conjugacy classes of small sizes in the linear and...

CONJUGACY CLASSES OF SMALL SIZESIN THE LINEAR AND UNITARY GROUPS

SHAWN T. BURKETT AND HUNG NGOC NGUYEN

Abstract. Using the classical results of Wall on the parametriza-tion and sizes of (conjugacy) classes of finite classical groups, wepresent some gap results for the class sizes of the general lineargroups and general unitary groups as well as their variations. Inparticular, we identify the classes in GLn(q) of size up to q4n−8 andclasses in GUn(q) of size up to q4n−9. We then apply these gapresults to obtain some bounds and limits concerning the zeta-typefunction encoding the conjugacy class sizes of these groups.

1. Introduction and results

The problem of determining the irreducible characters (includingboth complex and Brauer characters) of relatively small degrees of finiteclassical groups and more generally finite groups of Lie type has beenstudied extensively over the last two decades. The results obtained haveproved very useful in various applications. The irreducible complexcharacters of finite Lie-type groups of small degrees were studied in [12,16] and more recently in [15]. In these papers, the irreducible complexcharacters of degrees up to a certain bound are classified and it is provedthat there is a large gap between the degrees of these characters andthe next degree. For that reason, these results are often referred to asthe gap results on character degrees of finite groups.

Here we turn our attention to the problem of determining the con-jugacy classes of relatively small sizes of finite classical groups.

Conjugacy classes of finite classical groups have been studied sincethe classical work of Wall [18]. There indeed has been several waysto formulate and estimate the class sizes of finite classical groups. Forinstance, besides the well-known class size formulas of Wall, it is shownby Liebeck and Shalev [10] that the size of the class of an element g ina classical group G can be presented in terms of the codimension of thelargest eigenspace of g on the natural G-module. Burness in the series

Date: February 12, 2014.2010 Mathematics Subject Classification. Primary 20E45; Secondary 20D06.Key words and phrases. Linear group, unitary group, conjugacy class size.

1

2 SHAWN T. BURKETT AND HUNG NGOC NGUYEN

of papers [1, 2, 3, 4] has used this formulation to estimate the classsizes of prime order elements and then to study the fixed point ratiosin actions of finite classical groups. Recently, Fulman and Guralnick [8]have provided an upper bound for the largest class size in various finiteclassical groups.

We approach the problem of small size classes as follows. Let G bea finite classical group and let g ∈ G. Then g has a unique Jordandecomposition g = su = us where s ∈ G is semisimple and u ∈ Gis unipotent. That is, s is diagonalizable over the algebraic closureof the underlying field of G and all eigenvalues of u are equal to 1.Let clG(g) denote the conjugacy class of G containing g. The Jordandecomposition of conjugacy classes says that

clG(g)↔ (clG(s), clCG(s)(u))

is a one-to-one correspondence, where CG(s) is the centralizer of s inG. Furthermore,

|clG(g)| = |clG(s)||clCG(s)(u)|.The structure of the centralizers of semisimple elements in finite classi-cal groups are well-known and due to Carter [6]. These centralizers areisomorphic (or close) to a product of classical groups of lower dimen-sion. Therefore, in order to find small size classes, one should identifythe semisimple classes and unipotent classes of small size. The formulasdescribing the class sizes of finite classical groups are often complicated(see (2.1) and (2.2) below) and it is difficult to derive the classes of sizeup to a certain bound. However, when reducing to unipotent andsemisimple classes, we are able to estimate those formulas.

Our first main results are:

Theorem 1.1. Let n ≥ 6 and q be a prime power. Then GLn(q) has

(i) q − 1 (central ) classes of size 1,(ii) q − 1 classes of size (qn−1 − 1)(qn − 1)/(q − 1), and

(iii) q2 − 3q + 2 (semisimple) classes of size qn−1(qn − 1)/(q − 1).

Moreover, all other classes have sizes greater than q4n−8.

Theorem 1.2. Let n ≥ 5 and q be a prime power. Then GUn(q) has

(i) q + 1 (central ) classes of size 1,(ii) q + 1 classes of size (qn−1 − (−1)n−1)(qn − (−1)n)/(q + 1), and

(iii) q2 + q (semisimple) classes of size qn−1(qn − (−1)n)/(q + 1).

Moreover, all other classes have sizes greater than q4n−9.

Theorem 1.1 and Theorem 1.2 will be proved in Sections 3 and 5,respectively. We also derive in Sections 4 and 6 some corresponding

SMALL SIZE CLASSES IN THE CLASSICAL GROUPS 3

gap results for other variations of the linear and unitary groups. Wenote that the condition on n is necessary and it is indeed not difficultto work out all the class sizes of the classical groups in low dimension.The symplectic and orthogonal groups require more delicate treatmentand as the paper is long enough, we leave these groups to another time.

For a finite group G, let C(G) denote the set of all conjugacy classesof G. Following [11], we define the zeta-type function encoding theconjugacy class sizes of G as follows:

ζG(t) =∑

C∈C(G)

|C|−t for t ∈ R.

This function, together with its dual analogue for character degrees,have appeared in several contexts. For instance, it was used by Liebeckand Shalev in [11, §5 and §7] to obtain some probabilistic results onbase sizes for primitive actions of almost simple classical groups of di-mensions high enough. In [5, 10], a slightly different zeta-type functionwhere the sum is taken over all conjugacy classes of elements of primeorder in G was defined and applied to obtain upper bounds on basesizes for almost simple primitive permutation actions. For more detailson this, we refer the reader to [5, 10, 11] and references therein.

Let ks(G) denote the number of classes of G of size s. The zeta-typefunction can be rewritten as

ζG(t) =∑s∈N

ks(G)s−t.

When t is positive, we see that the main contribution of the abovesum is the terms where the size s is small. Therefore, knowing theconjugacy classes of small sizes of G will provide a good estimation forζG(t).

