On the minimal dimension of a faithful linear representation of a finite group
aa r X i v : . [ m a t h . G R ] F e b On the minimal dimension of a faithful linearrepresentation of a finite group byAlexander Moret´oDepartament de Matem`atiquesUniversitat de Val`encia46100 Burjassot. Val`encia [email protected]
Abstract.
The representation dimension of a finite group G is theminimal dimension of a faithful complex linear representation of G . Weprove that the representation dimension of any finite group G is at most p | G | except if G is a 2-group with elementary abelian center of order 8and all irreducible characters of G whose kernel does not contain Z ( G )are fully ramified with respect to G/Z ( G ). We also obtain bounds for therepresentation dimension of quotients of G in terms of the representationdimension of G , and discuss the relation of this invariant with the essentialdimension of G . AMS Subject Classification.
Primary 20C15, Secondary 14E07,12F10
Keywords and phrases. representation dimension, essencial dimen-sion, faithful representationResearch supported by Ministerio de Ciencia e Innovaci´on PID-2019-103854GB-100, FEDER funds and Generalitat Valenciana AICO/2020/298. I thank D.Holt, Z. Reichstein and G. Robinson for helpful comments.1
Introduction
Given a positive integer n , the study of the (finite complex) linear groups ofdegree n has been a classical theme of research in finite group theory. Forinstance, in 1878, C. Jordan proved that if G is a linear group of degree n ,then there exists A ≤ G abelian such that | G : A | ≤ j ( n ) for some integervalued function j ( n ) (see [30] for a modern classification-free proof of thistheorem and for a description of earlier proofs.) After the classificationof finite simple groups was completed, sharp bounds for the function j ( n )were found by M. Collins [8] in 2008, improving on an earlier unpublishedmanuscript by B. Weisfeiler.Following [7], let rdim( G ) be the minimal integer such that a finite group G embeds into GL(rdim( G ) , C ), i.e., rdim( G ) is the smallest integer n suchthat a finite group G is a linear group of degree n . This was called therepresentation dimension in [7]. Clearly, rdim( G ) ≤ | G | . Surprisingly, thisnatural invariant of a finite group has not been very studied from a group-theoretic point of view. Recently, it has been proven to be very relevant ina large number of areas outside finite group representation theory. See forinstance the Preface of [32] for its relevance in group cohomology theory or [5,14] for its relevance to show that certain Cayley graphs are expander graphs.All the nontrivial results we are aware of on rdim( G ) when G is not close toa simple group have been motivated by the so-called essential dimension of afinite group ed( G ). This concept was introduced in 1997 by J. Buhler and Z.Reichstein in [6] with motivations from algebraic geometry. Since then it hasfound applications in a large number of areas (see [25, 26]). It is known thated( G ) ≤ rdim( G ) (see Proposition 4.15 of [3]). Both ed( G ) and rdim( G )depend on the field over which we are considering the representations of G and are of interest over arbitrary fields. For simplicity, in this note we willrestrict ourselves to the field of complex numbers, although our argumentswork over any field with sufficiently many roots of unity. A major resultwas the proof by N. Karpenko and A. Merkurjev [20] that ed( G ) = rdim( G )when G is a p -group. This has motivated the study of rdim( G ) for severalfamilies of p -groups. See [27, 7, 1, 2].In this note, prompted by a question raised on the Math Overflow website, we study the problem of finding sharp bounds for rdim( G ) in terms of | G | . More precisely, the question asked was whether rdim( G ) ≤ p | G | . Aspointed out by D. Holt, C × C × C is a counterexample. Our first mainresult shows that all counterexamples are closely related to Holt’s example.In the following statement, Soc( G ) is the socle of G and Irr( G | Z ( G )) isthe set of irreducible characters of G that lie over a nonprincipal linearcharacter of the center of G . We refer the reader to Problem 6.3 of [18] forthe definition of fully ramified character.2 heorem A. Let G be a finite group. Then one of the following holds: (i) rdim( G ) ≤ p | G | ; (ii) G is a -group with socle Soc( G ) = Z ( G ) = C × C × C and allcharacters in Irr( G | Z ( G )) are fully ramified with respect to G/Z ( G ) .For any such group G , rdim( G ) = √ p | G | . We have also shown that the equality rdim( G ) = p | G | just holds ingroups that are similar to those in (ii) above. Theorem B.
