On the Pronormality of Subgroups of Odd Index in Finite Simple Groups
aa r X i v : . [ m a t h . G R ] J u l MSC 20D60, 20D06
ON THE PRONORMALITY OF SUBGROUPS OF ODD INDEXIN FINITE SIMPLE GROUPS
ANATOLY S. KONDRAT’EV, NATALIA V. MASLOVA, AND DANILA O. REVIN
Abstract.
A subgroup H of a group G is said to be pronormal in G if H and H g are conjugate in h H, H g i for every g ∈ G . Some problems infinite group theory, combinatorics, and permutation group theory weresolved in terms of pronormality. In 2012, E. Vdovin and the third authorconjectured that the subgroups of odd index are pronormal in finitesimple groups. In this paper we disprove their conjecture and discussa recent progress in the classification of finite simple groups in which thesubgroups of odd index are pronormal. Keywords: pronormal subgroup, odd index, finite simple group, Sylowsubgroup, maximal subgroup.
Throughout the paper we consider only finite groups, and thereby the term”group” means ”finite group”.According to P. Hall, a subgroup H of a group G is said to be pronormal in G if H and H g are conjugate in h H, H g i for every g ∈ G .Some of well-known examples of pronormal subgroups are the following: normalsubgroups; maximal subgroups; Sylow subgroups; Sylow subgroups of proper normalsubgroups; Hall subgroups of solvable groups. In 2012, E. Vdovin and the thirdauthor [23] proved that the Hall subgroups are pronormal in all simple groups.The following assertion give a connection between pronormality of subgroupsand properties of permutation representations of finite groups. Theorem 1. (Ph. Hall, 1960s).
Let G be a group and H ≤ G . H is pronormalin G if and only if in any transitive permutation representation of G , the subgroup N G ( H ) acts transitively on the set f ix ( H ) of fixed points of H . Pronormality is the universal property with respect to Frattini Argument. Indeed,it is not hard to prove the following proposition (see [5, Lemma 4]).
Proposition 1.
Let G be a group, A E G , and H ≤ A . Then the following statementsare equivalent : (1) H is pronormal in G ; (2) H is pronormal in A and G = AN G ( H ) ; (3) H is pronormal in A and H A = H G . Some problems in finite group theory, combinatorics, and permutation grouptheory were solved in terms of pronormality. For example, according to L. Babai [1], a group G is called a CI-group if between every two isomorphic relational structureson G (as underlying set) which are invariant under the group G R = { g R | g ∈ G } of right multiplications g R : x xg (where g, x ∈ G ), there exists an isomorphismwhich is at the same time an automorphism of G . Babai [1] proved that a group G is a CI-group if and only if G R is pronormal in Sym ( G ) . In particular, if G isa CI-group, then G is abelian. With using mentioned Babai’s result, P. Palfy [21]obtained a classification of CI-groups.Thus, the following problem naturally arises. General Problem.
Given a finite group G and H ≤ G , is H pronormal in G ? Ch. Praeger [22] investigated pronormal subgroups of permutation groups. Sheproved the following theorem.
Theorem 2.
Let G be a transitive permutation group on a set Ω of n points, andlet K be a non-trivial pronormal subgroup of G . Suppose that K fixes exactly f points of Ω . Then f ≤ ( n − , and if f = ( n − , then K is transitive on itssupport in Ω , and either G ≥ Alt ( n ) , or G = GL d (2) acting on the n = 2 d − non-zero vectors, and K is the pointwise stabilizer of a hyperplane. Thus, if in some permutation representation of G , | f ix ( H ) | is too big, then H is not pronormal in G . Therefore, it is interesting to consider pronormalityof the subgroups of a group G containing a subgroup S which is pronormal in G . In particular, it is interesting to consider pronormality of overgroups of Sylowsubgroups. Note that the subgroups of odd index in a finite group G are exactlyovergroups of Sylow -subgroups of G .First of all, we will concentrate on the question of pronormality of subgroups ofodd index in non-abelian simple groups. In 2012, E. Vdovin and the third author[23] formulated the following conjecture. Conjecture 1.
