Positions of characters in finite groups and the Taketa inequality
PPositions of characters in finite groups and the Taketa inequality
Tobias Kildetoft ∗ Aarhus UniversityAugust 6, 2018
Mathematics Subject Classification (2010): 20C15.Keywords: Solvable groups, Derived length, Character degrees.
Abstract
We define the position of an irreducible complex character of a finite group as an alternative tothe degree. We then use this to define three classes of groups: PR-groups, IPR-groups and weakIPR-groups. We show that IPR-groups and weak IPR-groups are solvable and satisfy the Taketainequality (ie, that the derived length of the group is at most the number of degrees of irreduciblecomplex characters of the group), and we show that any M-group is a weak IPR-group. We alsoshow that even though PR-groups need not be solvable, they cannot be perfect.
Let G be a finite group and let cd( G ) = { χ (1) | χ ∈ Irr( G ) } be the set of (irreducible) characterdegrees of G . It is a conjecture that if G is solvable then dl( G ) ≤ | cd( G ) | where dl( G ) denotes thederived length of G . This inequality is called the Taketa inequality.One of the first results in the direction of the above inequality was the theorem of Taketa ([Isa76,Theorem 5.12]) that if G is an M-group then G is solvable and satisfies the Taketa inequality. Someother conditions under which a finite solvable group is known to satisfy the Taketa inequality are | G | odd ([MW93, Corollary 16.7]) and | cd( G ) | ≤ G is a finite solvable group which does not satisfy any of the above conditions, there are stilltwo known bounds on dl( G ) in terms of | cd( G ) | , namely dl( G ) ≤ | cd( G ) | − G ) ≤ | cd( G ) | + 24log ( | cd(G) | ) + 364 ([Kel03, Theorem 3.6]). The latter is the better of thetwo bounds when | cd( G ) | ≥ G as an alternative to thedegree, and use these positions to define certain classes of groups, which we call position reduciblegroups (PR-groups), inductively position reducible groups (IPR-groups) and weak IPR-groups.We then show that any IPR-group is a weak IPR-group and that any weak IPR-group is solvableand satisfies the Taketa inequality. We also show that if G is an M-group then G is a weak IPR-group, and we conjecture that in fact G is an IPR-group. ∗ Supported in part by the center of excellence grant ’Center for Quantum Geometry of Moduli Spaces’ from theDanish National Research Foundation (DNRF95) a r X i v : . [ m a t h . G R ] D ec R-groups need not be solvable, but we show that if G is a PR-group, then the derived subgroupof G is not perfect (and hence neither is G ). We also show that if G is a solvable PR-group with atleast 6 character degrees, then dl( G ) ≤ | cd( G ) | − Acknowledgements
I would like to thank Jørn B Olsson for acting as advisor on my master’s thesis, in which the ideasof this paper first emerged.I would also like to thank Mark L. Lewis for reading an early version of the paper and providinghelpful comments.
In this paper, G is a finite group, and character means complex character. We will use the followingnotation. • Irr( G ) is the set of irreducible characters of G . • Lin(G) is the set of linear characters of G (i.e. the characters of degree 1). • Irr( G | G (cid:48) ) is the set of non-linear irreducible characters of G (following the notation of [IK98]). • cd( G ) = { χ (1) | χ ∈ Irr( G ) } is the set of character degrees of G . • ϕ G is the character of G induced from ϕ when ϕ is a character of H for some H ≤ G . • χ H is the restriction of χ to H when χ ∈ Irr( G ) and H ≤ G . • [ χ, ψ ] is the usual normalized inner product of the characters χ and ψ of G .The following lemmas will be used several times in the paper. Lemma 2.1.
Let H ≤ G and let ϕ be a character of H .Then we have ker( ϕ G ) = (cid:92) g ∈ G ker( ϕ ) g In particular, we have ker( ϕ G ) ≤ H .Proof. This is Lemma 5.11 in [Isa76]
Lemma 2.2.
