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A subgroup \( H \) in a group \( G \) is pronormal if the subgroups \( H \) and \( H^{g} \) are conjugate in \( \langle{}H,H^{g}\rangle{} \) for all \( g\in G \).

Pronormal subgroups were introduced by Hall in [1] as a natural generalization of normal subgroups. Pronormality is closely related to the argument widely used in group theory known as the Frattini argument. Pronormality plays an important role both in group theory itself and its combinatorial applications. There is an extensive literature on the problems related to this property (see the surveys [2,3,4]).

Finding pronormal subgroups is of a special interest in simple groups, where proper nontrivial normal subgroups are absent. Alongside normal subgroups, in every (in particular, simple) group, maximal subgroups are always pronormal. Together with usual maximal subgroups, which are inclusion maximal among proper subgroups, under consideration are the elements of the lattice of subgroups maximal among \( {\mathfrak{X}} \)-subgroups, where \( {\mathfrak{X}} \) is a class of groups and by an \( {\mathfrak{X}} \)-groups we mean a group belonging to \( {\mathfrak{X}} \). Such subgroups are \( {\mathfrak{X}} \)-maximal. The problems of searching such subgroups stem from the classical works of Galois and Jordan and are the contents of Wielandt’s program proposed in 1979 at the famous Santa Cruz Conference on Finite Groups [5]. An important aspect of this program is connected with the study of the properties, in particular, the pronormality, \( {\mathfrak{X}} \)-maximal subgroups, and their generalizations.

In what follows, we will assume like in Wielandt’s talk that the class \( {\mathfrak{X}} \) is complete [6, Definition 11.1]; i.e., closed under subgroups, homomorphic images, and extensions. Refer as a relatively maximal subgroup to a subgroup \( {\mathfrak{X}} \)-maximal for some class \( {\mathfrak{X}} \). The typical examples of complete classes are as follows: the class of soluble groups and the class of \( \pi \)-groups for every set \( \pi \) of primes, where, as usual, a \( \pi \)-group is a group whose all prime divisors of the order belong to \( \pi \).

On the one hand, maximal and relatively maximal subgroups are substantially different notions. For instance, if \( G \) is a nonidentical finite \( p \)-group for a prime \( p \) then maximal subgroups in \( G \) are exactly subgroups of index \( p \), whereas a maximal \( {\mathfrak{X}} \)-subgroup in a complete class \( {\mathfrak{X}} \) is always unique; this is either \( G \) or \( 1 \) depending on whether \( {\mathfrak{X}} \) contains a group of order \( p \) or not.

On the other hand, in a finite simple group \( G \), maximal subgroups are exactly \( {\mathfrak{X}} \)-maximal subgroups, where \( {\mathfrak{X}} \) is the class of all groups for which the orders of composition factors are strictly less than the order of \( G \) (it is clear that \( {\mathfrak{X}} \) is complete). Therefore, the following problem seems natural:

Problem 1

Are the relatively maximal subgroups of finite simple groups always pronormal?


In 2018 in [4, Problem 5.20] and [7, Problem 6], the problem was posed even more widely, as the problem of the pronormality of the so-called \( {\mathfrak{X}} \)-submaximal subgroups which include \( {\mathfrak{X}} \)-maximal subgroups. The works [7,8,9] and the survey [4] can give an impression about \( {\mathfrak{X}} \)-submaximal subgroups and the role of the pronormality problems in their study, especially for the groups that are simple or close to simple. An additional reason for studying Problem 1 is the pronormality of Hall subgroups (proved in [10]), i.e., of subgroups for which the index and the order are coprime; or, equivalently, of \( {\mathfrak{X}} \)-Hall subgroups in simple groups, i.e., of \( {\mathfrak{X}} \)-subgroups whose indices do not divide by primes \( p \) if only \( {\mathfrak{X}} \) contains a group of order \( p \). Obviously, \( {\mathfrak{X}} \)-Hall subgroups are always \( {\mathfrak{X}} \)-maximal.

In the recent work [11], there is given a negative solution of Problem 1. Basing on rather subtle examples by Wilson (see [12]), which in turn serve as counterexamples to one of Aschbacher’s Conjectures [13, Question 8.2], the authors proved the existence of several series of triples \( ({\mathfrak{X}},G,H) \), where \( {\mathfrak{X}} \) is a complete class of finite groups, \( G \) is a finite simple group, and \( H \) is its \( {\mathfrak{X}} \)-maximal subgroup such that \( H \) is not pronormal in \( G \). In all such triples arising from Wilson’s examples, the class \( {\mathfrak{X}} \) is chosen in a very whimsical way, includes nonsoluble groups; and, moreover, a \( {\mathfrak{X}} \)-maximal subgroup of \( H \) itself turns out to be nonsoluble. This gave the authors a basis for posing the following problem:

Problem 2 [11, Problem 3]

Are the \( {\mathfrak{X}} \)-maximal subgroups of finite simple groups always pronormal if the complete class \( {\mathfrak{X}} \) consists only of soluble groups or, more narrowly, only of groups of odd order?


In the present article, we solve this problem in the negative and provide a much more direct and natural example presents a negative solution of Problem 1.