The subgroups of odd index are pronormal in all simple groups.
In this paper, we disprove Conjecture 1 and discuss a recent progress in theclassification of non-abelian simple groups in which the subgroups of odd index arepronormal.In Section 10, we discuss the question of pronormality of subgroups of odd indexin a non-simple group. Note that an answer to this question for a group G is veryweakly connected with answers to this question for subgroups of G , even for normalsubgroups of G . For example, it could be proved that there exist infinitely manynon-abelian simple groups in which the subgroups of odd index are pronormal whilea direct product of any two of them contains a non-pronormal subgroup of odd index(see Example 1). 1. Terminology and Notation
Our terminology and notation are mostly standard and can be found in [3, 9].For a group G and a subset π of the set of all primes, O π ( G ) and Z ( G ) denotethe π -radical (the largest normal π -subgroup) and the center of G , respectively. Asusual, π ′ stands for the set of those primes that do not lie in π . If n is a positiveinteger, then n π is the largest divisor of n whose all prime divisors lie in π . Also,for a group G , it is common to write O ( G ) instead of O ′ ( G ) . The set of Sylow N THE PRONORMALITY OF SUBGROUPS OF ODD INDEX 3 p -subgroups of G is denoted by Syl p ( G ) . The socle of G is denoted by Soc ( G ) .Recall that G is almost simple if Soc ( G ) is a non-abelian simple group.We use the following notation for non-abelian simple groups: Alt ( n ) for alternatinggroups; P SL εn ( q ) , where ε = + for P SL n ( q ) and ε = − for P SU n ( q ) ; E ε ( q ) , where ε = + for E ( q ) and ε = − for E ( q ) .Let m and n be non-negative integers with m = P ∞ i =0 a i · i and n = P ∞ i =0 b i · i ,where a i , b i ∈ { , } . We write m (cid:22) n if a i ≤ b i for every i and m ≺ n if, in addition, m = n .2. Verification of Conjecture 1 for many families of non-abeliansimple groups
An important role in the verification of Conjecture 1 is played by the followingeasy assertion (see [23, Lemma 5]), which is a consequence of Theorem 1.
Lemma 1.
Suppose that G is a group and H ≤ G . Assume also that H contains aSylow subgroup S of G . Then the following statements are equivalent : (1) H is pronormal in G ; (2) The subgroups H and H g are conjugate in h H, H g i for every g ∈ N G ( S ) . Note that Sylow -subgroups in non-abelian simple groups are usually self-normalized, and all the exceptions were described by the first author in [10]. Heproved the following proposition (see [10, Corollary of Theorems 1-3]). Proposition 2.
Let G be a non-abelian simple group and let S ∈ Syl ( G ) . Then N G ( S ) = S excluding the following cases: (1) G ∼ = J , J , Suz or F and | N G ( S ) : S | = 3 ; (2) G ∼ = G (3 n +1 ) or J and N G ( S ) ∼ = ( C ) ⋊ ( C ⋊ C ) < Hol (( C ) ) ; (3) G is a group of Lie type over a field of characteristic and N G ( S ) is a Borelsubgroup of G ; (4) G ∼ = P SL ( q ) , where < q ≡ ± and N G ( S ) ∼ = Alt (4) ; (5) G ∼ = P Sp n ( q ) , where n ≥ , q ≡ ± , n = 2 s + · · · + 2 s t for s > · · · > s t ≥ and N G ( S ) /S is the elementary abelian group of order t ; (6) G ∼ = E η ( q ) where η ∈ { + , −} , q is odd, and | N G ( S ) : S | = ( q − η ′ / ( q − η , = 1;(7) G ∼ = P SL ηn ( q ) , where n ≥ , η ∈ { + , −} , q is odd, n = 2 s + · · · + 2 s t for s > · · · > s t > , and N G ( S ) ∼ = S × C × · · · × C t − > S, where C , . . . C t − , and C t − are cyclic subgroup of orders ( q − η ′ , . . . , ( q − η ′ , and ( q − η ′ / ( q − η , n ) ′ ,respectively. Thus, using Lemma 1 for the verification of Conjecture 1, it is sufficient toconsider simple groups from items (1) – (7) of Proposition 2.In [11] we considered simple groups from items (1) – (4) and, with taking intoaccount of simple groups with self-normalized Sylow -subgroups, we have provedthe following theorem. Theorem 3.