Let H ≤ G , let ϕ be a character of H and assume that N is a normal subgroup of G with N ≤ ker( ϕ ) .Then N ≤ ker( ϕ G ) .Proof. This follows directly from Lemma 2.1. 2
Positions of characters
In this section, we will define the position of an irreducible character of G as an alternative tothe degree. We will then use this to associate certain numbers to arbitrary characters and also tosubgroups. Definition 3.1 (Position of a character) . Let cd( G ) = { f , f , . . . , f n } with f < f < · · · < f n .Let χ ∈ Irr( G ) with χ (1) = f i . Then the position of χ is defined to be pos( χ ) = i .Note that an alternative (but equivalent) way to define the position of an irreducible character χ is pos( χ ) = |{ i ∈ cd( G ) | i ≤ χ (1) }| .Clearly we have pos( χ ) ≤ χ (1).Given some group G , if we know cd( G ), then for any χ ∈ Irr( G ), we have that χ (1) and pos( χ )provide the same information. But if we look at some character “in isolation”, then the two numbersgive us different information.Of course, we have that pos( χ ) = 1 if and only if χ (1) = 1, and if χ (1) = 2 then pos( χ ) = 2. Butpos( χ ) = 2 need not imply that χ (1) = 2 (in fact, χ (1) can be arbitrarily large in this situation).The concept of position of irreducible characters has in fact been used several places in theliterature already, though without giving a name or a notation to it. Examples include the preciseformulation of the theorem of Taketa on the solvability of M-groups already mentioned, as well asthe normal series D i ( G ) (4.11).One motivation for looking at positions rather than degrees of the characters is that it allowsus to make the following definitions much easier. Definition 3.2 (Taketa-character) . Let χ ∈ Irr( G ) with pos( χ ) = i . We say that χ is a Taketa-character if G ( i ) ≤ ker( χ ).It is clear that any linear character is a Taketa-character. Definition 3.3 (Taketa-group) . G is said to be a Taketa-group if all the irreducible characters of G are Taketa-characters.The mentioned theorem of Taketa can now be stated as “If G is an M-group then G is aTaketa-group”, and this is the reason for the choice of the name.Clearly, if G is a Taketa-group then G is solvable and G satisfies the Taketa inequality (sincethe intersection of the kernels of all the irreducible characters is trivial).Unlike degrees, it does not make sense to speak of the position of a character if it is notirreducible. There are, however, two distinguished “positions” of any character, which we will beinterested in. Definition 3.4 (Maximal and minimal position) . Let χ be a character of G . Let ψ, ϕ ∈ Irr( G ) beconstituents of χ of largest and smallest degrees, respectively. Then we define pos max ( χ ) = pos( ψ )and pos min ( χ ) = pos( ϕ ), the maximal and minimal position of χ .We can also use these positions to assign a number to any subgroup of G . This number willbehave a bit like the index of the subgroup, though with some notable exceptions. Definition 3.5 (Positional index of a subgroup) . Let H ≤ G . We define[ G : H ] pos = min ψ ∈ Irr( H ) { pos max ( ψ G ) } and call it the positional index of H in G . 3hus, [ G : − ] pos is a function from { H | H ≤ G } to { , , . . . , | cd( G ) |} . It has the followingproperties. Proposition 3.6.
