On the commuting probability for subgroups of a finite group
aa r X i v : . [ m a t h . G R ] F e b On the commuting probability forsubgroups of a finite group
Eloisa Detomi and Pavel Shumyatsky
Abstract.
The probability that two randomly chosen elements ofa finite group G commute is denoted by P r ( G ). If K ≤ G , we de-note by P r ( K, G ) the probability that an element of G commuteswith an element of K . A well known theorem, due to P. M. Neu-mann, says that if G is a finite group such that P r ( G ) ≥ ǫ , then G has a nilpotent normal subgroup T of class at most 2 such thatboth the index [ G : T ] and the order | [ T, T ] | of the commutatorsubgroup of T are ǫ -bounded.The main purpose of this paper is to establish the followingstronger forms of P. M. Neumann’s theorem.1. Let ǫ >
0, and let K be a subgroup of a finite group G suchthat P r ( K, G ) ≥ ǫ . Let H = h K G i be the normal closure of K in G . Then G has a nilpotent normal subgroup T ≤ H of nilpotencyclass at most 2 such that both the index [ H : T ] and the order | [ T, T ] | of the commutator subgroup are ǫ -bounded.2. Let ǫ >
0, and let H be a normal subgroup of a finite group G containing γ k ( G ) for some k ≥
1. Suppose that
P r ( H, G ) ≥ ǫ .Then G has a nilpotent normal subgroup T of nilpotency classat most k + 1 such that both the index [ G : T ] and the order of γ k +1 ( T ) are ǫ -bounded.We also deduce a number of corollaries of these results.
1. Introduction
The probability that two randomly chosen elements of a finite group G commute is given by P r ( G ) = |{ ( x, y ) ∈ G × G : xy = yx }|| G | . Mathematics Subject Classification.
Key words and phrases.
Commuting degree, Conjugacy classes, Nilpotentsubgroups.The second author was supported by FAPDF and CNPq.
The above number is called the commuting probability (or the commu-tativity degree ) of G . This is a well studied concept. In the literatureone can find publications dealing with problems on the set of possiblevalues of P r ( G ) and the influence of P r ( G ) over the structure of G (see [
5, 7, 8, 11, 12 ] and references therein). The reader can consult[
13, 19 ] and references therein for related developments concerningprobabilistic identities in groups.P. M. Neumann [ ] proved the following theorem (see also [ ]). Theorem 1.1.
Let ǫ > , and let G be a finite group such that P r ( G ) ≥ ǫ . Then G has a nilpotent normal subgroup T of nilpotencyclass at most such that both the index [ G : T ] and the order | [ T, T ] | of the commutator subgroup of T are ǫ -bounded. Throughout the article we use the expression “( a, b, . . . )-bounded”to mean that a quantity is bounded from above by a number dependingonly on the parameters a, b, . . . .If K is a subgroup of G , write P r ( K, G ) = |{ ( x, y ) ∈ K × G : xy = yx }|| K || G | . This is the probability that an element of G commutes with an elementof K (the relative commutativity degree of K in G ).This notion has been studied in several recent papers (see in par-ticular [
6, 14 ]). The main purpose of this paper is to establish thefollowing stronger form of P. M. Neumann’s theorem.
Theorem 1.2.
Let ǫ > , and let K be a subgroup of a finite group G such that P r ( K, G ) ≥ ǫ . Let H = h K G i . Then G has a nilpotentnormal subgroup T ≤ H of nilpotency class at most such that boththe index [ H : T ] and the order | [ T, T ] | of the commutator subgroup are ǫ -bounded. Here, as usual, h K G i denotes the normal closure of K in G . Theo-rem 1.1 can be obtained from the above result taking K = G .Theorem 1.2 has some interesting consequences. Surprizingly, thetheorem shows that however “insignificant” a subgroup K of G mightbe, the probability P r ( K, G ) has strong impact on the global structureof G . For example, it is straightforward that if G is simple, then theorder of G is bounded in terms of P r ( K, G ). Several other noteworthycorollaries of the theorem will be given in Section 4.The proof of Theorem 1.2 will be given in Section 3. In the nextsection we establish another extension of P. M. Neumann’s theorem.Throughout, γ i ( G ) stands for the i th term of the lower central seriesof G . HE COMMUTING PROBABILITY 3
Theorem 1.3.
