) A non-cyclic elementary abelian 2-subgroup of G is TI if and only if it is of combinatorial depth 3.
If K 2 n is TI, then it is of combinatorial depth 3 by Remark 2.9. Let n > 1. If K 2 n Ä F is not TI, then we have that there exists an element x 2 … N G .K 2 n / such that K x 2 2 n \ K 2 n ¤ ¹1º. Let x 1 2 F . Then x 1 centralizes K 2 n and we cannot find an element y with K 2 n \ K x 1 2 n \ K x 2 2 n D K 2 n \ K y 2 n such that y also centralizes the intersection since then y 2 F and it also centralizes K 2 n . Hence the combinatorial depth of K 2 n in G is bigger than 3.

(8) If L < F is not elementary abelian, then L is not TI.
Suppose by contradiction that L is TI and not elementary abelian. Let I be the subgroup generated by the involutions of L. If jI j D 2, then since F does not contain quaternion subgroups, L ' C 4 ; hence it is not TI by step (6). Let jI j > 2.
Suppose that I \ I x ¤ ¹1º; then L \ L x ¤ ¹1º and x 2 N G .L/ Ä N G .I /. Thus I is also TI; hence it is the center of a Sylow 2-subgroup of a simple Suzuki subgroup of G or conjugate to it. If x 2 N G .I / n N G .L/, then If S 1 2 Syl 2 .G 1 / for a Suzuki subgroup G 1 Ä G and jZ.S 1 /j D 2 n , then the number of H -conjugates of Z.S 1 / is jH W N H .Z.S 1 //j D .2 f 1/ .2 n 1/ , which is strictly smaller than the above number if n < f . Thus there are some subgroups, Z.S 1 / and its H -conjugates, which are TI, and the remaining subgroups of Z.F / of order 2 n are not TI.
Proof of Corollary 1.6. By part (7) (b) in the proof of Theorem 1.1, we have that S h2H Z.F 1 / h D Z.F /, and the subgroups Z.F 1 / h for h 2 H intersect trivially. Let a 2 F n Z.F / such that a 2 F 1 . Since H acts sharply 1-transitively on the nontrivial elements of F=Z.F /, each coset of Z.F / contains an element of a conjugate of F 1 . By the proof of [17, Lemma 5.9, Chapter XI], the element a has q=2 D jF W C G .a/j D jF W Z.F /haij conjugates and a 1 also. The F -conjugates of these elements all lie in Z.F /a. Thus these give all the elements of Z.F /a since this coset has q elements. Hence each element of F is in a conjugate of F 1 . Since F is a TI set, the conjugating elements are in N G .F /.
For the second part, see [14,Proposition 4.13] or the following shorter argument. Suppose that x 2 G n N G .F 1 / and a 2 F 1 \ F x 1 is an element of order 4. Then there exists an element b 2 F 1 such that b x D a. By [17, Lemma 11.7, Chapter XI], we have that, in G, there are two conjugacy classes of elements of order 4. We know that K G .a/ ¤ K G .a 1 /; otherwise, an element of H would centralize a 2 ; similarly, K G .b/ ¤ K G .b 1 /. They are also in different conjugacy classes in G 1 . Thus a and b are conjugate also in G 1 . Hence they are conjugate in N G 1 .F 1 /. Thus there exists an element y 2 N G .F 1 / such that b y D a. Hence we have xy 1 2 C G .b/. Since x … N G .F 1 / and y 2 N G .F 1 /, then xy 1 … N G .F 1 /. However, xy 1 2 C G .b/ D Z.F /hbi Ä N G .F 1 /. This contradiction shows that there cannot be an order 4 element in two conjugates of F 1 . If F 1 \ F x 1 ¤ ¹1º, then they have common elements of order 2; hence Z.F 1 / \ Z.F 1 / x ¤ ¹1º. We have seen that Z.F 1 / is a TI subgroup of G; hence Z.F 1 / D Z.F 1 / x . Since F 1 and F x 1 do not have common elements of order 4 and all their elements of order 2 are central, hence