Let K and H be subgroups of G .(1) If K ≤ H then [ G : K ] pos ≥ [ G : H ] pos .(2) [ G : G ] pos = [ G : G (cid:48) ] pos = 1 .(3) [ G : H ] pos = 1 ⇔ G (cid:48) ≤ H .(4) [ G : { } ] pos = | cd ( G ) | .(5) If H < G then [ G : H ] pos < [ G : H ] .Proof. (1) Let ϕ ∈ Irr( K ) minimize pos max (( − ) G ) and let ψ ∈ Irr( H ) be a constituent of ϕ H . Now ψ G is a constituent of ϕ G so pos max ( ψ G ) ≤ pos max ( ϕ G ) which proves the claim.(2) By (1) it is enough to show that [ G : G (cid:48) ] pos = 1. But we have that (1 G (cid:48) ) G = (cid:80) χ ∈ Lin(G) χ so thisis clear.(3) One direction is clear by (1) and (2). So assume that [ G : H ] pos = 1. This means that there issome ϕ ∈ Irr( H ) such that ϕ G is a sum of linear characters. But then G (cid:48) ≤ ker( ϕ G ) ≤ H byLemma 2.1 as claimed.(4) This is clear since (1 { } ) G is the regular character of G which has all irreducible characters asconstituents.(5) This follows because (1 H ) G has degree [ G : H ] and has 1 G as a constituent, so the degree ofany constituent is strictly less than [ G : H ] and hence so is the position.We will be interested in when it is possible to find a subgroup H such that [ G : H ] pos is smallcompared to the positions of certain characters of H (made precise below). Note that given some ϕ ∈ Irr( H ), the number pos max ( ϕ G ) is smallest if ϕ G splits into as many constituents as possible.When calculating [ G : H ] pos it would be nice if we knew something about which characters ϕ ∈ Irr( H ) minimize pos max ( ϕ G ). Unfortunately, it is not easy to say much about this. It is forexample not always the case that it is minimized by a linear character. An example of this can befound by looking at the group G = SL (3) × C , which has a subgroup H of order 8 isomorphic to Q such that pos max ( ϕ G ) is minimized by a character ϕ ∈ Irr( H ) with pos( ϕ ) = 2 (and it is strictlylarger for all the linear characters of H ). Definition 3.7 (Position reducing tuple (PRT)) . Let
H < G , χ ∈ Irr( G ) and ϕ ∈ Irr( H ) be aconstituent of χ H .( G, H, χ, ϕ ) is said to be a position reducing tuple (PRT) if pos( ϕ ) + [ G : H ] pos ≤ pos( χ ).We will also call ( G, H, χ ) a PRT if there is some constituent ϕ ∈ Irr( H ) of χ H such that( G, H, χ, ϕ ) is a PRT. Note that to check whether (
G, H, χ ) is a PRT it is enough to check whetherpos min ( χ H ) + [ G : H ] pos ≤ pos( χ ). The reason we also need to consider PRTs with the constituent ϕ of χ H specified is that sometimes, it will be useful to be able to put extra requirements on thisconstituent. 4lso note that allowing H = G in the definition of PRT would not change anything, as ( G, G, χ )can never be a PRT. We have chosen to require
H < G for practical reasons that will be apparentwhen we define IPR-groups (Definition 4.3).Part of the motivation behind the definition of a PRT is that if (
G, H, χ ) is a PRT, then insome respects, the character χ behaves as if it was induced from a linear character of H (see alsoProposition 3.9). Definition 3.8 (Position reducible character (PR-character)) . We say that χ ∈ Irr( G ) is a positionreducible character (PR-character) if there is some H < G such that (
G, H, χ ) is a PRT.
Proposition 3.9. If χ ∈ Irr ( G | G (cid:48) ) is monomial, then χ is a PR-character.Proof. Let H ≤ G and ϕ ∈ Lin(H) with ϕ G = χ . We then see that ( G, H, χ, ϕ ) is a PRT, since wehave pos( ϕ ) = 1 and [ G : H ] pos ≤ pos max ((1 H ) G ) < pos( χ ) since 1 G is a constituent of (1 H ) G andthe latter has the same degree as χ .As mentioned, it will sometimes be useful to put a further condition on the constituent ϕ of χ H in the definition of a PRT. The precise requirement we will need in this paper is the following. Definition 3.10 (Taketa-PR-character) . χ ∈ Irr( G ) is said to be a Taketa-PR-character if thereis a PRT ( G, H, χ, ϕ ) such that ϕ is a Taketa-character.Since any linear character is a Taketa-character, we see that any monomial χ ∈ Irr( G | G (cid:48) ) is alsoa Taketa-PR-character (by the proof of Proposition 3.9).The following two results say something about when the derived subgroup can be used to forma PRT. Lemma 3.11.
Let χ ∈ Irr ( G | G (cid:48) ) and ϕ ∈ Irr ( G (cid:48) ) be a constituent of χ G (cid:48) . Then ( G, G (cid:48) , χ, ϕ ) is aPRT if and only if pos ( ϕ ) < pos ( χ ) .Proof. This follows directly from the fact that [ G : G (cid:48) ] pos = 1.The following result has proven surprisingly useful in determining whether specific groups wereIPR-groups or weak IPR-groups (see later definitions). In specific cases, this result will often besufficient to prove that a group is a PR-group, and iterating it by taking further derived subgroupsuntil one gets a nilpotent group will then sometimes allow one to conclude that the group is anIPR-group (using Proposition 4.4). Proposition 3.12.
