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Bounding the Number of Classes of a Finite Group in Terms of a Prime

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Bounding the Number of Classes of a Finite Group in Terms of a Prime J. Group Theory 23 (2020), 471–488 DOI 10.1515/jgth-2019-0144 © de Gruyter 2020 Bounding the number of classes of a finite group in terms of a prime Attila Maróti and Iulian I. Simion Communicated by Michael Giudici Abstract. Héthelyi and Külshammer showed that the number of conjugacy classes k.G/ of any solvable finite group G whose order is divisible by the square of a prime p is at least .49p 1/=60. Here an asymptotic generalization of this result is established. It is proved thatC there exists a constant c > 0 such that, for any finite group G whose order is divisible by the square of a prime p, we have k.G/ cp. 1 Introduction Let k.G/ denote the number of conjugacy classes of a finite group G. This is also the number of complex irreducible characters of G. Bounding k.G/ is a funda- mental problem in group and representation theory. Let G be a finite group and p a prime divisor of the order G of G. In this j j paper, we discuss lower bounds for k.G/ only in terms of p. Pyber observed that results of Brauer [1] imply that G contains at least 2pp 1 conjugacy classes provided that p2 does not divide G . Building on works of j j Héthelyi and Külshammer [8], Malle [17], Keller [12], Héthelyi, Horváth, Keller and Maróti [7], it was shown in [19] that k.G/ 2pp 1 for any finite group G and any prime p dividing G , with equality if and only if pp 1 is an integer, Ì j j G Cp Cpp 1 and CG.Cp/ Cp. D D The objective of the current paper is to provide a stronger lower bound for k.G/ in the case where p2 divides G . Héthelyi and Külshammer [9] showed that, for j j any finite solvable group G and any prime p such that p2 divides G , the number j j The work of the first author leading to this application has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation pro- gramme (grant agreement No. 741420). The first author was supported by the National Research, Development and Innovation Office (NKFIH) Grant No. K132951 and No. K115799 and by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. 472 A. Maróti and I. I. Simion of conjugacy classes of G is at least .49p 1/=60. This bound is sharp [9] for C infinitely many primes p; however, it does not generalize [8] to arbitrary finite groups since there are infinitely many non-solvable groups G and primes p with k.G/ 0:55p 0:05. D The main result of this paper is the following. Theorem 1.1. There exists a constant c > 0 such that, for any finite group G whose order is divisible by the square of a prime p, we have k.G/ cp. Questions of Pyber and the papers [8,9] of Héthelyi and Külshammer motivated our result. Let B be a p-block of a finite group G, and let D be a defect group of B. The number k.B/ of complex irreducible characters of G associated to the block B is a lower bound for k.G/. A recent result of Otokita [20, Corollary 4] states that k.B/ .pm p 2/=.p 1/, where pm denotes the exponent of the center C of D. Finally, note that Kovács and Leedham-Green [14] have constructed, for every odd prime p, a finite p-group G of order pp with k.G/ 1 .p3 p2 p 1/ D 2 C C (see also [21]). 2 Affine groups The purpose of this section is to prove Proposition 2.2. For this, we need the fol- lowing lemma. The base of the logarithms in this paper is always 2. Lemma 2.1. Let H be a finite group and V a finite, faithful, completely reducible H-module over a finite field of characteristic p. Assume that H has no composi- tion factor isomorphic to an alternating group of degree larger than .log p/3 and has no composition factor isomorphic to a simple group of Lie type defined over a field of characteristic p. Put pn V . Then H has an abelian subgroup of 7.n 1/ D j j index at most .c1 log p/ for some universal constant c1 > 1. Note that, once