Using the obtained results on conjugacy classes of relatively smallsizes of the linear and unitary groups, we prove in Section 7 someresults on upper bounds and limits involving their associated zeta-typefunctions. One such result is the following:

Theorem 1.3. Let G = GLn(q) or GUn(q) where n ≥ 6. If t > 1/3,then

ζG(t)

q→ 1 as q →∞.

Moreover, the convergence is uniformly in n.


2. Preliminaries

2.1. Linear groups. Let V be a vector space of dimension n over afield Fq of q elements. Let G be the group of invertible linear transfor-mations of V , i.e. G is the general linear group GLn(q). Let Fq(x) bethe ring of polynomials over Fq with variable x. For each element g inG, define an action of Fq(x) on V by v · x = vg for all v ∈ V . Thismakes V become an Fq(x)-module. Denote this module by Vg. It iseasy to see that two elements g and h of G are conjugate if and only ifVg and Vh are isomorphic as Fq(x)-modules, see [13].

Let M be the set of monic irreducible polynomials in Fq[x] dif-ferent from x and let P be the set of partitions. Since Fq(x) is aprinciple domain, the Fq(x)-module Vg is a direct sum of modulesof the form Fq(x)/(f)m where m ≥ 1 and (f) is the ideal generatedby f ∈ M. In other words, Vg ∼=

⊕f∈M,i∈N Fq(x)/(f)λi(f), where

λ(f) = (λ1(f), λ2(f), ...) is a partition. Therefore, we get a functionλ :M→ P such that

∑f∈M deg(f)|λ(f)| = n. Denoting this function

λ by λg, we now obtain a one-to-one correspondence cl(g) 7→ λg be-tween the set of conjugacy classes of G and the set of partition-valuedfunctions λ such that ∑

f∈M

deg(f)|λ(f)| = n.

For reader convenience, we recall some standard notations and ter-minologies of partitions. Let α = (α1, α2, ...) be a partition, i.e. α1 ≥α2 ≥ · · ·. Each positive number αi is called a part of α. The size of α,which is α1 +α2 + · · · will be denoted by |α|. Let mi(α) be the numberof parts of α of size i and let α′ be the dual partition of α in the sensethat α′i = mi(α) +mi+1(α) + · · ·. If a part a in a partition is repeatedk times, we will write (..., ak, ...) instead of (..., a, a, ..., a, ...).

It is well known that each conjugacy class of G is determined byits rational canonical form. The one-to-one correspondence betweenrational canonical forms and partition-valued functions onM is known,see [7] for instance. From this correspondence, we see that an elementg ∈ G is semisimple if and only if all parts of partitions λg(f), f ∈M,are at most 1. Also, g ∈ G is unipotent if and only if |λg(f)| = 0 for allf ∈M except when f is the polynomial x−1. In particular, unipotentclasses of G are parameterized by partitions of n.

Suppose that a is a nonzero real number, m is a positive integer, andα = (α1, α2, ...) is a partition. Let (a)m denote (1−a)(1−a2) · · · (1−am)

and h(a, α) denote aΣi(α′i)

2∏i(1/a)mi(α). We also set (a)0 = 1. If λ = λg

is the partition-valued function corresponding to g ∈ G, the size of the


centralizer CG(g) is (see [7, 9, 14]):

(2.1)∏f∈M

qdeg(f)∑

i(λ(f)′i)2∏i

(1/qdeg(f))mi(λ(f)) =∏f∈M

h(qdeg(f), λ(f)).

A lower bound for h(a, α) where α ∈ P is obtained by Fulman andGuralnick in [8, Lemma 6.1]. Since we are working on conjugacy classesof small sizes, we need to maximize h(a, α).

Lemma 2.1. Let α be a partition of size k ≥ 4. If a ≥ 2 and α is not(1k) or (2, 1k−2), then h(a, α) ≤ h(a, (22, 1k−4)).

Proof. Suppose that α = (α1, α2, ..., αm) 6= (1k) where α1 ≥ α2 ≥· · ·αm ≥ 1 and i is the largest index so that t := αi ≥ 2. Consider thepartition β = (β1, β2, ..., βm, 1) of size k such that βj = αj for all j 6= iand βi = αi − 1. We will show as follows that h(a, β) > h(a, α). Ift = 2, then

h(a, β)

h(a, α)=aΣi(β

′i)

2(1/a)m1(α)+2(1/a)m2(α)−1

aΣi(α′i)2(1/a)m1(α)(1/a)m2(α)

≥ a2(1− 1/am1(α)+1)(1− 1/am1(α)+2)

1− 1/am2(α)

> a2(1− 1/a)2 ≥ 1.

Similarly, if t > 2 then

h(a, β)

h(a, α)=aΣi(β

′i)

2(1/a)m1(α)+1(1/a)mt(α)−1(1/a)mt−1(α)+1

aΣi(α′i)2(1/a)m1(α)(1/a)mt(α)(1/a)mt−1(α)

≥ a2(1− 1/am1(α)+1)(1− 1/amt−1(α)+1)

1− 1/amt(α)

> a2(1− 1/a)2 ≥ 1.

Since α is not (1k) or (2, 1k−2), after some transforms similar to theone above, we will get to either the partition (22, 1k−4) or (3, 1k−3).Therefore,

h(a, α) ≤ max{h(a, (22, 1k−4)), h(a, (3, 1k−3))}.Observe that

h(a, (22, 1k−4)) = ak2−4k+8(1/a)k−4(1− 1/a)(1− 1/a2)

=a2(1− 1/a2)

1− 1/ak−3h(a, (3, 1k−3))

> a2(1− 1/a2)h(a, (3, 1k−3))

> h(a, (3, 1k−3))


and hence the lemma follows. �

2.2. Unitary groups. The general unitary group GUn(q) is the sub-group of all transformations of GLn(q2) that fix a non-singular Her-mitian form on Fq2 . The following description of conjugacy classesof GUn(q) and their sizes is due to Wall [18]. Let N be the set ofmonic irreducible polynomials in Fq2 [x] different from x. For eachf(x) = xdeg(f) + adeg(f)−1x

deg(f)−1 + · · · + a1x + a0 in N , the conju-

gate polynomial of f , denoted by f , is defined to be

f(x) = xdeg(f) + (a1/a0)qxdeg(f)−1 + · · ·+ (adeg(f)−1/a0)qx+ (1/a0)q.