The subgroups of odd index are pronormal in the following simplegroups : Alt ( n ) , where n ≥ ; sporadic groups ; groups of Lie type over fields ofcharacteristic ; P SL ε n ( q ) , where ε ∈ { + , −} ; P Sp n ( q ) , where q
6≡ ± ; orthogonal groups ; exceptional groups of Lie type not isomorphic to E ε ( q ) , where ε ∈ { + , −} . ANATOLY S. KONDRAT’EV, NATALIA V. MASLOVA, AND DANILA O. REVIN Maximal subgroups of odd index in simple groups
In this paper, the complete classification of maximal subgroups of odd index insimple classical groups is a crucial tool in view of the following evident lemma.
Lemma 2.
Suppose that H and M are subgroups of a group G and H ≤ M . Then (1) if H is pronormal in G , then H is pronormal in M ; (2) if S ≤ H for some Sylow subgroup S of G , N G ( S ) ≤ M , and H is pronormalin M , then H is pronormal in G . M. Liebeck and J. Saxl [16] and, independently, W. Kantor [7] proposed aclassification of primitive permutation groups of odd degree. It is considered to beone of remarkable results in the theory of finite permutation groups. In particular,both papers [16] and [7] contain lists of subgroups of simple groups that can turn outto be maximal subgroups of odd index. However, in the cases of alternating groupsand of classical groups over fields of odd characteristics, neither in [16] nor in [7] itwas described which of the specified subgroups are precisely maximal subgroups ofodd index. Thus, the problem of the complete classification of maximal subgroupsof odd index in simple groups was remained open.The classification was finished by the second author in [18, 20]. In [18], the authorreferred to results obtained by P. Kleidman [8] and by P. Kleidman and M. Liebeck[9]. However, there are a number of inaccuracies in Kleidman’s PhD thesis [8]. Theseinaccuracies have been corrected in [2]. Due to uncovered circumstances, in [19] thesecond author revised the main result of [18] taking into account the results of [2].In particular, changes in statements of items (6) , (10) , and (21) of [18, Theorem 1]were made. 4. Some tools for disproving Conjecture 1
In [12], we disproved Conjecture 1. The aim of this section is to demonstratesome tools for this.A consequence of well-known Schur–Zassenhaus theorem (see, for example, [6,Theorems 3.8 and 3.12]) is the following proposition [6, Ch. 4, Lemma 4.28].
Proposition 3. If V is a normal subgroup of a group G and H is a subgroup of G such that ( | H | , | V | ) = 1 , then, for any H -invariant subgroup U of V , the equality U = C U ( H )[ H, U ] holds. We proved that the following more general statement [12, Proposition 2] holds.
Proposition 4. If V is a normal subgroup of a group G and H is a pronormalsubgroup of G , then, for any H -invariant subgroup U of V , the equality U = N U ( H )[ H, U ] holds. It is easy to see that, in the case when the subgroups H and V from Proposition4 have trivial intersection, the equality N U ( H ) = C U ( H ) holds for any H -invariantsubgroup U of V . Therefore, Proposition 3 is a special case of Proposition 4.We showed that the statement converse to Proposition 4 holds when the group V is abelian and G = HV (i. e., H is a supplement to the subgroup V in G ). Wehave proved the following theorem (see [12, Theorem 1]). Theorem 4.