Let ǫ > , and let H be a normal subgroup of a finitegroup G containing γ k ( G ) for some k ≥ . Suppose that P r ( H, G ) ≥ ǫ .Then G has a nilpotent normal subgroup T of nilpotency class at most k + 1 such that both the index [ G : T ] and the order of γ k +1 ( T ) are ǫ -bounded. It is easy to see that P. M. Neumann’s theorem is a particular caseof the above result (take k = 1).The main technical tools employed in the proofs of Theorem 1.2 andTheorem 1.3 are provided by the recent results [
1, 2, 3, 4 ] strength-ening B. H. Neumann’s theorem [ ] that says that the commutatorsubgroup of a BFC-group is finite.
2. Theorem 1.3
Our aim in this section is to furnish a proof of Theorem 1.3, butwe start with a couple of remarks on the result. So, let G and T beas in Theorem 1.3. The fact that both the index [ G : T ] and theorder of γ k +1 ( T ) are ǫ -bounded implies that for any x , . . . , x k ∈ T thecentralizer of the long commutator [ x , . . . , x k ] has ǫ -bounded indexin G . Therefore there is an ǫ -bounded number e such that G e – thesubgroup generated by all e th powers of elements of G – centralizes allcommutators [ x , . . . , x k ] where x , . . . , x k ∈ T . Then G = G e ∩ T isa nilpotent normal subgroup of nilpotency class at most k with G/G of ǫ -bounded exponent.If G is additionally assumed to be m -generator for some m ≥
1, then G has a nilpotent normal subgroup of nilpotency class at most k and( ǫ, m )-bounded index. Indeed, we know that for any x , . . . , x k ∈ T thecentralizer of the long commutator [ x , . . . , x k ] has ǫ -bounded index in G . An m -generator group has only ( i, m )-boundedly many subgroupsof any given index i [ , Theorem 7.2.9]. Therefore G has a subgroup J of ( ǫ, m )-bounded index that centralizes all commutators [ x , . . . , x k ]with x , . . . , x k ∈ T . Then J ∩ T is a nilpotent normal subgroup ofnilpotency class at most k and ( ǫ, m )-bounded index in G .These observations are in parallel with Shalev’s results on proba-bilistically nilpotent groups [ ].Given an element x ∈ G and a subgroup H ≤ G , we write x H forthe set of conjugates of x by elements from H . Our proof of Theorem1.3 requires the following result from [ ]. Theorem 2.1.
Let G a group such that | x γ k ( G ) | ≤ n for any x ∈ G .Then γ k +1 ( G ) has finite ( k, n ) -bounded order. ELOISA DETOMI AND PAVEL SHUMYATSKY
A proof of the following lemma can be found in Eberhard [ , Lemma2.1]. Lemma 2.2.
Let G be a finite group and X a symmetric subset of G containing the identity. Then h X i = X r provided ( r + 1) | X | > | G | . We can now prove Theorem 1.3.
Proof of Theorem 1.3.