Let G be a finite group. Then rdim( G ) = p | G | if and only ifone of the following holds: (i) G is a -group with socle Soc( G ) = Z ( G ) = C × C and all charactersin Irr( G | Z ( G )) are fully ramified with respect to Z ( G ) . (ii) G is a -group with socle Soc( G ) = Z ( G ) = C × C × C × C and allcharacters in Irr( G | Z ( G )) are fully ramified with respect to Z ( G ) . It is interesting to note that the groups with rdim( G ) as large as it can bein comparison with | G | turn out to be 2-groups. There are nonabelian groupsamong those that appear in Theorem A (ii) and in Theorem B: consider forinstance the Sylow 2-subgroups of GL(3 ,
8) and GL(3 ,
4) and GL(2 , G with rdim( G ) arbitrarily close to p | G | : consider the Frobeniusgroups of order ( p − p for any prime p .There is a related invariant that has been more studied with a group-theoretic motivation. This is the smallest dimension of a faithful permuta-tion representation, denoted by µ ( G ). It was shown by P. Neumann [28] thatthere are groups G with normal subgroups N such that µ ( G/N ) > µ ( N ).L.Kov´acs and C. Praeger [21] showed that µ ( G/N ) ≤ µ ( G ) holds whenever G/N does not have nontrivial abelian normal subgroups. Holt and J. Walton[15]proved that µ ( G/N ) ≤ (4 . µ ( G ) − . As, for instance, the double coverof M shows, it is not true that rdim( G/N ) ≤ rdim( G ) even when G/N does not have nontrivial abelian normal subgroups. As a consequence of Jor-dan’s theorem, it is easy to obtain the following variation of the Holt-Waltontheorem for representation dimension.
Theorem C.
Let G be a finite group and N a normal subgroup of G . Write rdim( G ) = n . Then rdim( G/N ) ≤ nj ( n ) , where j ( n ) is any bound in Jordan’s theorem.
3t is an old conjecture of Praeger and Easdown [10] that if N E G and G/N is abelian, then µ ( G/N ) ≤ µ ( G ). This conjecture still remainsopen. In the case when G is a p -group with an abelian maximal subgroupit was proved in [12] that µ ( G/G ′ ) ≤ µ ( G ). We will see that it is not truethat rdim( G/G ′ ) ≤ rdim( G ) even in this case. However, we can obtain thefollowing bound. Theorem D.
Let G be a finite group and N a normal subgroup of G with G/N abelian. Write rdim( G ) = n . Then rdim( G/N ) ≤ n/ . This bound in Theorem D in the only result in this paper that dependson the classification of finite simple groups. Without the CFSG, we canprove that rdim(
G/N ) ≤ Kn / log n for some universal constant K . Bothversions of this result are straightforward consequences of bounds on thenumber of generators of a linear group.As a consequence of Theorem C and [29], which relies on a deep resultin Mori theory [4], we can obtain a new result on the essential dimension ofan arbitrary finite group. It was asked in [19] whether ed( G/N ) ≤ ed( G ) forany finite group G and any N E G . A negative answer to this question wasgiven in Theorem 1.5 of [27]. In fact, the example of A. Meyer and Reichsteinshows that we cannot hope for bounds better than exponential in TheoremC, even if we assume that G is a p -group. We obtain the following boundfor ed( G/N ) in terms of ed( G ). Corollary E.
Let G be a finite group and N E G . Then ed( G/N ) ≤ ed( G ) j (ed( G )) j (ed( G ) j (ed( G )) , where j is the bounding function in Jordan’s theorem. We close this Introduction with a remark on the style that we have usedin this paper. It is a paper on character theory of finite groups that, wehope, will be of interest to other areas outside group theory, particularly tothose areas where the essential dimension of a finite group plays a role. Forthis reason, we have decided to include some details in our proofs that wewould not have included in a paper addressed exclusively to group theorists.
Our approach will be character-theoretic. Our notation follows [18]. If χ is acharacter of a finite group then χ can be decomposed as a sum of irreducible4haracters, called the irreducible constituents, and it is easy to see thatthe kernel of χ , Ker χ , is the intersection of the kernels of the irreducibleconstituents. (Lemma 2.21 of [18]). We thus have the following elementaryresult. Lemma 2.1.