Let H and V be subgroups of a group G such that V is an abeliannormal subgroup of G and G = HV . Then the following statements are equivalent: (1) the subgroup H is pronormal in G; N THE PRONORMALITY OF SUBGROUPS OF ODD INDEX 5 (2) U = N U ( H )[ H, U ] for any H -invariant subgroup U of V . With using Theorem 4, we have proved the following proposition (see [12, Corollaryof Theorem 1]).
Proposition 5.
Let G = A ≀ Sym ( n ) = HV be the natural permutational wreathproduct of an abelian group A and the symmetric group H = Sym ( n ) , where V denotes the base of the wreath product. Then the following statements are equivalent: (1) the subgroup H is pronormal in G; (2) ( | A | , n ) = 1 . A series of examples disproving Conjecture 1
The aim of this section is to construct a series of examples disproving Conjecture 1.We prove the following theorem (see [12, Theorem 2]).
Theorem 5.
The simple group
P Sp n ( q ) for any q ≡ ± contains anon-pronormal subgroup of odd index. Sketch of proof.
Let q ≡ ± be a prime power and n be a positiveinteger. It is well known that a Sylow -subgroup S of the group T = Sp ( q ) = SL ( q ) is isomorphic to Q , and N T ( S ) ∼ = SL (3) ∼ = Q : 3 . We have the followingchain of embeddings: Q ≀ Sym (3 n ) ≤ L = Sp (3) ≀ Sym (3 n ) ≤ Sp ( q ) ≀ Sym (3 n ) ≤ G = Sp n ( q ) . It could be proved by direct calculations that | G : L | is odd. It is easy to see that L/O ( L ) ∼ = C ≀ Sym (3 n ) . In view of Proposition 5, the group L/O ( L ) contains anon-pronormal subgroup T of odd index. Let H be the preimage of T in L . Then H is a non-pronormal subgroup of odd index in L in view of the following proposition. Proposition 6. (see [17, Lemma 3] and [4, Chapter I, Proposition (6.4)] ) Supposethat H is a subgroup and N is a normal subgroup of a group G . Let ¯: G → G/N be the natural epimorphism. The following statements hold : (1) if H is pronormal in G , then H is pronormal in G ; (2) H is pronormal in G if and only if HN is pronormal in G and H is pronormalin N G ( HN ) ; (3) if N ≤ H and H is pronormal in G , then H is pronormal in G . In particular,a subgroup H of odd index is pronormal in G if and only if H/O ( G ) is pronormalin G/O ( G ) . In view of Lemma 2, H is a non-pronormal subgroup of G , and it is easy to seethat | G : H | is odd. Note that O ( G ) ≤ H . Thus, H/O ( G ) is a non-pronormalsubgroup of odd index in G/O ( G ) ∼ = P Sp n ( q ) in view of Proposition 6.6. Classification problem
The following problem naturally arises.
Problem 1.
Classify non-abelian simple groups in which the subgroups of oddindex are pronormal.
To solve Problem 1 it remains to consider the following simple groups: (1)
P Sp n ( q ) , where q ≡ ± and does not divide n ; (2) E ε ( q ) , where ε ∈ { + , −} and q is odd; (3) P SL εn ( q ) , where ε ∈ { + , −} , q is odd, and n = 2 w . ANATOLY S. KONDRAT’EV, NATALIA V. MASLOVA, AND DANILA O. REVIN
In this paper, we consider in some details a solution of Problem 1 for symplecticgroups, briefly discuss a solution of Problem 1 for groups E ε ( q ) , and formulate aconjecture for groups P SL εn ( q ) .7. Simple symplectic groups containing non-pronormal subgroups ofodd index
In fact, Theorem 5 permits us to investigate Problem 1 within a much widerfamily of symplectic groups.Let G = Sp n ( q ) , where q is odd, and V be the natural n -dimensional vectorspace over the field F q with a non-degenerate skew-symmetric bilinear form associatedwith G . Let us look to the list of maximal subgroups of odd index of G (this listcan be found in [19]). Proposition 7.