Theorem

The simple group \( Sp_{4}(4) \) of order

$$ 2^{8}\cdot 3^{2}\cdot 5^{2}\cdot 17=979200 $$

has a nonpronormal \( \{3,5\} \)-maximal subgroup.

Proof

By [14], the list of maximal subgroups in \( Sp_{4}(4) \) is exhausted up to conjugacy by the groups \( M \) in Table 1.

Table 1 The maximal subgroups of \( Sp_{4}(4) \)
Full size table

It is known that the groups \( A_{5} \) and \( S_{6} \) do not contain elements and hence the subgroups of order \( 15=3\cdot 5 \) and \( S_{6} \) does not contain subgroups of order \( 3^{2}\cdot 5 \) either (see, for instance, [14]). It follows that every subgroup of order \( 15 \) in \( Sp_{4}(4) \) is a maximal \( \{3,5\} \)-subgroup. Now, in the maximal subgroup

$$ M=(A_{5}\times A_{5}):2\cong Sp_{2}(4)\wr S_{2}, $$

belonging to the Aschbacher class \( {\mathcal{C}}_{2} \) [15, Table 8.14], take a subgroup \( H \) of order \( 15 \) from the base

$$ B=A_{5}\times A_{5}\cong Sp_{2}(4)\times Sp_{2}(4) $$

of the wreath product \( Sp_{2}(4)\wr S_{2} \) generated by an element of order \( 3 \) in the first factor and an element of order \( 5 \) in the second factor. Every element \( g\in M\setminus B \) permutes direct factors in \( B \). Moreover, the subgroup \( H^{g} \) lies in \( B \) and hence is such that its projection to the first direct factor has order \( 5 \), and its projection to the second factor has order \( 3 \). Thus, the projections of \( H \) and \( H^{g} \), say, to the first factor are not only nonconjugate but also have different orders. Hence, the subgroups \( H \) and \( H^{g} \) in \( B \) are not conjugate in \( B \) and a fortiori not conjugate in the subgroup \( \langle{}H,H^{g}\rangle{} \) of \( B \). By definition, \( H \) is not pronormal in \( Sp_{4}(4) \).  ☐


Apparently, each example of a nonpronormal relatively maximal subgroup of odd order in a finite simple group is not exceptional and gives rise to a series of examples of nonpronormal relatively maximal subgroups of odd order in the groups \( Sp_{2n}(q) \), where \( n\geq 2 \) can be arbitrary and \( q \) arbitrarily large. We will find exposition just to describing the idea of how we can transfer the finding of the subgroup \( H \) in the proof of the theorem to a more general situation.

Observe that there exist infinitely many primes \( q \) such that the number of prime divisors of \( q^{2}-1 \) (counted with multiplicity) does not exceed \( 20 \) [16, Corollary D]. Therefore, for every \( n \), we can find a prime \( q \) such that both numbers \( q-1 \) and \( q+1 \) have some prime divisors \( r \) and \( s \) respectively that are greater than \( n \). Then in the description of maximal subgroups [15, Table 8.2] it is clear that the group \( Sp_{2}(q)\cong SL_{2}(q) \) does not include groups of order \( rs \).

Consider the wreath product

$$ M=Sp_{2}(q)\wr S_{n}, $$

which is a subgroup in \( Sp_{2n}(q) \)—the stabilizer of the decomposition of the natural module into an orthogonal sum of nondegenerate subspaces of dimension 2. In the base \( B \) for this wreath product, take a subgroup \( H \) of order \( r^{n-1}s \) whose projections to all but the last factors have order \( r \) and whose projection to the last factor has order \( s \). Then in \( M \) there exists an element \( g \) such that the subgroup \( H^{g} \), lying in \( B \), has a projection to the first factor in \( B \) of order \( s \) and projections to the remaining factors of order \( r \). The subgroups \( H \) and \( H^{g} \) are not conjugate in \( B \) since have projections of different orders to the first factor. Hence, \( H \) is not pronormal in \( Sp_{2n}(q) \).

From similar considerations, we can deduce that the image of \( H \) is not pronormal in \( PSp_{2n}(q) \). Since every \( \{r,s\} \)-group is soluble by Burnside’s Theorem, the maximal \( \{r,s\} \)-subgroup in \( Sp_{2n}(q) \) into which \( H \) can be immersed lies in some element of one of the Aschbacher classes \( {\mathcal{C}}_{1} \)–\( {\mathcal{C}}_{8} \) by Aschbacher’s Theorem [17]. Exhausting these elements [15, the table in Section 8] and [18, Table 3.5C] and using the fact that \( r,s>n \), it is apparently not hard to show that \( H \) is a maximal \( \{r,s\} \)-subgroup in \( Sp_{2n}(q) \) and its image is a maximal \( \{r,s\} \)-subgroup in \( PSp_{2n}(q) \).

The detailed justification of these considerations would require additional efforts and would substantially increase the article while adding little to the general picture which is clear in principle from the Theorem.

In spite of the negative solution of Problem 1, advances in studying the problem of the pronormality of subgroups of odd index (see [3, 4]) give a reason to hope that at least maximal \( {\mathfrak{X}} \)-subgroups of odd index in finite simple groups are always pronormal [11, Problem 2].