Let χ ∈ Irr ( G | G (cid:48) ) . If ( G, G (cid:48) , χ ) is not a PRT then there exists a ϕ ∈ Irr ( G (cid:48) ) such that pos ( ϕ ) ≥ pos ( χ ) and ϕ (1) t divides χ (1) where t is the index of the stabilizer of ϕ in G .Proof. If (
G, G (cid:48) , χ ) is not a PRT, then by Lemma 3.11 we have that pos min ( χ G (cid:48) ) ≥ pos( χ ) so thereis a ψ ∈ Irr( G (cid:48) ) which is a constituent of χ G (cid:48) and such that pos( ψ ) ≥ pos( χ ).Since G (cid:48) is normal in G , however, we know from Clifford’s Theorem that χ G (cid:48) = e (cid:80) ti =1 ψ i where e = [ χ G (cid:48) , ψ ] and the ψ i are the different conjugates of ψ in G . In particular, we have that χ (1) = χ G (cid:48) (1) = etψ (1), so ψ (1) t divides χ (1) as claimed.Another consequence of Lemma 3.11 is Proposition 3.13.
Let χ ∈ Irr ( G | G (cid:48) ) be given and assume that G (cid:48)(cid:48) ≤ ker( χ ) . Then ( G, G (cid:48) , χ ) is aPRT. roof. By Lemma 3.11 we just need to show that pos min ( χ G (cid:48) ) < pos( χ ). In fact we claim thatpos min ( χ G (cid:48) ) = 1.Since χ G (cid:48)(cid:48) is a multiple of 1 G (cid:48)(cid:48) we get that all irreducible constituents of χ G (cid:48) are constituents of(1 G (cid:48)(cid:48) ) G (cid:48) , but these are all linear, which completes the proof.In this paper, there are three main types of characters considered: PR-characters, Taketa-PR-characters and Taketa-characters. The following results shows that for a character of position 2,these concepts coincide. Proposition 3.14.
Let χ ∈ Irr ( G ) with pos ( χ ) = 2 . Then the following are equivalent:(1) χ is a PR-character(2) χ is a Taketa-PR-character(3) χ is a Taketa-characterProof. (1) = ⇒ (2): Assume that χ is a PR-character and let ( G, H, χ, ϕ ) be a PRT. We then havepos( ϕ ) + [ G : H ] pos ≤ pos( χ ) = 2 so the only possibility is that pos( ϕ ) = [ G : H ] pos = 1 whichmeans that ϕ is linear and thus a Taketa character, so χ is a Taketa-PR-character.(2) = ⇒ (3): Assume that χ is a Taketa-PR-character and let ( G, H, χ, ϕ ) be a PRT with ϕ aTaketa-character. As above, we get pos( ϕ ) = [ G : H ] pos = 1 so by Proposition 3.6 (3) we must have G (cid:48) ≤ H and hence H (cid:69) G so since pos( ϕ ) = 1 this holds for all constituents of χ H and thus also forall constituents of χ G (cid:48) . Hence restricting χ to G (cid:48) gives only linear constituents, and since the linearcharacters of G (cid:48) have G (cid:48)(cid:48) in their kernel, this shows that G (cid:48)(cid:48) ≤ ker( χ ), so χ is a Taketa-character.(3) = ⇒ (1): Assume that χ is a Taketa-character. Since pos( χ ) = 2 this means that G (cid:48)(cid:48) ≤ ker( χ ) so by Proposition 3.13 we have that ( G, G (cid:48) , χ ) is a PRT, and hence χ is a PR-character asclaimed. In this section we will turn our attention to properties of the group G related to the previouslydefined properties of the characters of G .Since we have defined a certain type of character for G , it is natural to look at groups where allcharacters are of this type. Definition 4.1 (Position reducible group (PR-group)) . G is said to be a position reducible group(PR-group) if all χ ∈ Irr( G | G (cid:48) ) are PR-characters.Note that we only require the non-linear irreducible characters to be PR-characters. This isbecause a linear character can never be a PR-character.This also means that any abelian group will automatically be a PR-group, as it has no non-linearirreducible characters.We now have three definitions with names containing PR (PRT, PR-character and PR-group).The PR part means essentially the same thing in all of these, and the concepts are very muchrelated. To summarize, a PR-group is one where all non-linear irreducible characters are PR-characters, and PR-characters are those irreducible characters can can be part of a PRT. The twodifferent “versions” of PRT that we have (one with four entries and one with 3 entries) are thesame thing, where in one of them, we suppress part of the information.6 orollary 4.2. If G is an M-group then G is a PR-group.Proof. This is clear from Proposition 3.9.One problem with PR-groups is that subgroups of PR-groups need not be PR-groups themselves.The same is true for M-groups, but where one can often say nice things about M-groups becausethe characters one considers for the subgroups are linear, this is not the case for PR-groups.One could remedy this for PR-groups by requiring the character ϕ in a PRT ( G, H, χ, ϕ ) to belinear. But this would be unnecessarily restrictive, as we can still say interesting things withoutthis. Instead, we will consider the following groups.