Similar to the general linear groups, there is a one-to-one correspon-dence cl(g) 7→ λg between the set of conjugacy classes of GUn(q) andthe set of partition-valued functions λ : N → P such that

∑f∈N deg(f)|λ(f)| =

n and λ(f) = λ(f). The order of the corresponding centralizer CGUn(q)(g)is (see [8, 18]):

(2.2)

∏f∈N ,f=f q

deg(f)∑

i(λ(f)′i)2∏

i(−1/qdeg(f))mi(λ(f))

·(∏

f∈N ,f 6=f qdeg(f)

∑i(λ(f)′i)

2∏i(1/q

deg(f))mi(λ(f)))1/2

q 7→q2 ,

where q 7→ q2 means that all occurrences of q are replaced by q2. Ifl(a, α) denotes aΣi(α

′i)

2∏i(−1/a)mi(α), then the above centralizer order

is ∏f∈N ,f=f

l(qdeg(f), λ(f)) ·∏

f∈N ,f 6=f

(h(qdeg(f), λ(f)))1/2

q 7→q2 .

A lower bound for l(a, α) where α ∈ P is given in [8, Lemma 6.5].Here again we need to maximize l(a, α).

Lemma 2.2. Let α be a partition of size k ≥ 4. If a ≥ 2 and α is not(1k) or (2, 1k−2), then l(a, α) ≤ l(a, (22, 1k−4)).

Proof. This is similar to the proof of Lemma 2.1. �

We also need the following known result on the centralizers of semisim-ple elements in the unitary groups.

Lemma 2.3 ([6, 17]). Let s be a semisimple element of GUn(q) withcharacteristic polynomial given by the product

t∏i=1

(fi fi)ai

r∏i=1

gbii

where each fi, fi are distinct monic irreducible polynomials over Fq2of degree ki and each gi is a distinct monic irreducible polynomial over


Fq2 of degree 2mi − 1. Then

CGUn(q)(s) ∼= GLa1(q2k1)×GLa2(q

2k2)× · · · ×GLat(q2kt)×

GUb1(q2m1−1)×GUb2(q

2m2−1)× · · · ×GUbr(q2mr−1),

where n =∑bi(2mi − 1) +

∑2aiki.

Remark 2.4. The number of GLai(q2k) appearing in the product is at

most one half of the number of monic irreducible polynomials f over Fq2of degree d such that f 6= f and the number of GUbi(q

2m−1) appearingin the product is at most the number of self-conjugate monic irreduciblepolynomials over Fq2 of degree 2m− 1.

3. The general linear groups

First, we determine unipotent classes of small sizes of GLn(q).

Lemma 3.1. Let u be a unipotent element of GLn(q) with n ≥ 5. Thenone of the following holds:

(i) |cl(u)| = 1 (one class),(ii) |cl(u)| = (qn−1 − 1)(qn − 1)/(q − 1) (one class),

(iii) |cl(u)| > q4n−8.

Proof. As mentioned above, since u is unipotent, λu(f) is empty for allf 6= x − 1. Suppose that λu(x − 1) = α = (α1, α2, ...), a partition ofsize n. Formula (2.1) gives us the size of CG(u):

|CG(u)| = q∑

i(α′i)

2∏i

(1/q)mi(α) = h(q, α).

Remark that the partitions (1n) and (2, 1n−2) correspond respectivelyto the conjugacy class of the identity element of size 1 and a conjugacyclass of size

|GLn(q)|h(q, (2, 1n−2))

=(qn−1 − 1)(qn − 1)

q − 1.

If α is different from (1n) and (2, 1n−2), then Lemma 2.1 implies that

|GLn(q)|h(q, α)

≥ |GLn(q)|h(q, (22, 1n−4))

=qn(n−1)/2(q − 1)(q2 − 1) · · · (qn − 1)

q(n−2)2+22(1/q)n−4(1/q)2

=q(qn−3 − 1)(qn−2 − 1)(qn−1 − 1)(qn − 1)

(q − 1)(q2 − 1)> q4n−8

for every n ≥ 5, as desired. �


Remark 3.2. 1) From the correspondence between Jordan canonicalforms and partition-valued functions, we see that the Jordan canonicalform associated to the conjugacy class of size (qn−1−1)(qn−1)/(q−1)in Lemma 3.1 is

diag(J(2, 1), J(1, 1), . . . , J(1, 1)).

2) By [8, Lemma 6.1], h(q, α) ≥ q|α|(1 − 1/q). Moreover, followingthe proof, we see that the equality happens if and only if α = (|α|).Therefore, GLn(q) has a unique conjugacy classes of unipotent elementsof maximal size

GLn(q)

h(q, (n))= q(n−1)2(q2 − 1)(q3 − 1) · · · (qn − 1).

We now determine semisimple classes of small sizes of GLn(q).

Lemma 3.3. Let s be a semisimple element of G := GLn(q) withn ≥ 6. Then one of the following holds:

(i) |cl(s)| = 1 (q − 1 classes),(ii) |cl(s)| = qn−1(qn − 1)/(q − 1) (q2 − 3q + 2 classes),(iii) |cl(s)| > q4n−8.

Proof. The structure of centralizers of semisimple elements in the clas-sical groups is described in [6] and in more detailed in [17]. If thecharacteristic polynomial of s is a product

∏ti=1 f

aii (x), where each fi

is a distinct monic irreducible polynomial over Fq of degree ki, then

CG(s) ∼= GLa1(qk1)×GLa2(q

k2)× · · · ×GLat(qkt),

where∑t

i=1 aiki = n and the number of GLa(qk) appearing in the

product is at most the number of monic irreducible polynomials overFq of degree k.