Let G be a finite group. Then rdim( G ) = min { r X i =1 χ i (1) | r ∈ Z + , χ i ∈ Irr( G ) for every i = 1 , . . . , r , r \ i =1 Ker χ i = 1 } . If G has a faithful irreducible character χ , then it follows from Corollary2.7 of [18] that χ (1) < p | G | , so Theorem A holds in this case. The problemof which finite groups have faithful irreducible characters is therefore rele-vant for our purposes. This problem has been studied since the beginning ofthe 20th century and there are several, perhaps not very well-known, char-acterizations of these groups. We refer the reader to Section 2 of [31] for anice description of the history of this problem.The socle of G is very relevant in these characterizations. Recall thatit is the product of all the minimal normal subgroups of G . We will writeSoc( G ) to denote it. Recall that Soc( G ) = A ( G ) × T ( G ), where A ( G ) = A × · · · × A t is a direct product of some elementary abelian minimal normalsubgroups of G and T ( G ) is the direct product of all the nonabelian minimalnormal subgroups of G (see Definition 42.6 and Lemma 42.9 of [17]). Inthe remaining of this article, we will use the notation introduced in thisparagraph without further explicit mention. In particular, t = t ( G ) is thenumber of elementary abelian minimal normal subgroups of G that appear ina decomposition of A ( G ) as a direct product of minimal normal subgroups.We will use the following consequence of Gasch¨utz’s characterization(Theorem 42.7 of [17]) of finite groups with a faithful irreducible character. Theorem 2.2.
If for every prime p every simple GF( p ) G -module appearsat most with multiplicity one in A ( G ) , then G has a faithful irreduciblecharacter.Proof. This is Theorem 42.12(a) of [17].
Lemma 2.3.
Let G be a finite group. Then T ( G ) has a faithful irreduciblecharacter. In particular, if t = 0 (or, equivalently, if G does not haveany nontrivial abelian normal subgroup) then G has a faithful irreduciblecharacter and rdim( G ) < p | G | .Proof. Note that T ( G ) is a direct product of nonabelian simple groups. ByProblem 4.3 of [18], for instance, the product ϕ of nonprincipal charactersof each of the factors is a faithful irreducible character of T ( G ).5f χ ∈ Irr( G ) lies over ϕ then χ T ( G ) is a sum of conjugates of ϕ , so byLemma 2.21 of [18], χ T ( G ) is faithful. This implies that1 = Ker χ ∩ T ( G ) = Ker χ ∩ Soc( G ) , so χ is a faithful irreducible character of G . Now, rdim( G ) ≤ χ (1) < p | G | ,by Corollary 2.7 of [18]. Lemma 2.4.
Let G be a finite group without nonabelian minimal normalsubgroups. Let χ be a faithful character of G with rdim( G ) = χ (1) . Thenfor every ψ ∈ Irr( G ) irreducible constituent of χ , Soc( G ) Ker ψ .Proof. Assume not. Then Soc( G ) = A ( G ) ≤ Ker ψ for some ψ ∈ Irr( G )irreducible constituent of χ . Consider ∆ = χ − ψ . Since ψ is an irreducibleconstituent of χ , ∆ is a character of G and ∆(1) < χ (1) = rdim( G ). Thus∆ is not faithful. Let ψ , . . . , ψ s be the remaining irreducible constituentsof χ . Since ∆ is not faithful, the intersection of the kernels of the ψ i ’s isnot trivial. But since Ker ψ contains the whole socle, we deduce that theintersection of the kernels of all the irreducible constituents of χ is not trivial.This contradicts the hypothesis that χ is faithful.We write d ( G ) to denote the rank of a group G . Recall that it is theminimal number of generators of G . The following result is well-known. Lemma 2.5.
Let G be a finite abelian group. Then d ( G ) = rdim( G ) .Proof. Set d ( G ) = m . By the fundamental theorem of abelian groups, G = C × · · · × C m is a direct product of m cyclic groups C i = h x i i . Let λ i be agenerator of Irr( C i ) and let µ i ∈ Irr( G ) be the linear character determinedby means of µ i ( x i ) = ε , where ε is an o ( x i )th primitive root of unity, and µ i ( x j ) = 1 for j = i . Notice that Ker µ i = C · · · C i − C i +1 · · · C m . Put µ = µ + · · · + µ m . By Lemma 2.21 of [18], Ker µ = 1, i.e., µ is faithful.Since µ (1) = m we deduce rdim( G ) ≤ µ (1) = m = d ( G ).Conversely, let χ be any faithful character of G . Decompose χ = a χ + · · · + a s χ s as a sum of irreducible (linear) characters χ i . Since G/ Ker χ i iscyclic for every i and the intersection of the kernels of the characters χ i istrivial, we deduce that G is isomorphic to a subgroup of the direct productof the cyclic groups G/ Ker χ i . Write Γ to denote this group. Since d (Γ) = s and Γ is abelian, we deduce that d ( G ) ≤ d (Γ) = s ≤ χ (1) ≤ rdim( G ). Theresult follows.The next result, in conjunction with Lemma 2.1 lies at the core of ourproof of Theorems A and B. 6 emma 2.6. Let G be a finite group. Assume that t > . For i = 1 , . . . , t ,write B i = A × · · · × A i − × A i +1 × · · · × A t . Then (i) For every i = 1 , . . . , t , there exists χ i ∈ Irr( G ) such that Ker χ i ∩ Soc( G ) = B i . Furthermore, χ i (1) ≤ | G/B i : Z ( G/B i ) | / . (ii) We have T ti =1 Ker χ i = 1 . In particular, if χ = χ + · · · + χ t , then rdim( G ) ≤ χ (1) .Proof. Let λ i ∈ Irr( A i ) be nonprincipal for i = 1 , . . . , t and ϕ ∈ Irr( T ( G ))be faithful. Recall that A ( G ) = A i × B i . Put µ i = λ i × B i × ϕ ∈ Irr(
Soc ( G )) . Let χ i ∈ Irr( G ) lying over µ i . Since µ i is an irreducible constituent of( χ i ) Soc( G ) , Ker χ i ∩ Soc( G ) ≤ core G (Ker µ i ) = B i . Since ( µ i ) B i is a multiple of the principal character, we clearly have that B i ≤ Ker χ i . The first claim of part (i) follows. The second claim holds byCorollary 2.30 of [18].By the definition of the subgroups B i , their intersection is trivial. Thus1 = t \ i =1 (Ker χ i ∩ Soc( G )) = t \ i =1 Ker χ i ! ∩ Soc( G ) . Since T ti =1 Ker χ i is a normal subgroup of G , we deduce that it has to bethe trivial subgroup. The inequality rdim( G ) ≤ χ (1) follows from Lemma2.1.Now, we can obtain our first approximatiom to Theorem A when t > χ i and λ i will be the charactersthat have appeared in the statement of Lemma 2.6 and its proof. Lemma 2.7.
Let G be a finite group. Assume that t ≥ . Write | A i | = a i for every i = 1 , . . . , t . Then rdim( G ) < p | G | t X j =1 Y k = j √ a k . roof. Since χ i ∈ Irr(
G/B i ), we note that χ i (1) < s | G | a · · · a i − a i +1 · · · a t , (It suffices to observe that | B i | = a · · · a i − a i +1 · · · a t .)Hence, if χ = χ + · · · + χ t ,rdim( G ) ≤ χ (1) < t X i =1 s | G | a · · · a i − a i +1 · · · a t = p | G | t X j =1 Y k = j √ a k , as desiredLemma 2.7 implies that Theorem A holds when P tj =1 Q k = j √ a k <
1. Inthe next elementary lemma we see that this is the case most of the times.
Lemma 2.8.
Let t ≥ be an integer and let a ≥ · · · ≥ a t ≥ be t integers.If t X j =1 Y k = j √ a k ≥ then one of the following holds: (i) t = 2 and ( a , a ) ∈ { ( x, , ( y, | ≤ x ≤ , ≤ y ≤ } . (ii) t = 3 and ( a , a , a ) ∈ { ( x, , , (4 , , , (3 , , | ≤ x ≤ } . (iii) t = 4 and ( a , a , a , a ) ∈ { ( x, , , , (3 , , , | ≤ x ≤ } . (iv) t = 5 and ( a , a , a , a , a ) ∈ { (2 , , , , , (3 , , , , } .Proof. Notice that in the expression P tj =1 Q k = j √ a k we have t summandsand the denominator of each of the summands is at least √ t − . Therefore,each of the summands is at most 1 / ( t − / so t ( t − / ≥ t X j =1 Y k = j √ a k > t ≤
5. The possible values for( a , · · · , a t ) for each of the possibilities for t can also be obtained in anelementary way. We omit the details.8ow, we can complete the proof of Theorems A and B by analyzing theexceptional cases that appear in Lemma 2.8. We will use several times thatif G is a finite group and χ ∈ Irr( G ) then χ (1) ≤ | G : Z ( G ) | / (by Corollary2.30 of [18]). Groups with an irreducible character χ such that χ (1) = | G : Z ( G ) | / are called groups of central type have been rather studied. By acelebrated theorem of Howlett and Isaacs [16] they are solvable. We willnot need the Howlett-Isaacs theorem, but we will use a more elementarypreviousresult that says that if G is a group of central type then the set ofprimes that divide | Z ( G ) | coincides with the set of primes that divide | G | (see Theorem 2 of [9]).The next result includes both Theorem A and Theorem B. As usual, if N E G and λ ∈ Irr( N ), we write Irr( G | λ ) to denote the set of irreduciblecharacters of G that lie over λ . Theorem 2.9.