Maximal subgroups of odd index in Sp n ( q ) = Sp ( V ) , where n > and q is odd, are the following : (1) Sp n ( q ) , where q = q r and r is an odd prime ; (2) Sp m ( q ) × Sp n − m ) ( q ) , where m ≺ n ; (3) Sp m ( q ) ≀ Sym ( t ) , where n = mt and m = 2 k ; (4) 2 .Alt (5) , where n = 2 and q ≡ ± is a prime. Note that if q is odd, then | Z ( Sp n ( q )) | = 2 . Thus, in view of Proposition 6, thesubgroups of odd index are pronormal in Sp n ( q ) if and only if the subgroups of oddindex are pronormal in P Sp n ( q ) , i. e., there is no difference between investigationof Problem 1 for Sp n ( q ) or for P Sp n ( q ) .Let q ≡ ± and n is not of the form w or w (2 k + 1) . Then the -adicrepresentation n = P ∞ i =0 s i · i either has two s in positions s and s of differentparity, or three s in positions s , s , and s of the same parity. Define m = 2 s +2 s or m = 2 s + 2 s + 2 s , respectively. It is easy to see that m ≺ n and divides m .Let M be the stabilizer in G of a non-degenerate m -dimensional subspace of V .It is easy to see that M = M × M , where M ∼ = Sp m ( q ) and M ∼ = Sp n − m ) ( q ) .Note that the index | G : M | is odd by Proposition 7. Thus, if H is a subgroupof odd index in M , then H is a subgroup of odd index in G too. Since divides m , it follows by Theorem 5 that M /Z ( M ) = M /O ( M ) has a non-pronormalsubgroup H /O ( M ) of odd index. Then H is a non-pronormal subgroup of oddindex in M in view of Proposition 6. Now it is easy to see that therefore, H × M is a non-pronormal subgroup of odd index in M × M . So, H × M is a non-pronormal subgroup of odd index in G . Thus, we proved the following theorem (see[13, Theorem 1]). Theorem 6.
Let G = P Sp n ( q ) , where q ≡ ± and n is not of the form w or w (2 k + 1) . Then G has a non-pronormal subgroup of odd index. Now in view of Theorems 3 and 6, to finish a classification of simple symplecticgroups in which the subgroups of odd index are pronormal, it remains to considergroups
P Sp n ( q ) , where q ≡ ± and n is of the form w or w (2 k + 1) .In the next section we will prove that the subgroups of odd index are pronormal inthese groups. N THE PRONORMALITY OF SUBGROUPS OF ODD INDEX 7 Classification of simple symplectic groups in which the subgroupsof odd index are pronormal
In this section we prove the following theorem, whose proof was recently finishedby the authors (see [13, Theorem 2] for the case when n is of the form w and [14]for the case when n is of the form w (2 k + 1) ). Theorem 7.
Let G = P Sp n ( q ) . Then each subgroup of odd index is pronormal in G if and only if one of the following statements holds: (1) q
6≡ ± ; (2) n is of the form w or w (2 k + 1) . Sketch of proof.
