Groups of p-central type

A finite group G with center Z is of central type if there exists a fully ramified character λ∈Irr(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in \textrm{Irr}(Z)$$\end{document}, i. e. the induced character λG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda ^G$$\end{document} is a multiple of an irreducible character. Howlett–Isaacs have shown that G is solvable in this situation. A corresponding theorem for p-Brauer characters was proved by Navarro–Späth–Tiep under the assumption that p≠5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\ne 5$$\end{document}. We show that there are no exceptions for p=5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=5$$\end{document}, i. e. every group of p-central type is solvable. Gagola proved that every solvable group can be embedded in G/Z for some group G of central type. We generalize this to groups of p-central type. As an application we construct some interesting non-nilpotent blocks with a unique Brauer character. This is related to a question by Kessar and Linckelmann.


Introduction
For an irreducible character χ of a finite group G it is easy to show that χ(1) 2 ≤ |G : Z| where Z := Z(G) denotes the center of G (see [13,Corollary 2.30]).If equality holds for some χ ∈ Irr(G), we say that G has central type.It was conjectured by Iwahori-Matsumoto and eventually proved by Howlett-Isaacs [11], using the classification of finite simple groups (CFSG), that all groups of central type are solvable (see also [23,Chapter 8]).Apart from this there are no restrictions on the structure of groups of central type.In fact, Gagola [6,Theorem 1.2] has shown that every solvable group can be embedded into G/Z for some group G of central type.
It is well-known that G is of central type if there exists some fully ramified character λ ∈ Irr(Z), i. e. λ G = eχ for some integer e and some χ ∈ Irr(G) (in this case χ(1) 2 = e 2 = |G : Z|; see [23,Lemma 8.2]).This can be carried over to Brauer characters over an algebraically closed field of characteristic p > 0. We call λ ∈ IBr(Z) fully ramified if λ G = eφ for some integer e and φ ∈ IBr(G).In this situation, Navarro-Späth-Tiep [24, Theorem A] proved (again relying on the CFSG) that G is solvable unless p = 5.Using the structure of a minimal counterexample for p = 5 (as described in [24]), we are able to eliminate this exceptional case.
Theorem 1.Let G be a finite group, Z ⊴ G and λ ∈ IBr(Z) be G-invariant.Suppose that λ G = eφ for some integer e and φ ∈ IBr(G).Then G/Z is solvable.
Notice that Theorem 1 generalizes the Howlett-Isaacs theorem by choosing a prime which does not divide |G|.Assuming that G is (p-)solvable, there exists a fully ramified Brauer character in Z = Z(G) if and only if φ(1) 2 = |G : Z| p ′ for some φ ∈ IBr(G) where n p ′ denotes the p ′ -part of an integer n (see Proposition 6 below).If this is the case, we call G a group of p-central type.Our second objective is to carry over Gagola's theorem as follows.
Theorem 2. Let G be a solvable group and p be a prime.Then there exists a group H of p-central type such that G is isomorphic to a subgroup of H/Z(H).
Further properties of the group H in Theorem 2 are stated in Theorem 7 below.Theorems 1 and 2 are proved in the next section.
In Section 3 we apply our results to p-blocks B of G with defect group D. The omnipresent nilpotent blocks introduced by Broué-Puig [3] have a unique irreducible Brauer character, i. e. l(B) := |IBr(B)| = 1.We are interested in non-nilpotent blocks with l(B) = 1.These occur far less frequent, e. g. they cannot be principal blocks.Indeed, it was speculated by Kessar and Linckelmann that all such blocks are Morita equivalent to their Brauer correspondent in N G (D) (if D is abelian, this is equivalent to Broué's conjecture by [27,Theorem 3]).Using Külshammer's reduction [17] and Theorem 1, all such blocks should occur in solvable groups up to Morita equivalence.
Concrete examples can be build as follows: If H is a non-trivial p ′ -group of central type acting on an elementary abelian p-group V with kernel 23), which first appeared in Kiyota [15].Kessar [14] has shown that every non-nilpotent block with l(B) = 1 and defect group C 2 3 is Morita equivalent to this block (see also [16]).Similar examples with abelian defect group were investigated in [1,2,10,12,18].All these blocks arise from groups with a normal abelian Sylow p-subgroup.As an application of Theorem 2, we illustrate that non-nilpotent blocks with l(B) = 1 occur in arbitrarily "complicated" solvable groups.
Theorem 3. Let p be a prime and l ≥ 1 an integer.Then there exists a solvable group G of p-length l p (G) ≥ l such that G has a non-nilpotent p-block B of maximal defect with l(B) = 1.
On the other hand, we construct some non-solvable examples using the following recipe.Theorem 4. Let b be a p-block of a finite group H such that l(b) is not a p-power (in particular l(b) > 1).Suppose that there exists an automorphism group A ≤ Aut(H) such that A acts regularly on IBr(b).Let Q be a p-group upon A acts non-trivially.Then the group G = (H × Q) ⋊ A where A acts diagonally has a non-nilpotent p-block B with l(B) = 1.
Finally we study lifts of Brauer characters.By the Fong-Swan theorem, every irreducible Brauer character φ of a p-solvable group G is the restriction of an ordinary character to the set of p-regular elements.This is no longer true in non-solvable groups.However, if IBr(B) = {φ}, then Malle-Navarro-Späth [20] proved (using the CFSG) that φ still has such a lift.For nilpotent blocks with defect group D, the number of these lifts is |D : D ′ |.In particular, φ has at least p 2 lifts unless |D| ≤ p.For nonnilpotent blocks, it was conjectured by Malle-Navarro [19] (in combination with Olsson's conjecture) that the number of lifts is strictly less than |D : D ′ |.Answering a question of G. Navarro, we construct (non-nilpotent) blocks with IBr(B) = {φ} such that φ has a unique lift.