Recall that H is a normal subgroup ofthe finite group G such that γ k ( G ) ≤ H and P r ( H, G ) ≥ ǫ . Let X bethe set of all elements x ∈ G such that | x H | ≤ /ǫ , that is: X = { x ∈ G | | C H ( x ) | ≥ ( ǫ/ | H |} . Note that ǫ | G || H | = |{ ( x, y ) ∈ G × H | xy = yx }|≤ X x ∈ X | H | + X x ∈ G \ X ǫ | H |≤ | X || H | + ( | G | − | X | ) ǫ | H | ≤ | X || H | + ǫ | G || H | . Therefore ( ǫ/ | G | < | X | . Set L = h X i . Clearly, | L | ≥ | X | > ( ǫ/ | G | and so the index of L in G is ǫ -bounded. Moreover, L is normal in G . As X is symmetricand (2 /ǫ ) | X | > | G | , it follows from Lemma 2.2 that every element of L is a product of at most 6 /ǫ elements of X . Since | x H | ≤ /ǫ for every x ∈ X , we conclude that | y H | ≤ (2 /ǫ ) /ǫ for every y ∈ L . Thus, as γ k ( G ) ≤ H , we have | y γ k ( L ) | ≤ (2 /ǫ ) /ǫ for every y ∈ L . Now Theorem 2.1 implies that γ k +1 ( L ) has ǫ -boundedorder.Let T = C L ( γ k +1 ( L )). As γ k +1 ( L ) has ǫ -bounded order, T has ǫ -bounded index in L , hence ǫ -bounded index in G . Moreover, γ k +1 ( T ) ≤ γ k +1 ( L ) is central in T . It follows that T is nilpotent of class at most k + 1. This completes the proof. (cid:3)
3. Theorem 1.2
We start this section by showing that if K is a subgroup of a finitegroup G and N is a normal subgroup of G , then P r ( KN/N, G/N ) ≥ P r ( K, G ). More precisely, we establish the following lemma.
Lemma 3.1.
Let N be a normal subgroup of a finite group G , andlet K ≤ G . Then P r ( K, G ) ≤ P r ( KN/N, G/N ) P r ( N ∩ K, N ) . HE COMMUTING PROBABILITY 5
This is an improvement over [ , Theorem 3.9] where the result wasobtained under the additional hypothesis that N ≤ K . Proof.
In what follows ¯ G = G/N and ¯ K = KN/N . Write ¯ K for the set of cosets ( N ∩ K ) h with h ∈ K . If S = ( N ∩ K ) h ∈ ¯ K ,write S for the coset N h ∈ ¯ K . Of course, we have a natural one-to-onecorrespondence between ¯ K and ¯ K .Write | K || G | P r ( K, G ) = X x ∈ K | C G ( x ) | = X S ∈ ¯ K X x ∈ S | C G ( x ) N || N | | C N ( x ) |≤ X S ∈ ¯ K X x ∈ S | C ¯ G ( xN ) || C N ( x ) | = X S ∈ ¯ K | C ¯ G ( S ) | X x ∈ S | C N ( x ) | == X S ∈ ¯ K | C ¯ G ( S ) | X y ∈ N | C S ( y ) | . If C S ( y ) = ∅ , then there is y ∈ C S ( y ) and so S = ( N ∩ K ) y .Therefore C S ( y ) = ( N ∩ K ) y ∩ C G ( y ) = C N ∩ K ( y ) y , whence | C S ( y ) | = | C N ∩ K ( y ) | . Conclude that | K || G | P r ( K, G ) ≤ X S ∈ ¯ K | C ¯ G ( S ) | X y ∈ N | C N ∩ K ( y ) | . Observe that X S ∈ ¯ K | C ¯ G ( S ) | = | K || N ∩ K | | G || N | P r ( ¯ K, ¯ G )and X y ∈ N | C N ∩ K ( y ) | = | N ∩ K || N | P r ( N ∩ K, N ) . It follows that
P r ( K, G ) ≤ P r ( ¯ K, ¯ G ) P r ( N ∩ K, N ), as required. (cid:3)
The following lemma will be useful (see for example [ , Lemma 2.1]for the proof). Lemma 3.2.
Let i, j be positive integers and G a group having asubgroup K such that | x G | ≤ i for each x ∈ K . Suppose that | K | = j .Then h K G i has finite ( i, j ) -bounded order. The next result partially explains why the probability
P r ( K, G ) hasstrong impact on the global structure of a finite group G even whenthe subgroup K is “unimportant”. ELOISA DETOMI AND PAVEL SHUMYATSKY
Lemma 3.3.
Let K be a subgroup of a finite group G and supposethat P r ( K, G ) = ǫ . Let H be the normal closure of K in G . Then theorder of H is bounded in terms of | K | and ǫ only. Proof.
We assume that K = 1 and use induction on | K | . Let m = min {| x G | | x ∈ K \ { }} . Thus 1 /m ≥ | C G ( x ) | / | G | for all x ∈ K \ { } . From ǫ | K | = X x ∈ K | C G ( x ) || G | = X x ∈ K \{ } | C G ( x ) || G | + 1 ≤ | K | − m + 1we deduce that ǫ | K | 6 = 1 and m ≤ | K | − ǫ | K | − . So there is a nontrivial element y ∈ K such that | y G | = m ≤ | K |− ǫ | K |− .Note that the order of y divides that of K . By Lemma 3.2 the orderof h y G i is bounded in terms of | K | and ǫ only. Taking into accountLemma 3.1 we now pass to the quotient G/ h y G i and the result followsby induction. (cid:3) The following theorem is taken from [ ]. It plays a crucial role inthe proof of Theorem 1.2. Theorem 3.4.
Let n be a positive integer, G a group having asubgroup K such that | x G | ≤ n for each x ∈ K , and let H = h K G i .Then the order of the commutator subgroup [ H, H ] is finite and n -bounded. We are now ready to prove Theorem 1.2.
Proof of Theorem 1.2.
Recall that K is a subgroup of the group G such that P r ( K, G ) ≥ ǫ . Set X = { x ∈ K ; | x G | ≤ /ǫ } and U = h X i . Note that ǫ | K || G | = ≤ |{ ( x, y ) ∈ K × G | xy = yx }| = X x ∈ K | C G ( x ) |≤ X x ∈ X | G | + X x ∈ K \ X ǫ | G |≤ | X || G | + ( | K | − | X | ) ǫ | G | . Therefore ǫ | K | ≤ | X | + ( ǫ/ | K | − | X | ), whence ( ǫ/ | K | < | X | and sothe index of U in K is ǫ -bounded. As X is symmetric and (2 /ǫ ) | X | > HE COMMUTING PROBABILITY 7 | K | , it follows from Lemma 2.2 that every element of U is a product ofat most 6 /ǫ elements of X . Therefore | u G | ≤ (2 /ǫ ) /ǫ for every u ∈ U .Let V = h U G i . Theorem 3.4 tells us that the commutator subgroup[ V, V ] has ǫ -bounded order. Recall that H = h K G i and, by Lemma3.1, P r ( KV /V, G/V ) ≥ P r ( K, G ) ≥ ǫ . Since the index [ K : U ] is ǫ -bounded, we deduce from Lemma 3.3 that so is the order of H/V .Now set T = C V ([ V, V ]). As [
V, V ] has ǫ -bounded order, T has ǫ -bounded index in V , hence ǫ -bounded index in H . Moreover, since[ T, T ] ≤ [ V, V ], we see that T is nilpotent of class at most 2 and | [ T, T ] | is ǫ -bounded. The proof is complete. (cid:3)
4. Some corollaries of Theorem 1.2
In this section we will record several easy corollaries of Theorem 1.2.Strong conclusions about the group G can be drawn once informationon the commuting probability of a Sylow subgroup is given. Theorem 4.1.
Let P be a Sylow p -subgroup of a finite group G such that P r ( P, G ) = ǫ . Then G has normal subgroups L ≤ H suchthat L is a p -subgroup of class at most , G/H is a p ′ -group, and boththe index [ H : L ] and the order | [ L, L ] | are ǫ -bounded. Proof.
Let H be the normal closure of P in G . Obviously, G/H is a p ′ -group. Theorem 1.2 tells us that G has a nilpotent normalsubgroup T ≤ H of class at most 2 such that both the index [ H : T ]and the order | [ T, T ] | are ǫ -bounded. Let L be the Sylow p -subgroupof T . Set ¯ G = G/L and let ¯ P be the image of P in G . Obviously, theorder of ¯ P is at most the index [ H : T ], which is ǫ -bounded. In viewof Lemma 3.1 P r ( ¯
P , ¯ G ) ≥ ǫ . Taking into account Lemma 3.3 deducethat the image of H in ¯ G has ǫ -bounded order. Hence the result. (cid:3) Curiously, once we have information on the commuting probabilityof all Sylow subgroups of G , the result is as strong as in P.M. Neumann’stheorem. Theorem 4.2.