Proof of Theorem 1.3
Proof of Theorem 1.3. (i) As by [14], except for N G .F /, all maximal subgroups have a disjoint conjugate, this holds also for their subgroups; thus any such nontrivial subgroup has ordinary depth 3.
(ii) This holds since F is TI.
(iii) Now we have to consider the subgroups of N G .F / D FH . We want to prove that if the subgroup does not contain F , then it has a disjoint conjugate.
Let U be such a subgroup. Let us suppose now that U D F 1 K, where F 1 < F and K Ä H . Then by [24,Theorem 17.3], we have that U is a Frobenius group with kernel F 1 D F \ U and complement K. Moreover, since K is a characteristic subgroup in the cyclic group H , and H is a TI set in G, so K is also a TI set Since F is TI, we have that x 2 N G .F /, which is not the case. Then jU x \ U j is a divisor of jKj. Since U is solvable, by Hall's theorem, U x \ U is contained in a complement of F 1 , and it can be conjugated in U , moreover, in F 1 , to a subgroup of any other complement of F 1 in U . We may suppose that U x \ U Ä K Ä H . Otherwise, let s 2 F 1 be an element such that .U x \ U / s Ä K. Then K .U x \ U / s D U xs \ U and xs … N G .F /. Thus we can, if necessary, exchange x to xs to get U x \ U Ä K. Let K 1´U x \ U . So we may suppose that K 1 Ä K. The Frobenius complements of U x are of the form K f 1 x for some f 1 The subgroup FK \ .FK/ f 1 x contains K; however, it cannot contain any elements of F ; otherwise, as F is TI, this would imply that f 1 x 2 N G .F /, which is not the case.
Thus a nontrivial element of F x normalizes a Frobenius complement of F x K, which cannot happen. Thus K \ .F x 1 K l / D ¹1º, and so Let us suppose now that U D FK, where K < H . We want to prove that d.U; G/ D 5. We know, see the proof of [14,Proposition 4.3], that there exist elements x 1 ; x 2 2 G with FH \ .FH / x 1 \ .FH / x 2 D ¹1º. Hence, by the second part of Theorem 2.6, we have that d.U; G/ Ä 5. Suppose by contradiction that d.U; G/ Ä 4. Then, by Definition 2.5, we have that m. / Ä 1 for each irreducible character 2 Irr.G/. We will prove that, for D 1 G , this is not true. For let us take a nontrivial irreducible character 2 Irr.FK=F /. We will prove that d. ; 1 FK / D 2.
Since can be extended to a nontrivial irreducible character Â of N G .F / containing F in its kernel, by the proof of [ [14], we proved that subgroups of type (i) and (ii) have disjoint conjugates. In (iii), we proved above this property for subgroups of type (iii). Conversely, if a non-normal subgroup has a disjoint conjugate, then by the second part of Theorem 2.6, it is of ordinary depth 3.

Final remarks
Since for subgroups L of ordinary depth 3 in an arbitrary group G, it is not true in general that L has a disjoint conjugate, it is natural to ask the following.
Question 5.1. Let L Ä G be a subgroup of a finite group G of ordinary depth 3. Is it true that Core G .L/ D L \ L x 1 \ L x 2 for suitable elements x 1 and x 2 ?
We are also interested in a possible converse of Corollary 1.6. The following group is defined in [16, Example 6.7, Chapter VIII].  (b) (i) either S is isomorphic to a group A.n; Â / for some non-identity automorphism Â of GF .2 n / of odd order and jSj D jZ.S/j 2 D 2 2n , (ii) or jS j D jZ.S/j 3 .
Remark 5.4. Sylow 2-subgroups of simple Suzuki groups Sz.q/ are isomorphic to certain A.2m C 1; Â/, where Â 2 Aut.GF .2 2mC1 // (acting as xÂ D x 2 mC1 ), which is of odd order, thus falls into the category (i) in the previous theorem. However, not every A.n; Â/ is isomorphic to a Sylow 2-subgroup of a simple Suzuki group.
Question 5.5. Let S be a 2-group, and let H Ä Aut.S /. Suppose that S 1 Ä S is a Suzuki 2-group, which is isomorphic to a Sylow 2-subgroup of a simple Suzuki group Sz.q/. Let H 1 Ä Aut.S 1 / be a solvable group of automorphisms, acting on the involutions of S 1 transitively, and let H 1 Ä H . Let us consider the semidirect product SH . Suppose further that S D S x2SH S x 1 and S 1 \ S x 1 is either trivial or the common center of them. Is it true that S is a Suzuki 2-group?
Remark 5.6. From Theorem 5.3, it follows that, in Question 5.5, Aut.S/ acts transitively on the involutions of S , so this conjecture is a weakened form of a conjecture of Gross on finite 2-groups S with more than 1 involution admitting a group of automorphisms that transitively permutes the involutions of S , called 2-automorphic 2-groups; see [11]. According to the results of Gross, these groups fall into three classes: (a) homocyclic, (b) of exponent 4 and class 2 of order jZ.S/j 2 or jZ.S/j 3 and S 0 D Ã 1 .S/ D Z.S / has exponent 2, or (c) of exponent 8 and class 3, S 0 is homocyclic of order jZ.S/j 2 and exponent 4, and S 0 D Ã 1 .S/ D Z.S / D OES; S 0 has exponent 2. The conjecture was that the groups of type (b) are Suzuki 2-groups and those of type (c) do not occur. In the paper [26], it was proved that 2-automorphic 2-groups are of class at most 2, so case (c) does not occur. In the paper [5], it was proved that 2-automorphic 2-groups of type (b) are Suzuki 2-groups if jZ.S/j 3 D jS j. The other case in type (b) is still open. Thus, in Question 5.5, the group S must be of type (b), and in the case jZ.S /j 3 D jS j, the question has a positive answer. In [5], it was also proved that the automorphism group of a Suzuki 2-group is always solvable. In [27], it was proved that, in Suzuki 2-groups S of order jZ.S/j 2 , any two elements of the same order are conjugate in Aut.S/. Now we summarize the conjectures. Taking into consideration the results of Higman, Shult, Gross, Bryukhanova and Wilkens, the reformulated conjecture is the following: if a finite group G acts on a non-abelian finite 2-group S with more than 1 involution and G is transitive on the involutions of S, then G is solvable.
So far, it has been proved that if a finite group G acts on a non-abelian finite 2-group S with more than 1 involution, then if there is a solvable subgroup G 1 Ä G that acts on the involutions of S transitively, then G is solvable. We ask that if S has some more properties, namely we suppose that S has a special kind of covering of Question 5.5, then G is solvable.