Let G = Sp n ( q ) , where q ≡ ± and n is of the form w ≥ . Suppose that the claim of the theorem is false, and let q be the smallestprime power congruent to ± modulo such that G has a non-pronormal subgroup H of odd index. Let S ≤ H be a Sylow -subgroup of G . Note that | N G ( S ) /S | = 3 in view of Proposition 2.Pick g ∈ N G ( S ) . Without loss of generality we can assume that | g | = 3 . Put K = h H, H g i , and we can suppose that K < G . Then there exists a maximalsubgroup M of G such that K ≤ M .Using Proposition 7, we conclude that one of the following cases arises.1. M ∼ = Sp n ( q ) , where q = q r and r is an odd prime. In this case it could beproved that N G ( S ) ≤ M , and we use inductive reasonings and Lemma 2 to provethat H is pronormal in G .2. M ∼ = Sp m ( q ) ≀ Sym ( t ) , where n = mt and m = 2 k , and M is choosen of suchtype so that m is as small as possible. In this case we can prove that H/O ( M ) ispronormal in M/O ( M ) ∼ = P Sp m ( q ) ≀ Sym ( t ) with using two following propositionsand additional inductive reasonings. Proposition 8. (see [13, Lemma 15] and [5, Lemma 9] ) Let Q be a subgroup ofodd index in a group L = L × L × . . . × L n , where L i are groups, and let π i bethe projection from L to L i . If there is i such that L i is almost simple, L i /Soc ( L i ) is a -group, and Q π i = L i , then L i ≤ Q . Proposition 9. (see [13, Lemma 17] ) Let G = A ≀ Sym ( n ) = LH be the naturalpermutation wreath product of a non-abelian simple group A and H = Sym ( n ) ,where n = 2 w , L = L × . . . × L n , and for each i ∈ { , . . . , n } , L i ∼ = A and π i : L → L i is the projection from L to L i . If K is a subgroup of odd index in G , K = K ∩ L ,and M ≤ L such that N L ( K π ) ≤ M , then K ≤ U ∼ = M ≀ Sym ( n ) ≤ G . Moreover, it could be proved that in this case N G ( S ) ≤ M . Now, in view ofProposition 6 and Lemma 2, we conclude that H is pronormal in G .3. M ∼ = 2 .Alt (5) , where n = 4 and q is a prime. In this case it could be easyproved that N G ( S ) ≤ M . Now we conclude that H is pronormal in G in view ofProposition 6, Lemma 2, and Theorem 3.In the case when n is of the form w (2 k + 1) , the scheme of the proof is similar,however we can not use Proposition 9. The proof of Proposition 9 is based on a factthat K contains a regular subgroup of some conjugate of H . And the fact could befalse if n is not of the form w .To investigate this case the second and the third authors in a joint work withW. Guo [5] obtained the following useful criterion of the pronormality of subgroupsof odd index in extensions of groups (see [5, Theorem 1]). ANATOLY S. KONDRAT’EV, NATALIA V. MASLOVA, AND DANILA O. REVIN
Theorem 8.
Let G be a group, A E G , the subgroups of odd index are pronormal in A , and Sylow -subgroups of G/A are self-normalized. Let T be a Sylow -subgroupof A . Then the following statements are equivalent : (1) The subgroups of odd index are pronormal in G ; (2) The subgroups of odd index are pronormal in N G ( T ) /T . However, there are difficulties in a direct application of Theorem 8 to the maximalsubgroups of the group Sp n ( q ) , where n is of the form w (2 k + 1) , in view of thefollowing fact: if the subgroups of odd index are pronormal in both groups G and G , it does not imply that the subgroups of odd index are pronormal in the group G × G , even if G and G are non-abelian simple. It is clear from the followingexample (see [5, Proposition 1]). Example 1.
Consider Frobenius groups H i = L i ⋊ K i ∼ = C ⋊ C for i ∈ { , } and H = H × H . Note that any proper subgroup of H i is its Sylow subgroup, and henceis pronormal in H i . Let L = O ( H ) = L × L ∼ = C × C and D = { ( x, x ) | x ∈ C } . There exists k ∈ K ×{ } such that D k = { (2 x, x ) | x ∈ C } . Hence h D, D k i = L is abelian and D is a non-pronormal subgroup ( of odd index ) in H .Let G , G ∈ { J } ∪ { G (3 m +1 ) | m ≥ } and S i ∈ Syl ( G i ) . In view ofTheorem 3, the subgroups of odd index are pronormal in G i for i ∈ { , } . Andin view of Proposition 2, N G i ( S i ) /S i for i ∈ { , } is isomorphic to the Frobeniusgroup C ⋊ C . Using previous reasonings it is easy to construct a non-pronormalsubgroup of odd index in G = G × G . Recently basing on Theorem 8, the second and the third authors, and W. Guo [5]obtained the following pronormality criterion for subgroups of odd index in groupsof the type Q ti =1 ( A ≀ Sym ( n i )) , where A is an abelian group and all the wreathproducts are natural permutation (see [5, Theorem 2]). Theorem 9.