Definition 4.3 (Inductively position reducible group (IPR-group)) . G is said to be an inductivelyposition reducible group (IPR-group) if for each χ ∈ Irr( G | G (cid:48) ) there exists an H < G such that H is an IPR-group and ( G, H, χ ) is a PRT.Note that since we require
H < G , the recursive nature of the definition is not a problem (andjust as for PR-groups, we can see that any abelian group is vacuously an IPR-group).We see that if P is some property such that any group satisfying P is a PR-group and suchthat P is inherited by subgroups, then any group satisfying P is an IPR-group. In particular, byProposition 3.9 we see that if P is a property such that any group satisfying P is an M-group andsuch that P is inherited by subgroups, then any group satisfying P is an IPR-group. A special caseof this is the following: Proposition 4.4. If G is supersolvable, then G is an IPR-group. In particular, if G is nilpotent,then G is an IPR-group.Proof. That a supersolvable group is an M-group follows from [Isa76, Theorem 6.22], so the claimfollows from the comments preceding the proposition, since any subgroup of a supersolvable groupis itself supersolvable.Emulating the definition of a PR-group, now using Taketa-PR-characters instead, we get thefollowing.
Definition 4.5 (Weak IPR-group) . G is said to be a weak IPR-group if all χ ∈ Irr( G | G (cid:48) ) areTaketa-PR-characters.The justification for the term weak IPR-groups is given in Corollary 4.7.By the proof of Proposition 3.9, we see that any M-group is also a weak IPR-group (since alllinear characters are Taketa-characters).The following result is one of the main motivations for studying IPR-groups. Theorem 4.6. If G is an IPR-group then G is a Taketa-group.Proof. Let χ ∈ Irr( G ) with pos( χ ) = i . We then need to show that G ( i ) ≤ ker( χ ).The proof will proceed by induction on i and | G | (in the lexicographic ordering of N × N ). If | G | = 1 the result is trivial and if i = 1, χ is linear, so the result also holds in this case. Assumetherefore that | G | > i > H < G be given such that H is an IPR-group and let ϕ ∈ Irr( H ) such that ( G, H, χ, ϕ ) isa PRT. This means that there is some ψ ∈ Irr( H ) such that pos max ( ψ G ) + pos( ϕ ) ≤ i , so let sucha ψ be given. In particular, we then have pos max ( ψ G ) < i .7e now get ker( ψ G ) ≤ H by Lemma 2.1, but since this kernel is the intersection of the kernelsof its irreducible constituents and since we have pos max ( ψ G ) < i , we get by induction that G ( k ) ≤ H where k = pos max ( ψ G ) (since it must be contained in the kernel of each irreducible constituent of ψ G ).Let n = pos( ϕ ). Since | H | < | G | and H is an IPR-group, we get by induction that H ( n ) ≤ ker( ϕ ).Since we now have that G ( k ) ≤ H and H ( n ) ≤ ker( ϕ ) we get that G ( n + k ) ≤ ker( ϕ ). Since G ( n + k ) is normal in G , we thus get that G ( n + k ) ≤ ker( ϕ G ) by Lemma 2.2. Since ϕ is a constituent of χ H ,we also have that χ is a constituent of ϕ G by Frobenius reciprocity, so ker( ϕ G ) ≤ ker( χ ). Thus, G ( n + k ) ≤ ker( χ ), and since we have n + k = pos( ϕ ) + pos max ( ψ G ) ≤ i this completes the proof.The following corollary justifies the use of the term weak IPR-group. Corollary 4.7. If G is an IPR-group then G is a weak IPR-group.Proof. This is clear from Theorem 4.6.