Lemma 2.1 is proved, it may be extended by a theorem of Chermak and Delgado [11, Theorem 1.41] as follows. Under the conditions of Lemma 2.1, the group H contains a characteristic abelian subgroup of index at 14.n 1/ most .c1 log p/ for some universal constant c1 > 1 . Proof of Lemma 2.1. Assume first that V is a primitive and irreducible H -module. We use the following structure result which is implicit in the proofs of [6] (see for example the proof of [6, Theorem 9.1]). Let F be the largest field such that Bounding the number of classes in terms of a prime 473 H embeds in LF .V /. Let C be the subgroup of non-zero elements in F . We 7.n 1/ claim that H=.H C/ .c1 log p/ for some universal constant c1 > 1. j \ j Ä For this, we may assume that C H . Ä Let H0 be the centralizer of C in H, and let R be a normal subgroup of H contained in H0 minimal with respect to not being contained in C (if such exists). There are two possibilities for R. It is of symplectic type and R=Z.R/ r2a for j j D some prime r and integer a such that r divides F 1 or R is a central product j j of t isomorphic quasisimple groups. Choose a maximal collection J1;:::;Jm of such non-cyclic normal subgroups in H0 which pairwise commute (if such exist). Let J be the central product of the subgroups J1;:::;Jm. Then H0=.C Sol.J // embeds in the direct product of the automorphism groups of Ji =Z.Ji /, where Sol.J / denotes the solvable radical of J . (Note that, in the proof of [6, Theorem 9.1], it was falsely asserted that H0=C embeds in the direct product of the automorphism groups; however, this did not affect the proof of [6, Theorem 9.1] nor [6, Theorem 10.1].) Let W be an irreducible constituent of V for the normal subgroup J of H (provided that J is non-trivial). Since H is primitive on V , it follows that J acts homogeneously on V by Clifford’s theorem. Let E EndFJ .W /. Now we D have W U1 Um, where Ui is an absolutely irreducible EJi -module by Š ˝ ˝ [13, Lemma 5.5.5]. Notice that E may be viewed as a subfield of EndFJ .V /, and since J is normal in H, the multiplicative group of E is normalized by H. Our choice of F implies that E F . If Ji is of symplectic type with Ji =Z.Ji / 2ai aiD of order r , then dim Ui r . If Ji is a central product of t isomorphic qua- i D i sisimple groups Qi;j with 1 j t, then Ui Ui;1 Ui;t , where Ui;j is Ä Ä Š ˝ ˝ an absolutely irreducible (faithful) FQi;j -module for every j with 1 j t, by Ä Ä [13, Lemma 5.5.5 and Lemma 2.10.1]. f Let F p and d dimF V . The product of the orders of all abelian com- j j D D position factors in any composition series of the factor group H=C is less than 2 log d 3 2 log n 4 f d C n C by [6, Theorem 10.1] and its proof. This is at most n 1 Ä .c2 log p/ for some constant c2 > 2. We may now assume that J 1 and ¤ n > 1. Let b.X/ denote the product of the orders of all non-abelian composition factors n 1 in any composition series of a finite group X. Since H=C .c2 log p/ b.H /, j j Ä we proceed to bound b.H /. Without loss of generality, assume that J1;:::;Jk are groups of symplectic 2ai type with k 0 and Ji =Z.Ji / r for some primes ri and integers ai , and j j D i assume that Jk 1;:::;Jm are groups not of symplectic type. For each ` with k 1 ` m,C let J be a central product of t copies, say Q ;:::;Q , of C Ä Ä ` ` `;1 `;t` a quasisimple group Q . In this case, U U U , where U is an ` ` Š `;1 ˝ ˝ `;t` `;j irreducible (faithful) Q -module for every j with 1 j t . Using this nota- `;j Ä Ä ` 474 A. Maróti and I. I. Simion tion, we may write the following: k m ÂY à  Y à n d dim V dim W dim Ui dim U D D ` i 1 ` k 1 D D C k m k  à  à  à Pm Y ai Y t` Y ai t` ri .dim U / ri 2 ` k 1 : (2.1) D `;1 D C i 1 ` k 1 i 1 D D C D Since H=H0 and C Sol.J / are solvable, b.H / b.H0=.C Sol.J ///. Recall D from the third paragraph of this proof that the group H0=.C Sol.J // embeds in the direct product of the automorphism groups of the Ji =Z.Ji /.
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