Let
G > be a finite group. Then one of the following holds: (i) rdim( G ) < p | G | ; (ii) G is a -group with socle Soc( G ) = Z ( G ) = C × C × C and allcharacters in Irr( G | Z ( G )) fully ramified with respect to Z ( G ) . Forany such group G , rdim( G ) = √ | G | / . (iii) G is a -group with socle Soc( G ) = Z ( G ) = C × C and all charactersin Irr( G | Z ( G )) are fully ramified with respect to Z ( G ) . For any suchgroup G , rdim( G ) = p | G | . (iv) G is a -group with socle Soc( G ) = Z ( G ) = C × C × C × C andall characters in Irr( G | Z ( G )) are fully ramified with respect to Z ( G ) .For any such group G , rdim( G ) = p | G | .Proof. We have already seen that (i) holds if t = 0. This also holds when t = 1 by Theorem 2.2. Hence, using Lemmas 2.7 and 2.8, we may assumethat 2 ≤ t ≤
5. We consider these four cases separately.
Case 1:
Assume first that t = 2. We need to consider the values for( a , a ) that appear in Lemma 2.8. By Theorem 2.2, G has an irreduciblefaithful character if a = a , so it suffices to consider the cases ( a , a ) =(2 ,
2) and ( a , a ) = (3 , Subcase 1.1:
Suppose that ( a , a ) = (2 , χ ∈ Irr(
G/A ) and | A ( G ) /A | = 2, so A ( G ) /A is central in G . Hence, χ (1) ≤ | G/A : Z ( G/A ) | / ≤ | G : A ( G ) | / = | G | / / . Arguing analogously with χ , we obtain that χ (1) ≤ | G | / /
2, so χ (1) = χ (1) + χ (1) ≤ | G | / . χ (1) = | G | / then all inequalities so far are equalities. In particular, A ( G ) = A × A = Z ( G ) is a Klein 4-group and G is a group of central type.Since Z ( G ) is a 2-group, Theorem 2 of [9] implies that G is also a 2-group.Thus T ( G ) = 1 and Soc( G ) = A ( G ) = Z ( G ).On the other hand, χ and χ are fully ramified with respect to Z ( G ).Thus they are the unique irreducible characters of G lying over λ and λ ,respectively (recall that we are using the notation from the proof of Lemma2.6). Since | Z ( G ) | = 4, there is a third nonprincipal linear character λ ∈ Irr( Z ( G )). If λ is fully ramified with respect to G/Z ( G ), then (iii) holds. If λ is not fully ramified with respect to G/Z ( G ) then there exists χ ∈ Irr( G )lying over λ such that χ (1) < | G | / /
2. Notice that ( χ + χ )(1) < | G | / .If we see that χ + χ is faithful, then (i) holds. Note that( χ + χ ) Z ( G ) = χ (1) λ + χ (1) λ . The kernels of λ and λ are different subgroups of order 2 of Z ( G ), so theirintersection is trivial. Thus Ker( χ + χ ) ∩ Z ( G ) = 1 and we deduce that χ + χ is faithful, as desired. Subcase 1.2:
Now, we may assume that ( a , a ) = (3 , χ is an irreducible character of G that lies over a nonprincipal character of A ( G ) /A . Write C/A = C G/A ( A ( G ) /A ). Note that G/C is isomorphic toa subgroup of Aut( A ( G ) /A ) and since | A ( G ) /A | = 3, | G/C | ≤
2. Noticealso that A ( G ) /A is central in C/A . If C = G , then A ( G ) /A is central in G/A and χ (1) ≤ | G : A ( G ) | / = | G | / / . If | G : C | = 2 and γ ∈ Irr( C ) lies under χ , then γ (1) ≤ | C : A ( G ) | / . ByClifford theory, χ (1) ≤ γ (1) ≤ | C : A ( G ) | / = 2 ( | G | / / = | G | / (2 / √ . Thus, in both cases, χ (1) ≤ | G | / (2 / √ χ (1) ≤ | G | / (2 / √ χ (1) = χ (1) + χ (1) ≤ | G | / (4 / √ < | G | / , and (i) holds. Case 2:
Now, assume that t = 3. Using Theorem 2.2 again, togetherwith lemmas 2.7 and 2.8, we may suppose that ( a , a , a ) = ( x, ,
2) forsome 2 ≤ x ≤ a , a , a ) = (3 , , Subcase 2.1:
Suppose first that ( a , a , a ) = (3 , , χ (1) ≤ | G | / / , χ (1) ≤ | G | / /
3, and χ (1) ≤ | G | / / (3 √ χ (1) = χ (1) + χ (1) + χ (1) ≤ | G | / ( 13 + 13 + 13 √ < | G | / , Subcase 2.2:
Assume now that ( a , a , a ) = (7 , , χ is an irreducible character of G that lies over a nonprincipal character of A ( G ) /A A . Write C/A A = C G/A A ( A ( G ) /A A ). Note that G/C isisomorphic to a subgroup of Aut( A ( G ) /A A ) and since | A ( G ) /A A | = 7, | G/C | ≤
6. Arguing again as in previous cases, we have that the worsebound for χ (1) is obtained when | G : C | = 6 and in that case χ (1) ≤ | C : A ( G ) | / = 6 ( | G | / / = | G | / (3 / √ . Also, χ (1) ≤ | G | / / √ χ (1) ≤ | G | / / √
7. Thus χ (1) ≤ | G | / ( 3 √
42 + 12 √ √ < | G | / . We conclude that (i) holds too.The cases ( a , a , a ) = (5 , ,
2) and ( a , a , a ) = (3 , ,
2) are handledanalogously. We omit the details.