Theorem 4.8. If G is a weak IPR-group then G is a Taketa-group.Proof. Let χ ∈ Irr( G ) with pos( χ ) = i . We need to show that G ( i ) ≤ ker( χ ). We will proceed byinduction on i .If i = 1 then the statement is trivial, so assume i > H < G and ϕ ∈ Irr( H ) be given such that ϕ is a Taketa-character and ( G, H, χ, ϕ ) is aPRT. Let n = pos( ϕ ), so H ( n ) ≤ ker( ϕ ) since ϕ is a Taketa-character.Let ψ ∈ Irr( H ) such that pos( ϕ ) + pos max ( ψ G ) ≤ i . We have ker( ψ G ) ≤ H by Lemma 2.1 andsince pos max ( ψ G ) < i we have by induction that G ( k ) ≤ ker( ψ G ) ≤ H where k = pos max ( ψ G ).We now have that G ( n + k ) ≤ ker( ϕ ) and since G ( n + k ) is normal in G , this means that G ( n + k ) ≤ ker( ϕ G ) by Lemma 2.2. Since ϕ is a constituent of χ H we have that χ is a constituent of ϕ G byFrobenius reciprocity, so we get that G ( n + k ) ≤ ker( χ ), and since n + k = pos( ϕ ) + pos max ( ψ G ) ≤ i this completes the proof.An immediate consequence of Lemma 3.11 is the following. Proposition 4.9. If dl(G) ≤ then G is an IPR-group.Proof. Since G (cid:48) is abelian, it is an IPR-group, and for any χ ∈ Irr( G | G (cid:48) ) it is clear from Lemma3.11 that ( G, G (cid:48) , χ ) is a PRT.If G is a PR-group then G need not be solvable, but we do have the following result, showingthat it is at least not perfect (so in particular not simple). Corollary 4.10. If G is a non-abelian PR-group then G (cid:48) is not perfect. In particular, G is notperfect.Proof. Let χ ∈ Irr( G ) with pos( χ ) = 2. By Proposition 3.14 we know that G (cid:48)(cid:48) ≤ ker( χ ). But sincepos( χ ) (cid:54) = 1 we have that G (cid:48) (cid:54)≤ ker( χ ) so we cannot have G (cid:48)(cid:48) = G (cid:48) which proves the claim.We can also use the above to get a bound on the derived length of a solvable PR-group in termsof the number of character degrees. The bound we obtain is slightly better than what is knownfor arbitrary solvable groups as long as the number of character degrees is not too large (see theintroduction). First, we define a specific normal series for G and recollect some of the properties ofthis. 8 efinition 4.11. Let i ≥ i (G) = (cid:92) χ ∈ Irr(G) , pos( χ ) ≤ i ker( χ )If n = | cd( G ) | then it is now clear that we have a normal series { } = D n (G) ≤ D n − (G) ≤ · · · ≤ D (G) ≤ D (G) = Gwhere we use the convention that an empty intersection of subgroups of G is G itself. It is also clearthat we have D (G) = G (cid:48) and that all the D i (G) are normal in G . Some further properties of theD i (G) are listed in the following theorem: Theorem 4.12.