Groups of p-central type
For a finite group G we denote the Fitting subgroup by F(G) and the layer by E(G) (the product of all components of G).Then the generalized Fitting subgroup is a central product F * (G) = F(G) * E(G).Our notation for characters follows Navarro's book [21].
We need the following lemma from the theory of group extensions (see [8,Theorem 15.3.1]).
Lemma 5. Let N be a finite group, α ∈ Aut(N ) and m ∈ N. Then the following assertions are equivalent: (1) There exists a finite group (2) There exists n ∈ N such that α(n) = n and α m (x) = nxn −1 for all x ∈ N .
Proof of Theorem 1.By [24, Theorem A], we may assume that p = 5.Let G be a minimal counterexample with respect to |G : Z|.Then by [24, Theorem 8.1 and its proof], the following holds: (In the notation of [24] we have If a induces an inner automorphism on every T i , then we have the contradiction a ∈ F * (G)C G (F * (G)) ≤ F * (G).Thus, without loss of generality we may assume that a induces an outer automorphism α of T 1 .Then α(t 1 ) = t 1 and α 2 is the inner automorphism on T 1 induced by t 1 .Hence, by Lemma 5, there exists a group H with T 1 ⊴ H and ) is isomorphic to one of the three subgroups of Aut(A 6 ) of index 2, namely S 6 , PGL (2,9) or the Mathieu group M 10 .However, none of these three groups has a 6-fold Schur extension as can be checked with GAP [7].This contradiction shows that N = F * (G).Now the 2-group G/N is a transitive permutation group of degree m.Therefore, |G : N | is bounded by the order of a Sylow 2-subgroup of the symmetric group S m .This yields (2.1) By part (iv), there exists χ ∈ Irr(P ) such that χ(1) 2 = |P : Z 2 |.For a (linear) constituent µ ∈ Irr(A) of the restriction χ A we have χ(1) ≤ µ P (1) = |P : A|.Hence, and w ≥ 2 m .This contradicts (2.1).
The original conjecture by Iwahori-Matsumoto has been generalized by J. F. Humphreys (as mentioned in [9]) and independently by Navarro [22,Conjecture 11.1] to the following statement: Let N ⊴ G and λ ∈ Irr(N ) be G-invariant such that all irreducible constituents of λ G have the same degree.Then G/N is solvable.This conjecture is still open, but the corresponding version for Brauer characters does not hold.Indeed the Schur cover 6.A 6 considered in the proof above is a counterexample for p = 5.
Next, we prove the equivalent characterization of groups of p-central type mentioned in the introduction.
It has been remarked in [24] that Proposition 6 does not hold for the non-solvable group G = SL (2,17) when p = 17.Another counterexample with a different flavor is the direct product G = SL(2, 5) 2 for p = 5.Indeed, SL(2, 5) has Brauer characters of degree 3 and 4.So G has a Brauer character of degree 12.

Now we prove a strong version of Theorem 2.
Theorem 7. Let G be a solvable group and p be a prime.Then there exists a solvable H with the following properties: (i) Z := Z(H) has square-free p ′ -order.
(iii) G is isomorphic to a subgroup of H/Z.
Here we need to modify the construction along the lines of [6].Suppose first that |Z(K)| is not divisible by q.Let Q := ⟨x⟩ ∼ = C q and ρ ∈ Irr(Q) be faithful.We replace (K, µ) by . Thus, k ∈ Z(K) and µ(k) is an integral multiple of a root of unity.On the other hand, where Q n denotes the n-th cyclotomic field.This implies µ(k) = ±µ(1).In the case µ(k) = −µ(1) we have q ̸ = 2 and therefore ρ( Thus, µ 1 is faithful.From now on we may assume that q divides |Z(K)|.
Let z ∈ Z(K) be an element of order q and let Y := ⟨y⟩ ∼ = C q .It is easy to check that the map is a well-defined automorphism of order q.We define H := (U × Y ) ⋊ ⟨α⟩.Let h := ((x 1 , . . ., x q )W, y i , α j ) ∈ Z(H).