1) Case t = 1: We know that if CG(s) = GLn(q) then s ∈ Z(G) andhence |cl(s)| = 1. Since |Z(G)| = q − 1, G has q − 1 central classes ofsize 1. Now we can assume that CG(s) = GLa(q

k) with ak = n andk ≥ 2. In this case, we have(3.3)

|cl(s)| = |GLn(q)||GLa(qk)|

=qn(n−1)/2

∏nj=1(qj − 1)

qka(a−1)/2∏a

j=1(qkj − 1)>qn(n−1)/2

∏nj=1(qj − 1)

qka2.

Observe that, as q ≥ 2,

n∏j=1

(qj−1) =n∏j=1

qj(

1− 1

qj

)= qn(n+1)/2

n∏j=1

(1− 1

qj

)> qn(n+1)/2

∞∏j=1

(1− 1

qj

)


> qn(n+1)/2

(1− 1

q− 1

q2

)= qn(n+1)/2−2(q2 − q − 1) ≥ qn(n+1)/2−2.

Therefore, (3.3) implies that

|cl(s)| > qn(n−1)/2qn(n+1)/2−2

qka2=qn

2−2

qna= qn

2−na−2.

Since a ≤ n/2, it follows that |cl(s)| > qn2/2−2 ≥ q4n−8 for every n ≥ 6.

2) Case t = 2: First we consider the subcase CG(s) = GL1(q) ×GLn−1(q). This happens if and only if q ≥ 3 and the characteristicpolynomial of s is (x − λ1)(x − λ2)n−1 for some λ1 6= λ2 in Fq. Thisyields q2−3q+2 different classes of semisimple elements of size qn−1(qn−1)/(q − 1) and hence (ii) holds.

Next we assume CG(s) ∼= GL1(q)×GLa(qk), where k ≥ 2 and ak =

n− 1. We then have

|cl(s)| =qn(n−1)/2

∏nj=1(qj − 1)

(q − 1)qak(a−1)/2∏a

j=1(qkj − 1)>qn(n−1)/2

∏nj=1(qj − 1)

qka2+1

>qn(n−1)/2qn(n+1)/2−2

qka2+1≥ qn

2−2

q(n−1)2/2+1> q4n−8

for every n.Finally, we consider the subcase CG(s) ∼= GLa1(q

k1) × GLa2(qk2)

where a1k1 and a2k2 are greater than 1. Since |GLai(qki)| ≤ |GLaiki(q)|

for i = 1, 2, we obtain |CG(s)| ≤ |GLn1(q)×GLn2(q)| for some integersn1 and n2 such that n1 + n2 = n and 1 /∈ {n1, n2}. Then

|cl(s)| ≥ |GLn(q)||GLn1(q)||GLn2(q)|

=(qn − 1)(qn − q) · · · (qn − qn1−1)qn1(n−n1)

|GLn1(q)|

=(qn − 1)(qn−1 − 1) · · · (qn−(n1−1) − 1)

(qn1 − 1)(qn1−1 − 1) · · · (q − 1)qn1(n−n1)

> (qn−n1)n1qn1(n−n1) = q2n1(n−n1) ≥ q4n−8.

This last inequality comes from the observation that 2n1(n − n1) isminimized when n1 = 2 or n1 = n− 2.

3) Case t ≥ 3: Choose a subset T ⊆ {1, 2, · · · , t} so that∑

i∈T aiki ≥2 and

∑i/∈T aiki ≥ 2. We then have

|cl(s)| = |GLn(q)|∏i∈T |GLai(q

ki)|∏

i/∈T |GLai(qki)|≥ |GLn(q)||GL∑

i∈T aiki(q)||GL∑

i/∈T aiki(q)|

.

As proved in the previous case, the last fraction is greater than q4n−8,as desired. �


We are now ready to prove the first main theorem.

Proof of Theorem 1.1. Let g ∈ G := GLn(q) and let g = su = uswhere s ∈ G is semisimple and u ∈ G is unipotent be the Jordandecomposition of g. Since CG(g) = CG(s) ∩ CG(u), we get

|CG(g)| ≤ min{|CG(s)|, |CG(u)|}.

Assuming that |cl(g)| ≤ q4n−8, then we have |cl(s)| ≤ q4n−8. FromLemma 3.3, we consider two cases:

1) |cl(s)| = 1. Then s is one of the q − 1 central elements in G.Therefore, |cl(u)| = |cl(g)| ≤ q4n−8. By Lemma 3.1, u either is identityor belongs to the unipotent class of size (qn−1−1)(qn−1)/(q−1). Now(i) holds in the former case and (ii) holds in the latter case.

2) |cl(s)| = qn−1(qn − 1)/(q − 1). Then q ≥ 3 and, in particular,CG(s) ∼= GL1(q)×GLn−1(q). If u = 1 then we get q2−3q+2 conjugacyclasses of semisimple elements and hence (iii) holds. So we can assumeu 6= 1.

Since u commutes with s, we can consider u as a unipotent elementof CG(s) ' GL1(q)×GLn−1(q). Using Lemma 3.1, we have

|clCG(s)(u)| ≥ (qn−2 − 1)(qn−1 − 1)

q − 1.

It follows that

|cl(g)| = |GLn(q)||CG(g)|

=|GLn(q)||CG(s)|

· |CG(s)||CCG(s)(u)|

= |cl(s)| · |clCG(s)(u)|

≥ qn−1(qn − 1)

q − 1· (qn−2 − 1)(qn−1 − 1)

q − 1> q4n−8

for every n ≥ 6, as claimed. �

4. Other linear groups

In this section, we derive from Theorem 1.1 some corresponding gapresults on conjugacy class sizes of SLn(q), PGLn(q), and PSLn(q).