Let A be an abelian group and G = Q ti =1 ( A ≀ Sym ( n i )) , where allthe wreath products are natural permutation. Then the subgroups of odd index arepronormal in G if and only if for any positive integer m , the inequality m ≺ n i forsome i implies that g.c.d. ( | A | , m ) is a power of . Moreover, in [5, Theorem 3] the following theorem was proved.
Theorem 10.
Let G = Q ti =1 G i , where for any i ∈ { , ..., t } , G i ∼ = Sp n i ( q i ) , q i isodd, and n i is a power of . Then the subgroups of odd index are pronormal in G . Theorems 8, 9, and 10 became the main tools in the proof of Theorem 7 whichwas finished by the authors in [14].9.
Summary and further research on finite simple groups in whichthe subgroups of odd index are pronormal
Let G be a non-abelian simple group, S ∈ Syl ( G ) , and C = O ( C G ( S )) . In [15,Theorem 7] it was proved that C = 1 if and only if one of the following statementsholds: (1) G ∼ = E η ( q ) , where η ∈ { + , −} , q is odd, and C is a cyclic group of order ( q − η ′ / ( q − η , = 1 ; (2) G ∼ = P SL ηn ( q ) , where n ≥ , η ∈ { + , −} , q is odd, n = 2 s + · · · + 2 s t for s > · · · > s t > and t > , and C ∼ = C × · · · × C t − = 1 , where C , . . . C t − , and N THE PRONORMALITY OF SUBGROUPS OF ODD INDEX 9 C t − are cyclic subgroup of orders ( q − η ′ , . . . , ( q − η ′ , and ( q − η ′ / ( q − η , n ) ′ ,respectively.Thus, with taking into account of Theorems 3 and 7, we obtain the followingtheorem. Theorem 11.
Let G be a non-abelian simple group, S ∈ Syl ( G ) , and C = O ( C G ( S )) . If C = 1 , then exactly one of the following statements holds : (1) The subgroups of odd index are pronormal in G ; (2) G ∼ = P Sp n ( q ) , where q ≡ ± and n is not of the form w or w (2 k + 1) . Moreover, recently we have obtained a solution of Problem 1 for simple exceptionalgroups E ε ( q ) . We have proved the following theorem. Theorem 12.
Let G = E ε ( q ) , where q = p k , p is a prime, and ε ∈ { + , −} . Thenthe subgroups of odd index are pronormal in G if and only if does not divide q − ε and if p is odd and ε = + , then k is a power of . The scheme of our proof of Theorem 12 is similar as for symplectic groups and isbased on the classification of maximal subgroups of odd index in simple exceptionalgroups of Lie type obtained in [16, 7]. Moreover, we use some results on subgroupstructure of G = E ε ( q ) obtained in [15]. The most difficult case here is when apossibly non-pronormal subgroup H of odd index and its conjugate H g for some g ∈ G are both contained in a parabolic maximal subgroup of G . In this caseTheorem 8 is an useful tool.Thus, to solve Problem 1 it remains to consider the groups P SL εn ( q ) , where ε ∈ { + , −} , q is odd, and n = 2 w . We have the following conjecture. Conjecture 2.