Subcase 2.3:
Assume that ( a , a , a ) = (4 , , χ ∈ Irr(
G/A A ). As before, A ( G ) /A A is central in G/A A . Thus χ (1) ≤| G : A ( G ) | / = | G | / /
4. Assume first that χ (1) = | G : A ( G ) | / . Then Z ( G/A A ) = A ( G ) /A A and G/A A is a group of central type with χ fully ramified with respect to the center. Since Z ( G/A A ) is a 2-group and G/A A is of central type, we deduce that G/A A , and hence G , is a 2-group (by Theorem 2 of [9]). Notice that χ (1) = | G | / /
4. Analogously, wehave that χ (1) ≤ | G | / /
4. Next, we bound χ (1). As usual, let C/A A = C G/A A ( A ( G ) /A A ). Recall that G/C is isomorphic to a subgroup ofAut( A ( G ) /A A ) = Aut( C × C ) ∼ = S and since G is a 2-group, | G/C | ≤ χ (1) ≤ | C : A ( G ) | / = | G | / (1 / √ . We conclude that χ (1) < | G | / . Hence, we may assume that χ (1) < | G : A ( G ) | / . This implies that there exist at least two irreducible characters of G lying over the nonprincipal irreducible character λ of A ( G ) /A A . Since X χ ∈ Irr(
G/A A | λ ) χ (1) = | G : A ( G ) | , we deduce that for some of the characters χ in this sum, χ (1) ≤ | G : A ( G ) | / | G | /
32. Hence, we may assume that χ (1) ≤ | G | / / √
2. Re-peating the same reasoning, we may also assume that χ (1) ≤ | G | / / √ χ (1). With our usual notation and arguments, we mayassume that | G : C | = 6 and one can see that χ (1) ≤ | C : A ( G ) | / = 6 (cid:16) | G | / / √ (cid:17) = | G | / √ / . χ (1) ≤ | G | / ( √
64 + 14 √ √ < | G | / . In this case, (i) also holds.
Subcase 2.4:
Finally, we may assume that ( a , a , a ) = (2 , , A ( G ) ≤ Z ( G ). If A ( G ) < Z ( G ) then | Z ( G ) | ≥ and ψ (1) ≤| G : Z ( G ) | / ≤ | G | / / ψ ∈ Irr( G ). Since χ is the sum of 3irreducible characters of G , we deduce that χ (1) < | G | / and (i) holds.Thus we may assume that A ( G ) = Z ( G ). In particular, if ψ ∈ Irr( G ) then ψ (1) ≤ | G : Z ( G ) | / ≤ | G | / / √ G ) ≥ p | G | . Let λ ∈ Irr( Z ( G )) be a nonprincipalcharacter and K = Ker λ . We claim that Z ( G/K ) = Z ( G ) /K . We argue byway of contradiction. Assume that Z ( G/K ) > Z ( G ) /K . Let ψ ∈ Irr( G | λ ).Then ψ (1) ≤ | G/K : Z ( G/K ) | / ≤ | G | / /
4. We deduce that there existsa faithful character χ of G such that χ (1) ≤ | G | / / | G | / / √ | G | / / √ < | G | / . This is a contradiction. Hence, we have proved the claim. This argu-ment also shows that ψ (1) = | G : Z ( G ) | / = | G | / / √ ψ ∈ Irr( G | Z ( G )), as desired. We deduce that (ii) holds.The remaining two cases can be handled with the same techniques.Therefore, we will omit most details in Cases 3 and 4. Case 3:
Assume that t = 4. Again, using Theorem 2.2, together withLemma 2.7 and 2.8, we may suppose that ( a , a , a , a ) = (2 , , ,
2) or( a , a , a , a ) = (3 , , , χ i , i = 1 , . . . ,
4, that rdim( G ) < p | G | . Case 4:
Finally, assume that t = 5. Again, using Theorem 2.2, to-gether with Lemma 2.7 and 2.8, we may suppose that ( a , a , a , a , a ) =(2 , , , , A ( G ) is the direct product of 5 minimal normal sub-groups of order 2. Thus A ( G ) is central. Hence χ i (1) ≤ | G | / / √
32 for i = 1 , . . . , χ (1) < | G | / . We deduce that (i) holds.Now, it remains to determine rdim( G ) when G is one of the groupsthat appear in part (ii), (iii) or (iv). If G is one of the groups in (ii), weknow by Lemma 2.6 that G has a faithful character that is the sum of 3characters in Irr( G | Z ( G )). By hypothesis these characters are fully ramifiedwith respect to G/Z ( G ), so their degree is | G/Z ( G ) | / = | G | / / √