4.1. The special linear groups.

Corollary 4.1. Let n ≥ 6. Then SLn(q) has gcd(n, q − 1) (central )classes of size 1, gcd(n, q−1) classes of size (qn−1−1)(qn−1)/(q−1),q − 1− gcd(n, q − 1) (semisimple) classes of size qn−1(qn − 1)/(q − 1)and all other classes have sizes greater than q4n−9.


Proof. Let G := GLn(q) and S := SLn(q). It is clear that |Z(S)| =gcd(n, q− 1) and therefore S has exactly gcd(n, q− 1) classes of size 1.Let g ∈ S ≤ G and g is not central in G.

Suppose that g belongs to a G-conjugacy class of size (qn−1−1)(qn−1)/(q − 1). From the proof of Theorem 1.1, we know that g = au,where a ∈ F×q and u is unipotent with λu(x − 1) = (2, 1n−2) andλu(f) is empty for every f 6= x − 1. Without loss, assume that u =diag(J(2, 1), J(1, 1), . . . , J(1, 1)). Define

ϕ : CG(u) → F×qx 7→ det(x)

.

For each k ∈ F×q , the matrix diag(1, 1, k, 1, . . . , 1) commutes with u hasdeterminant k. Therefore ϕ is surjective and it follows that |CG(u)| =(q−1)|CS(u)|. Thus, |clS(u)| = (qn−1−1)(qn−1)/(q−1). We note thatthere are gcd(n, q− 1) choices for a and, hence, gcd(n, q− 1) classes ofS of size (qn−1 − 1)(qn − 1)/(q − 1).

Next we suppose that g belongs to aG-conjugacy class of size qn−1(qn−1)/(q− 1). Then q ≥ 3 and CG(g) ∼= GL1(q)×GLn−1(q). Thus, we seethat bk :=

(k 00 In−1

)∈ CG(g) for each k ∈ F×q and det(bk) = k. Utilizing

the determinant map once again, we obtain |CG(g)| = (q − 1)|CS(g)|and |clS(g)| = |clG(g)| = qn−1(qn − 1)/(q − 1). Recall that the charac-teristic polynomial of g is (x− λ1)(x− λ2)n−1 for some λ1 6= λ2 ∈ F×q .

We have λ1λn−12 = 1 and λn2 6= 1 as g is not central. Thus there are

q−1−gcd(n, q−1) choices for λ2, leaving only one choice for λ1, givingq− 1− gcd(n, q− 1) conjugacy classes of S of size qn−1(qn− 1)/(q− 1).

Finally, if g belongs to a G-conjugacy class of size greater than q4n−8

then

|clG(g)| = |G|/|CG(g)| ≤ (q − 1)|S|/|CS(g)| = (q − 1)|clS(g)|.Hence

|clS(g)| ≥ |clG(g)|/(q − 1) ≥ q4n−8/(q − 1) > q4n−9,

as desired. �

4.2. The projective linear groups.

Corollary 4.2. Let n ≥ 6. Then PGLn(q) has one (central ) class ofsize 1, one (unipotent ) class of size (qn−1 − 1)(qn − 1)/(q − 1), q − 2(semisimple) classes of size qn−1(qn − 1)/(q − 1) and all other classeshave sizes greater than q4n−8/ gcd(n, q − 1).

Proof. Let G := GLn(q), Z := Z(G), and G := PGLn(q) = G/Z. Foreach g = gZ ∈ G, let Cg denote the inverse image of CG(g) underthe canonical homomorphism from G to G. For each h ∈ Cg, we have


h ∈ CG(g) and it follows that hg = gh. Hence hg ∈ (gh)Z, whence itfollows that hg = (gh)zh for some zh ∈ Z. From this, we may definethe map

Tg : Cg → Zh 7→ zh = h−1g−1hg

.

It is easy to see that Tg is a homomorphism and zh = 1 if and only ifh ∈ CG(g). Therefore,

CgCG(g)

∼= Im(Tg) ≤ Z

Suppose that g belongs to a G-conjugacy class of size (qn−1−1)(qn−1)/(q− 1). Then g = ku, where u is unipotent with |clG(u)| = (qn−1 −1)(qn − 1)/(q − 1) and k ∈ F×q . Now let h ∈ Cg and suppose Tg(h) =

zh = a · 1. Then hg = (gh)zh implies huh−1 = uzh = au. Takingeigenvalues of both sides, we see that the eigenvalues of huh−1 are all1; whereas, the eigenvalues of au are all a. Thus, we must have a = 1.Hence Im(Tg) = 1. It then follows that

|clG(g)| = |clG(g)| = (qn−1 − 1)(qn − 1)/(q − 1).

Since an element from each G-conjugacy class of size (qn−1 − 1)(qn −1)/(q − 1) is contained in g, elements of this form contribute one G-conjugacy class of size (qn−1 − 1)(qn − 1)/(q − 1).

Next we suppose that g belongs to aG-conjugacy class of size qn−1(qn−1)/(q − 1). The characteristic polynomial of g is (x − λ1)(x − λ2)n−1

for some λ1 6= λ2 ∈ F×q . Now let h ∈ Cg and suppose Tg(h) = zh = a ·1.

Then hg = (gh)zh, from which it follows that hgh−1 = gzh = ag. Tak-ing eigenvalues of both sides, we see that the eigenvalues of hgh−1 areλ1 (with multiplicity 1) and λ2 (with multiplicity n− 1); whereas, theeigenvalues of ag are aλ1 (with multiplicity 1) and aλ2 (with multiplic-ity n− 1). Since n 6= 2, the only way this may be is that a = 1. HenceIm(Tg) = 1 and

|clG(g)| = |clG(g)| = qn−1(qn − 1)/(q − 1).

Since each scalar multiple of g has the same G-conjugacy class size asg and cannot be conjugate to g, the coset g contains q − 1 elements ofdifferent conjugacy classes of size qn−1(qn − 1)/(q − 1). Thus elementsof this form contribute (q2 − 3q + 2)/(q − 1) = q − 2 conjugacy classesof G of size qn−1(qn − 1)/(q − 1).