Let G = P SL εn ( q ) , where q is odd and ε ∈ { + , −} . The subgroupsof odd index are pronormal in G if and only if for any positive integer m , theinequality m ≺ n implies that g.c.d. ( m, q ε ( q − ε is a power of . Question of the pronormality of subgroups of odd index innon-simple groups
The Frattini Argument (see Proposition 1) and Proposition 6 are convenienttools to reduce General Problem to groups of smaller order. Assume that G is notsimple and A is a minimal non-trivial normal subgroup of G . Then A is a directproduct of pairwise isomorphic simple groups, and one of the following cases arises: (1) A ≤ H and, in view of Proposition 6, H is pronormal in G if and only if H/A is pronormal in
G/A . Note that | G/A | < | G | . (2) H ≤ A and, in view of Proposition 1, H is pronormal in G if and only if H is pronormal in A and G = AN G ( H ) . Note that | A | < | G | . Thus, the question ofpronormality of subgroups in direct products of simple groups is of interest. (3) Let H A and A H , and let N = N G ( HA ) . In view of Propositions 6 and 1, H is pronormal in G if and only if HA/A is pronormal in
G/A , N = AN N ( H ) , and H is pronormal in HA . Therefore, with using inductive reasonings, we can reducethis case to the subcase when G = HA .Suppose that G = HA , A is a minimal non-trivial normal subgroup of G , A H ,and | G : H | is odd. If | A | is odd, then A is abelian and H is pronormal in G . Indeed, if U is an H -invariant subgroup of A , then either U is trivial or U = A . Therefore, A ∩ H is trivial, and it is easy to see that U = C U ( H )[ H, U ] = N U ( H )[ H, U ] for every H -invariant subgroup U of A . Thus, in view of Theorem 4, H is pronormal in G .If A is a -group, then H is pronormal in G in view of Proposition 6.Suppose that A is a direct product of pairwise isomorphic non-abelian simplegroups. If the subgroups of odd index are pronormal in A , then we can use thefollowing criterion of the pronormality of subgroups of odd index in extensions ofgroups, which is a generalization of Theorem 8. Theorem 13.
Let G be a group, A E G , and the subgroups of odd index arepronormal both in A and in G/A . Let T be a Sylow -subgroup of A and let ¯: G → G/A be the natural epimorphism. (1)
Assume that H ≤ G , S ≤ H for some S ∈ Syl ( G ) , and T = A ∩ S . Let Y = N A ( H ∩ A ) and Z = N H ∩ A ( T ) . Then H is pronormal in G if and only if N H ( T ) /Z is pronormal in ( N H ( T ) N Y ( T )) /Z and N G ( H ) = N G ( H ) . (2) The subgroups of odd index are pronormal in G if and only if the subgroups ofodd index are pronormal in N G ( T ) /T and for every subgroup H ≤ G of odd index,we have N G ( H ) = N G ( H ) . Note that proof of Theorem 13 follows from proofs of Theorems 1 and 4 in [5].In view of Theorem 13, the following problem is of interest.
Problem 2.
Describe direct products of non-abelian simple groups in which thesubgroups of odd index are pronormal.
Note that Problem 2 is not equivalent to Problem 1 (see Example 1). However,in some cases the pronormality of subgroups of odd index in a direct product ofnon-abelian simple groups is equivalent to the pronormality of subgroups of oddindex in each factor. As an example, recently we have proved the following theorem.
Theorem 14.
Let G = Q ti =1 G i , where G i ∼ = P Sp n i ( q i ) and q i is odd for each i ∈ { , ..., t } . Then the subgroups of odd index are pronormal in G if and only if thesubgroups of odd index are pronormal in G i for each i . References [1] Laszlo Babai, Isomorphism Problem for a Class of Point-Symmetric Structures,
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E-mail address : [email protected] Natalia Vladimirovna MaslovaKrasovskii Institute of Mathematics and Mechanics of the Ural Branch of theRussian Academy of Science,16, S. Kovalevskaja Street,620990, Yekaterinburg, RussiaUral Federal University named after the first President of Russia B. N. Yeltsin,19, Mira Street,620002, Yekaterinburg, Russia
E-mail address : [email protected] Danila Olegovich RevinSobolev Institute of Mathematics of the Siberian Branch of the Russian Academyof Science,4, Koptuga Avenue,630090, Novosibirsk, RussiaNovosibirsk State University,1, Pirogova Street,630090, Novosibirsk, Russia
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