8. Wededuce that rdim( G ) ≤ √ | G | / . On the other hand, if we take a faithfulcharacter of G of minimal degree, then we know by Lemma 2.4 that allof its irreducible constituents lie over nontrivial irreducible characters of12 ( G ). By hypothesis, these irreducible constituents are fully ramified withrespect to G/Z ( G ) so they have degree | G | / √
8. The kernel of each of thesecharacters of Z ( G ) is a hyperplane, so we need to have at least 3 differentirreducible constituents, i.e., rdim( G ) ≥ | G | / / √ In this section we provide the short proofs of the remaining results. We startwith Theorem C.
Proof of Theorem C.
By Jordan’s theorem, there exists and abelian sub-group A ≤ G such that | G : A | ≤ j ( n ) for some function j . Since N A/N ∼ = A/A ∩ N is abelian, we deduce thatrdim( N A/N ) = d ( N A/N ) ≤ d ( A ) ≤ n. Thus
N A/N has a faithful character ∆ of degree at most n . Hence theinduced character ∆ G is faithful and has degree at most nj ( n ). The resultfollows.Now, we deduce Corollary E. Proof of Corollary E.
Write rdim( G ) = n . By Proposition 4.15 of [3],ed( G/N ) ≤ rdim( G/N ) . Furthermore, by Theorem C, rdim(
G/N ) ≤ nj ( n ). On the other hand, byTheorem 2 of [29], n ≤ ed( G ) j (ed( G )). The result follows.As we already mentioned in the Introduction, it is not true that rdim( G/N ) ≤ rdim( G ) when G/N does not have any nontrivial abelian normal subgroup.This is false even when G is a p -groups with an abelian maximal subgroup:consider G = SmallGroup ( , ). This group has an abelian maximal sub-group, it has faithful irreducible characters of degree 2, but rdim( G/G ′ ) = d ( G/G ′ ) = 3. Now, we prove Theorem D. Proof of Theorem D.
Since
G/N is abelian, rdim(
G/N ) = d ( G/N ) ≤ d ( G ).Now, the result follows from [22]. 13ote that the bound in [22] relies on the classification. Using the weaker(but classification-free) bound in [11], we get that rdim( G/N ) ≤ Kn / log n for some constant K . The Heisenberg groups mentioned in the Introduction are just one exampleof the nonabelian groups that appear in the statement of theorems A and B.Any semiextraspecial group with center of the specified order also satisfiesthose hypotheses. As discussed in [23], these form a rather large family ofgroups. However, all of them have class 2. This suggests the question ofwhether or not there are groups of nilpotence class larger than 2 among theexceptional groups in Theorem A and Theorem B. Using GAP [13] we havefound groups of order 2 and nilpotence class 3 with the properties of thosein the statement of Theorem B(i) (for instance, SmallGroup (2 , Question 4.1.
Do there exist -groups G with Z ( G ) elementary abelianof order and all characters in Irr( G | Z ( G )) fully ramified with respect to G/Z ( G ) of arbitrarily large nilpotence class? And if we replace the hypothesisthat Z ( G ) has order by | Z ( G ) | = 4 or | Z ( G ) | = 16 ? The proof of Theorem 1.5 of [27] shows that the best bounds one canhope for in Theorem C and Corollary E are exponential. The bounds wehave obtained, even with the help of results that depend on the classifica-tion of finite simple groups, are super-exponential. This, together with theknown bounds and examples known for the analog problem for permuta-tion representations mentioned in the Introduction, suggests the followingquestions. (Note, however, that we have already seen several differences be-tween results for minimal faithful permutation representations and minimaldimensions of faithful linear representations.)