Finally, suppose that g belongs to a G-conjugacy class of size greaterthan q4n−8. Let h ∈ Cg and suppose Tg(h) = zh. It follows thatgh = (hg)zh and, by taking determinants of both sides, we see that


det zh = 1. It follows that |Im(Tg)| ≤ gcd(n, q − 1). Therefore (q −1) |CG(g)| ≤ |CG(g)| gcd(n, q − 1), whence it follows that

|G||CG(g)|

≤ |G|(q − 1) |CG(g)|

gcd(n, q − 1).

Thus clG(g)| ≥ |clG(g)|/ gcd(n, q − 1) > q4n−8/ gcd(n, q − 1), andclaimed. �

4.3. The projective special linear groups.

Corollary 4.3. Let n ≥ 6. Then PSLn(q) has one (central ) classof size 1, one (unipotent ) class of size (qn−1 − 1)(qn − 1)/(q − 1),(q−1)/ gcd(n, q−1)−1 (semisimple) classes of size qn−1(qn−1)/(q−1)and all other classes have sizes greater than q4n−9/ gcd(n, q − 1).

Proof. Let S := SLn(q), Z := Z(S), and let S := PSLn(q) = S/Z.Let ϕ be the canonical homomorphism from S to S and let g = gZ.Also define Cg and the homomorphism Tg analogously as in the proofof Corollary 4.2. It is easy to see that zh = 1 if and only if h ∈ CS(g).Therefore

CgCS(g)

∼= Im(Tg) ≤ Z

Suppose that g belongs to an S-conjugacy class of size (qn−1−1)(qn−1)/(q − 1). Then g = ku, where u is unipotent with |clS(u)| = (qn−1 −1)(qn − 1)/(q − 1) and k ∈ F×q such that kn = 1. Using an eigenvalueargument similar to the one used in the proof of Corollary 4.2, weobtain that Im(Tg) = 1. Therefore

|clS(g)| = |clS(g)| = (qn−1 − 1)(qn − 1)/(q − 1).

Since an element from each S-conjugacy class of size (qn−1 − 1)(qn −1)/(q − 1) is contained in g, elements of this form contribute one S-conjugacy class of size (qn−1 − 1)(qn − 1)/(q − 1).

Now suppose that g belongs to an S-conjugacy class of size qn−1(qn−1)/(q − 1). Using the eigenvalue argument again, we have

|clS(g)| = |clS(g)| = qn−1(qn − 1)/(q − 1).

Since each scalar multiple k · g of g with kn = 1 has the same S-conjugacy class size as g and cannot be conjugate to g, the coset gcontains gcd(n, q − 1) elements of different conjugacy classes of sizeqn−1(qn − 1)/(q − 1). Thus elements of this form contribute (q − 1 −gcd(n, q−1))/ gcd(n, q−1) = (q−1)/ gcd(n, q−1)−1 conjugacy classesof S of size qn−1(qn − 1)/(q − 1).


Now suppose that g belongs to an S-conjugacy class of size greaterthan q4n−9. Since |CS(g)| ≤ |CS(g)|, it follows that

|clS(g)| ≥ |clS(g)|/ gcd(n, q − 1) > q4n−9/ gcd(n, q − 1),

and we are done. �

5. The general unitary groups

For the remainder of the paper, we let K :={k ∈ F×q2 | k

q+1 =

1}

. The unipotent and semisimple classes of small size of GUn(q) aredetermined in the next two lemmas.

Lemma 5.1. Let u be a unipotent element of G := GUn(q) with n ≥ 4.Then one of the following holds

(i) |cl(u)| = 1 (one class)(ii) |cl(u)| = (qn − (−1)n)(qn−1 − (−1)n−1)/(q + 1) (one class)(iii) |cl(u)| > q4n−9.

Proof. Similar to the proof of Lemma 3.1. �

Lemma 5.2. Let s be a semisimple element of G := GUn(q) withn ≥ 5. Then one of the following holds:

(i) |cl(s)| = 1 (q + 1 classes),(ii) |cl(s)| = qn−1(qn − (−1)n)/(q + 1) (q2 + q classes),(iii) |cl(s)| > q4n−9.

Proof. Recall from Lemma 2.3 that

CG(s) ∼=t⊗i=1

GLai(q2ki)×

r⊗i=1

GUbi(q2mi−1),

where∑

2aiki +∑bi(2mi − 1) = n.

1) Case t=0, r=1: Remark that CG(s) ∼= GUn(q) if and only ifs ∈ Z(G) and in that case |cl(s)| = 1. Since |Z(G)| = q + 1, the groupG has q+1 central classes of size 1 and (i) holds. If CG(s) ∼= GUb(q

2m−1)with b(2m− 1) = n and m ≥ 2, then we have for every n ≥ 5,

|cl(s)| =qn(n−1)/2

∏nj=1

(qj − (−1)j

)qn(b−1)/2

∏bj=1

(q(2m−1)j − (−1)j

) > qn(n−1)/2∏n

j=1

(qj − 1

)qn(b−1)/2

∏bj=1

(q(2m−1)j + 1

)>

qn(n−1)/2qn(n+1)/2−2

qn(b−1)/2q(2m−1)b(b+1)/2+2=qn

2−2

qnb+2= qn

2−nb−4

≥ qn2−n2/3−4 = q2n2/3−4 > q4n−9.