Let G be a finite group such that P r ( P, G ) = ǫ whenever P is a Sylow subgroup. Then G has a nilpotent normal sub-group T of nilpotency class at most such that both the index [ G : T ] and the order | [ T, T ] | of the commutator subgroup are ǫ -bounded. Proof.
For each prime p ∈ π ( G ) let H p denote the normal closureof a Sylow p -subgroup in G . Theorem 4.1 shows that G has a normal p -subgroup L p of class at most 2 such that both [ H p : L p ] and | [ L p , L p ] | are ǫ -bounded. Since the bounds on [ H p : L p ] and | [ L p , L p ] | do notdepend on p , it follows that there is an ǫ -bounded constant C such that ELOISA DETOMI AND PAVEL SHUMYATSKY H p = L p and [ L p , L p ] = 1 whenever p ≥ C . Obviously G = Q p ∈ π ( G ) H p .Set T = Q p ∈ π ( G ) L p . In view of the above T is of class at most 2 andboth [ G : T ] and | [ T, T ] | are ǫ -bounded. (cid:3) Recall that any finite soluble group G has a Sylow basis — a familyof pairwise permutable Sylow p i -subgroups P i of G , exactly one for eachprime divisor of the order of G , and any two Sylow bases are conjugate.The system normalizer (also known as the basis normalizer) of such aSylow basis in G is the intersection L = T i N G ( P i ). If G is a finitesoluble group and L is a system normalizer in G , then G = h L G i (see [ , Theorem 9.2.8]). A Carter subgroup of a group is a self-normalizing nilpotent subgroup. Every finite soluble group contains aCarter subgroup. In turn, every Carter subgroup of a finite solublegroup G contains a system normalizer of G .Therefore we deduce Theorem 4.3.
Let K be either a Carter subgroup or a system nor-malizer in a finite soluble group G . Suppose that P r ( K, G ) = ǫ . Then G has a nilpotent normal subgroup T of nilpotency class at most suchthat both the index [ G : T ] and the order | [ T, T ] | are ǫ -bounded. Proof.
This is immediate from Theorem 1.2 and the fact that G = h K G i . (cid:3) If φ is an automorphism of a group G , then the centralizer C G ( φ ) isthe subgroup formed by the elements x ∈ G such that x φ = x . In thecase where C G ( φ ) = 1 the automorphism φ is called fixed-point-free.A famous result of Thompson [ ] says that a finite group admittinga fixed-point-free automorphism of prime order is nilpotent. Higmanproved that for each prime p there exists a number h = h ( p ) dependingonly on p such that whenever a nilpotent group G admits a fixed-point-free automorphism of order p , it follows that G is nilpotent of class atmost h [ ]. Therefore the nilpotency class of a finite group admittinga fixed-point-free automorphism of order p is at most h .An automorphism φ of a finite group G such that ( | G | , | φ | ) = 1is called coprime. For coprime automorphisms we have the equality C G/N ( φ ) = C G ( φ ) N/N for each normal φ -invariant subgroup N of G . Theorem 4.4.
Let ǫ > and G be a finite group admitting acoprime automorphism φ of prime order p such that P r ( C G ( φ ) , G ) ≥ ǫ .Let h = h ( p ) . Then G has a nilpotent normal subgroup T of class atmost such that both [ T, T ] and γ h +1 ( G/T ) have ǫ -bounded order. Proof.
Let H be the normal closure of C G ( φ ). Theorem 1.2 tells usthat G has a nilpotent normal subgroup T ≤ H of class at most 2 such HE COMMUTING PROBABILITY 9 that both the index [ H : T ] and the order | [ T, T ] | of the commutatorsubgroup are ǫ -bounded. Note that φ acts fixed-point-freely on thequotient G/H and so
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