Assume G is solvable and let n = | cd ( G ) | .(1) For all ≤ i ≤ n we have dl(D i − (G) / D i (G)) ≤ (2) If dl(D i − (G) / D i (G)) = 3 for some ≤ i ≤ n then dl(D i − (G) / D i+1 (G)) ≤ (3) If i ≤ then dl(D n − i (G)) ≤ i Proof. (1) This follows from [MW93, Theorem 16.5].(2) This is [MW93, Theorem 16.8] (the conditions there are the same as here because of (1)).(3) This is [Kil12, Lemma 4.1].We also have the following corollary to Proposition 3.14.
Corollary 4.13.
Assume that G is a solvable PR-group. Then dl(G / D (G)) ≤ .Proof. This follows directly from Proposition 3.14.Note that we just need to assume that all irreducible characters of position 2 are PR-charactersfor the above proof to work.We can then combine these statements to get the following. The idea of the proof is the sameas in the proof of [Kil12, Theorem 1].
Theorem 4.14. If G is a solvable PR-group with | cd ( G ) | ≥ then dl(G) ≤ | cd (G) | − .Proof. Let n = | cd( G ) | . Let E = D (G) = G, E = D (G) = G (cid:48) , E = D (G) and define E i for3 ≤ i ≤ n − E i has already been defined by a previous step, skip to i + 1.If D i − (G) (cid:48)(cid:48) ≤ D i (G), set E i = D i (G).Otherwise, set E i = D i − (G) (cid:48)(cid:48) and E i +1 = D i+1 (G).Set E n − = D n − (G) (this is consistent with the previous rule), and E n − = D n − (G).We now have a new normal series for G given by { } ≤ E n − ≤ E n − ≤ · · · ≤ E ≤ E = G
9y Corollary 4.13 we have dl(E / E ) ≤
2, and by construction along with Theorem 4.12part 1, we get that dl( E i − /E i ) ≤ ≤ i ≤ n −
4. By Theorem 4.12 part 2, we also havedl( E n − /E n − ) ≤
3. Finally, Theorem 4.12 part 3 gives dl( E n − ) ≤ G ) ≤ dl( E /E ) + n − (cid:88) i =3 dl( E i − /E i ) + dl( E n − /E n − ) + dl( E n − ) ≤ n − · n − | cd( G ) | − G is a solvable PR-group with 5 character degrees,then dl(G) ≤ | cd(G) | . This is already known to hold for arbitrary solvable groups with 5 characterdegrees however, as mentioned in the introduction. As previously noted, if G is an M-group then G is a weak IPR-group. But looking at the groups upto order 384 in GAP, one can see that at least up to this order, all M-groups are in fact IPR-groups.This is of course not a very large order to compute up to, but unfortunately due to the recursivenature of the definition of an IPR-group, it takes a lot of time and memory to do this for largergroups.The reason it is hard to tell whether an M-group is necessarily an IPR-group is that not muchis known about what conditions guarantee a subgroup of an M-group to be an M-group itself.One thing to note is that the example given by Dade of an M-group which contains a normalsubgroup which is not itself an M-group ([Dad73]), is in fact an IPR-group, but so is the normalsubgroup which is not an M-group. This can be seen as follows: Let G be Dade’s example. Thencd( G ) = { , , , , } , cd( G (cid:48) ) = { , , } and cd( G (cid:48)(cid:48) ) = { , } . But | G (cid:48)(cid:48) | = 128 so this subgroup isnilpotent, and the claim now follows from applying Lemma 3.12 twice. The same type of argumentalso gives the claim for the normal subgroup which is not an M-group.Also worth noting is that all Taketa-groups of order at most 1000, except possible those of order768, are PR-groups (by GAP computations), and as noted in Proposition 3.14 there is at least someconnection between the properties.If it is indeed the case that any Taketa-group is a PR-group, then if P is a property such thatany group satisfying P is a Taketa-group and such that P is inherited by subgroups, then any groupsatisfying P will be an IPR-group. An interesting special case of this would be that any group ofodd order is an IPR-group, since these are Taketa-groups by the proof of [MW93, Corollary 16.7]. References [Dad73] Everett C. Dade. Normal subgroups of M -groups need not be M -groups. Math. Z. ,133:313–317, 1973. 10IK98] I. M. Isaacs and Greg Knutson. Irreducible character degrees and normal subgroups.
J.Algebra , 199(1):302–326, 1998.[Isa76] I. Martin Isaacs.
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