2) Case t=0, r=2: First we consider the subcase CG(s) ∼= GU1(q)×GUn−1(q). This happens if and only if the characteristic polynomial of


s is (x− λ1)(x− λ2)n−1 where λ1, λ2 ∈ K and λ1 6= λ2. Thus there areq2 + q different classes of size

|cl(s)| =qn(n−1)/2

∏nj=1

(qj − (−1)j

)(q + 1)q(n−1)(n−2)/2

∏n−1j=1

(qj − (−1)j

) =qn−1(qn − (−1)n)

q + 1

and (ii) holds.Next, we suppose that CG(s) ∼= GU1(q)×GUb(q

2m−1), where b(2m−1) = n− 1 and m ≥ 2. In this case, we have

|cl(s)| =qn(n−1)/2

∏nj=1

(qj − (−1)j

)(q + 1)q(n−1)(b−1)/2

∏bj=1

(q(2m−1)j − (−1)j

)≥

qn(n−1)/2∏n

j=1

(qj − 1

)(q + 1)q(n−1)(b−1)/2

∏bj=1

(q(2m−1)j + 1

)>

qn(n−1)/2qn(n+1)/2−2

(q + 1)q(n−1)(b−1)/2q(2m−1)b(b+1)+2

=qn

2−2

(q + 1)q(n−1)(b−1)/2q(n−1)(b+1)+2

=qn

2−(n−1)b−4

(q + 1)> qn

2−(n−1)2/3−6 ≥ q4n−9.

Finally, we consider the subcase CG(s) ∼= GUb1(q2m1−1)×GUb2(q

2m2−1)where b1(2m1 − 1) ≥ 2 and b2(2m2 − 1) ≥ 2. Since |GUbi(q

2mi−1)| ≤|GUbi(2mi−1)(q)| for i = 1, 2, we obtain |CG(s)| ≤ |GUn1(q)×GUn2(q)|for some positive integers n1 and n2 greater than 1 such that n1 +n2 =n. It follows that

|cl(s)| ≥ |GUn(q)||GUn1(q)||GUn2(q)|

= qn1(n−n1)

∏nj=1

(qj − (−1)j

)∏n1

j=1

(qj − (−1)j

)∏n2

j=1

(qj − (−1)j

)= qn1(n−n1)

∏nj=n−(n1−1)

(qj − (−1)j

)∏n1

j=1

(qj − (−1)j

) = q2n1(n−n1)

∏nj=n−(n1−1)

(1− (−1/q)j

)∏n1

j=1

(1− (−1/q)j

)≥ q2n1(n−n1)

[(1− 1/qn−(n1−1)

)(1 + 1/qn

)]n1/2[(1− 1/qn1

)(1 + 1/q

)]n1/2≥ q2n1(n−n1)

[1− 1/qn−(n1−1)

1 + 1/q

]n1/2

= q2n1(n−n1)

[qn−(n1−1)−1

q + 1

]n1/2 [q

qn−(n1−1)

]n1/2

≥ q2n1(n−n1)qn1/2

> q2n1(n−n1) ≥ q4n−8 > q4n−9.

3) Case t=0, r≥3: Then CG(s) =⊗r

i=1 GUbi(q2mi−1) where r ≥ 3

and∑bi(2mi − 1) = n. Choose a subset T ⊆ {1, 2, . . . , r} so that

16 SHAWN T. BURKETT AND HUNG NGOC NGUYEN∑i∈T bi(2mi − 1) ≥ 2 and

∑i/∈T bi(2mi − 1) ≥ 2. Then we have

|cl(s)| = |GUn(q)|∏i∈T |GUbi(q

2mi−1)|∏

i/∈T |GUbi(q2mi−1)|

≥ |GUn(q)||GU∑

i∈T bi(2mi−1)(q)||GU∑i/∈T bi(2mi−1)(q)|

> q4n−9

as was proved in the previous case.

4) Case t≥1, r≥0: First we assume that r = 0. Then n must beeven and we have

|cl(s)| = |GUn(q)|∏ti=1|GLai(q

2ki)|>|GUn(q)||GLn/2(q2)|

=qn/2

∏nj=1 (qj − (−1)j)∏n/2j=1 (q2j − 1)

>qn/2

∏nj=1 (qj − 1)

qn(n+2)/8>qn/2qn(n+1)/2−2

qn(n+2)/8

= q3n2/8+3n/4−2 > q4n−9.

Next, suppose that r ≥ 1. If∑r

i=1 bi(2mi − 1) = 1, then n is odd and

|cl(s)| = |GUn(q)||GU1(q)|

∏ti=1|GLai(q

2ki)|≥ |GUn(q)|

(q + 1)|GL(n−1)/2(q2)|

=q3(n−1)/2

∏nj=1 (qj − (−1)j)

(q + 1)∏(n−1)/2

j=1 (q2j − 1)>q3(n−1)/2

∏nj=1 (qj − 1)

(q + 1)q(n−1)(n+1)/4

>q3(n−1)/2qn(n+1)/2−2

(q + 1)q(n−1)(n+1)/4> qn

2/4+2n−21/4 ≥ q4n−9.

On the other hand, if∑r

i=1 bi(2mi − 1) ≥ 2, then

|cl(s)| = |GUn(q)|∏ti=1|GLai(q

2ki)|∏r

i=1|GUbi(q2mi−1)|

≥ |GUn(q)|∏ti=1|GUai(q

2ki)|∏r

i=1|GUbi(q2mi−1)|

≥ |GUn(q)||GU∑t

i=1 2aiki(q)||GU∑r

i=1 bi(2mi−1)(q)|> q4n−9

as was proved in the last subcase of 2). �

Proof of Theorem 1.2. Let g ∈ G := GUn(q) and let g = su = uswhere s ∈ G is semisimple and u ∈ G is unipotent be the Jordandecomposition of g. Since CG(g) = CG(s) ∩ CG(u), we get

|CG(g)| ≤ min{|CG(s)|, |CG(u)|}.


Assuming that |cl(g)| ≤ q4n−9, then we have |cl(s)| ≤ q4n−9. FromLemma 5.2, we have two cases:

1) |cl(s)| = 1. Then s is one of the q + 1 central elements in G.Therefore, |cl(u)| = |cl(g)| ≤ q4n−9. By Lemma 5.1, u either is identityor belongs to a class of size (qn−1− (−1)n−1)(qn− (−1)n)/(q+ 1). Now(i) holds in the former case and (ii) holds in the latter case.