Question 4.2.
Does there exist a constant c > such that if G is a finitegroup and N E G then rdim( G/N ) ≤ c rdim( G )1 ? Question 4.3.
Does there exist a constant c > such that if G is a finitegroup and N E G then ed( G/N ) ≤ c ed( G )2 ? eferences [1] M. Bardestani, K. Mallahi-Karai, H. Salmasian , Minimal di-mension of faithful representations for p -groups. J. Group Theory (2016), 589–608.[2] M. Bardestani, K. Mallahi-Karai, H. Salmasian , Kirillov’s or-bit method and polynomiality of the faithful dimension of p -groups. Compos. Math. (2019),[3]
G. Berhuy, G.. Favi , Essential dimension: a functorial point of view(after Merkurjev).
Doc. Math. (2003), 279–330.[4] C. Birkar , Singularities of linear systems and boundedness of Fanovarieties, arXiv:1609.05543[5]
J. Bourgain, A. Gamburd , Uniform expansion bounds for Cayleygraphs of SL ( F p ). Ann. of Math. (2008), 625–642.[6]
J. Buhler, Z. Reichstein , On the essential dimension of a finitegroup.
Comp. Math. (1997), 159–179.[7]
S. Cernele, M. Kamgarpour, Z. Reichstein , Maximal repre-sentation dimension of finite p -groups. J. Group Theory (2011),637–647.[8] M. Collins , On Jordan’s theorem for complex linear groups.
J. GroupTheory (2007), 411–423.[9] F. R. DeMeyer, G. J. Janusz , Finite groups with an irreduciblerepresentation of large degree.
Math. Z. (1969), 145–153.[10]
D. Easdown, C. Praeger , On minimal faithful permutation repre-sentations of finite groups.
Bull. Austral. Math. Soc. (1988), 207–220.[11] R. K. Fisher , The number of generators of finite linear groups.
Bull.London Math. Soc. (1974), 10–12.[12] C. Franchi , On minimal degrees of permutation representations ofabelian quotients of finite groups.
Bull. Austral. Math. Soc. (2011),408–413.[13] The GAP group , GAP - Groups, Algorithms, and Programming .Version 4.4, 2004, .[14]
W. Gowers , Quasirandom groups.
Combin. Probab. Comput. (2008), 363–387. 1515] D. Holt, J. Walton , Representing the quotient groups of a finitepermutation group.
J. Algebra (2002), 307–333.[16]
R. B. Howlett, I. M. Isaacs , On groups of central type.
Math. Z. (1982), 555–569.[17]
B. Huppert , Character Theory of Finite Groups . de Gruyter, Berlin,1998.[18]
I. M. Isaacs , Character Theory of Finite Groups . AMS-Chelsea,Providence, 2006.[19]
C. Jensen, A. Ledet, N. Yui , Generic polynomials.
Math. Sci. Res.Inst. Publ Cambridge University Press, Cambridge 2002.[20]
N. Karpenko, A. Merkurjev , Essential dimension of finite p -groups. Invent. Math. (2008), 491–508.[21]
L. Kov´acs, C. Praeger , On minimal faithful permutation represen-tations of finite groups.
Bull. Austral. Math. Soc. (2000), 311–317.[22] L. Kov´acs, G. Robinson , Generating finite completely reduciblelinear groups.
Proc. Amer. Math. Soc. (1991), 357–364.[23]
M. Lewis , Semi-extraspecial groups. In: Advances in algebra, 219-237, Springer Proc. Math. Stat.
Springer, 2019.[24]
A. Lucchini, F. Menegazzo, M. Morigi , On the number of gen-erators and composition length of finite linear groups.
J. Algebra (2001), 427–447.[25]
A. Merkurjev , Essential dimension: a survey
Transform. Groups (2013), 415–481.[26] A. Merkurjev , Essential dimension.
Bull. Amer. Math. Soc. (2017), 635–661.[27] A. Meyer, Z. Reichstein , Some consequences of the Kerpenko-Merkurjev theorem.
Doc. Math. (2010), 445–457.[28]
P. Neumann , Some algorithms for computing with finite permutationgroups. In: Groups-St. Andrews 1985, LMS Lecture Notes Series ,Cambridge University Press, 1987, 59–92.[29]
Z. Reichstein , The Jordan property of Cremona groups and essentialdimension.
Arch. Math. (2018), 449–455.[30]
G. Robinson , On linear groups.
J. Algebra (1990), 527–534.1631]
F. Szechtman , Groups having a faithful irreducible representation.
J. Algebra (2016), 292–307.[32]