2) |cl(s)| = qn−1(qn − (−1)n)/(q + 1). Then CG(s) ∼= GU1(q) ×GUn−1(q). If u = 1 then we get q2 + q conjugacy classes of semisimpleelements and hence (iii) holds. So we can assume u 6= 1.

Since u commutes with s, we can consider u as a unipotent elementof CG(s). Using Lemma 5.1, we have

|clCG(s)(u)| ≥ (qn−2 − (−1)n−2)(qn−1 − (−1)n−1)

q + 1.

It follows that

|cl(g)| = |cl(s)| · |clCG(s)(u)|

≥ qn−1(qn − (−1)n)

q + 1· (qn−2 − (−1)n−2)(qn−1 − (−1)n−1)

q + 1

>qn−1(qn − 1)(qn−1 − 1)(qn−2 − 1)

(q + 1)2> q4n−9,

as claimed. �

6. Other unitary groups

As for the linear groups, we can derive from Theorem 1.2 some gapresults on conjugacy class sizes for SUn(q), PGUn(q), and PSUn(q).We leave the proofs in this section to the reader as they are similar tothose in Section 4.

Corollary 6.1. Let n ≥ 5. Then SUn(q) has gcd(n, q + 1) (central )classes of size 1, gcd(n, q + 1) classes of size (qn−1 − (−1)n−1)(qn −(−1)n)/(q+1), q+1−gcd(n, q+1) (semisimple) classes of size qn−1(qn−(−1)n)/(q+1) and all other classes have sizes greater than q4n−9/(q+1).

Corollary 6.2. Let n ≥ 5. Then PGUn(q) has one (central ) class ofsize 1, one (unipotent ) class of size (qn−1−(1)n−1)(qn−(−1)n)/(q+1),q (semisimple) classes of size qn−1(qn − (−1)n)/(q + 1) and all otherclasses have sizes greater than q4n−9/ gcd(n, q + 1).

Corollary 6.3. Let n ≥ 5. Then PSUn(q) has one (central ) class ofsize 1, one (unipotent ) class of size (qn−1− (−1)n−1)(qn− (−1)n)/(q+1), (q + 1)/ gcd(n, q + 1) − 1 (semisimple) classes of size qn−1(qn −


(−1)n)/(q+1) and all other classes have sizes greater than q4n−9/((q+1) gcd(n, q + 1)).

7. Applications: Bounds and limits for the zeta-typefunction

We apply the results in previous sections to obtain some bounds andlimits for the zeta-type function encoding the conjugacy class sizes ofthe linear and unitary groups. We recall that the function associatedto a finite group G is

ζG(t) =∑

C∈C(G)

|C|−t =∑s∈N

ks(G)s−t

where ks(G) denotes the number of classes of G of size s.

Lemma 7.1. Suppose that n ≥ 6 and t > 0. Then

ζGLn(q)(t) < (q − 1) + (q − 1)2q−t(2n−2) + qn−t(4n−8).

Proof. Bounds for the number of classes in finite Chevalley groups aredetermined in [8]. In particular, the number of classes of GLn(q) isbounded above by qn (cf. Proposition 3.5 of [8]). This and Theorem 1.1imply that

ζGLn(q)(t) = (q − 1) + (q − 1)((qn−1 − 1)(qn − 1)

q − 1)−t

+(q2 − 3q + 2)(qn−1(qn − 1)

q − 1)−t +

∑s>q4n−8

ks(GLn(q))s−t

< (q − 1) + q−t(2n−2)(q − 1) + q−t(2n−2)(q2 − 3q + 2) + qnq−t(4n−8)

= (q − 1) + q−t(2n−2)(q − 1)2 + qn−t(4n−8).

�

Lemma 7.2. Suppose that n ≥ 5 and t > 0. Then

ζGUn(q)(t) < (q + 1) + (q + 1)2q−t(2n−3) + (qn + 16qn−1)q−t(4n−9).


Proof. The number of classes of GUn(q) is bounded above by qn +16qn−1 (cf. Proposition 3.9 of [8]). This and Theorem 1.2 imply that

ζGUn(q)(t) = (q + 1) + (q + 1)((qn−1 − (−1)n−1)(qn − (−1)n)

q + 1)−t

+(q2 + q)(qn−1(qn − (−1)n)

q + 1)−t +

∑s>q4n−9

ks(GUn(q))s−t

< (q + 1) + q−t(2n−3)(q + 1) + q−t(2n−3)(q2 + q) + (qn + 16qn−1)q−t(4n−9)

= (q + 1) + q−t(2n−3)(q + 1)2 + (qn + 16qn−1)q−t(4n−9).

�

Theorem 1.3 now is just a consequence of Theorems 7.1 and 7.2.

Proof of Theorem 1.3. Recall the hypothesis that n ≥ 6 and t > 1/3.Lemma 7.1 implies that

ζGLn(q)(t)

q< (1− 1

q) + (1− 2

q+

1

q2)q1−t(2n−2) + qn−1−t(4n−8).

As n ≥ 6 and t > 1/3, we have 1−t(2n−2) < 0 and n−1−t(4n−8) < 0.Therefore, the right side of the above inequality tends to 1 uniformly

in n as q →∞. As it is clear that ζGLn(q)(t)q

> q − 1, we conclude

ζGLn(q)

q→ 1 uniformly in n as q →∞.

Similarly, Lemma 7.2 yields

ζGUn(q)(t)

q< (1 +

1

q) + (1 +

2

q+

1

q2)q1−t(2n−3) + (qn−1 + 16qn−2)q−t(4n−9).

Now the proof proceeds as in the GLn(q) case. �

Acknowledgement

The authors acknowledge the support of a Faculty Research Grant(FRG 1747) from The University of Akron.

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Department of Mathematics, University of Colorado Boulder, Boul-der, Colorado 80309, U.S.A.

E-mail address: [email protected]

Department of Mathematics, The University of Akron, Akron, Ohio44325, U.S.A.

E-mail address: [email protected]

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