Applications to blocks
For N ⊴ G and λ ∈ IBr(N ) we define IBr(G|λ) as the set of irreducible constituents of λ G as usual.
Proof of Theorem 3. We may assume that l ≥ 2. By Theorem 7, there exists a solvable group H of p-central type with l p (H) ≥ l and Z := Z(H) a p ′ -group.Let H/Z act faithfully on an elementary abelian p-group V (for instance, the regular module) and define Let λ ∈ IBr(Z) be fully ramified in H. Since IBr(G) = IBr(H), λ is also fully ramified in G.By a theorem of Fong (see [21,Theorem 10.20]), there exists a block B of G of maximal defect with IBr(B) = IBr(G|λ).In particular, We give a concrete example starting with H := SmallGroup(54, 8) Moreover, G has a 2-block B of maximal defect and l(B) = 1.We remark that B covers the a non-principal block of V ⋊ 3 1+2 , which was investigated in [18].
with a non-nilpotent 3-block B with defect 2 and l(B) = 1.The same construction works with the simple group H = PSL (2,11).Similarly examples can be obtained by wreath products like where α ∈ Aut(H) and σ interchanges the two copies of H.By Kessar [14], such blocks are Morita equivalent to blocks of solvable groups.For p = 5, one can take H = PSL (2,19) (there is certainly an infinite family).As long as we take blocks b of H with cyclic defect group and conjugate Brauer characters in Aut(H), the Brauer tree of b is a star (otherwise there is no graph automorphism permuting the Brauer characters).Then b is a so-called inertial block, i. e. b is basically Morita equivalent to its Brauer correspondent in the normalizer of a defect group (see [25]).The same must be true for the block b ⊗ Q of H × Q.Now a theorem of Zhou [26,Corollary] implies that B is inertial.In particular, B is Morita equivalent to a block of a solvable group.
In general, the block of H ⋊ A covering b is often nilpotent.In this case, a theorem of Puig [25] shows that b is inertial.
There are examples where b has non-cyclic defect groups.For p = 2, we start with H := PSU (3,5) where b has defect group C 2 2 and l(b) = 3 (b is Morita equivalent to the principal block of S 4 ).We take Then G := H ⋊ A 4 has a 2-block with defect group C 4 2 .By Eaton [5] this block is again Morita equivalent to a block of a solvable group.
We now construct a block B with IBr(B) = {φ} such that φ has a unique lift.If a group H acts on a group V , we denote the stabilizer of v ∈ V in H by H v .Theorem 8. Let H be a p ′ -group of central type.Suppose that H/Z(H) acts faithfully on an elementary abelian p-group V such that |H : Proof.Let Z := Z(H) and choose a fully ramified character λ ∈ Irr(Z).Let λ H = eφ for some φ ∈ Irr(H).By Fong's theorem, there exists a block B of G such that Irr(B) = Irr(G|λ) and IBr(B) = IBr(G|λ).We may consider the inflation of φ as an ordinary character and as a Brauer character of B. It suffices to show that every χ ∈ Irr(B)\{φ} has degree χ(1) > φ(1) = e.There exists some non-trivial character µ ∈ Irr(V ) such that µ × λ is a constituent of χ V Z .Since H is a p ′ -group, the actions of H on V and on Irr(V ) are permutation-isomorphic (see [23,Corollary 2.12]).Thus, by Clifford theory and by the hypothesis we have The property |H : H v | > |H v : Z(H)| in Theorem 8 is fulfilled whenever H/Z(H) acts semiregularly on V , i. e. H v = 1 for all v ∈ V \ {1}.In this case, V ⋊ (H/Z(H)) is a Frobenius group with complement H/Z(H).In particular, the Sylow subgroups of H/Z(H) are cyclic or quaternion groups.But then H/Z(H) has trivial Schur multiplier and there is no group H of central type.Hence, |H/Z(H)| is a product of at least four (possibly equal) primes (keeping in mind that |H/Z(H)| is a square).Suitable groups H can be found with GAP [7].For instance, H = SmallGroup(128, 144) is of central type and acts on V ∼ = C 4  5 such that V ⋊ (H/Z(H)) ∼ = PrimitiveGroup(625, 166) with the desired property.The condition in Theorem 8 can be relaxed by taking the degrees of the characters in H µ (where µ ∈ Irr(V )) into account.This leads to a smaller example where H ∼ = SmallGroup(128, 138), V ∼ = C  acting on V ∼ = C 18  2 .We use the opportunity to construct yet another unusual class of blocks.Let Z ⊴ H and λ ∈ Irr(Z).A theorem of Gallagher asserts that |Irr(H|λ)| equals the number of so-called λ-good conjugacy classes of H/Z (see [23,Section 5.5]).If λ is fully ramified, then 1 is the unique λ-good conjugacy class.One may ask when the opposite situation occurs, where all conjugacy classes of H/Z are good.This is holds for instance whenever λ extends to G (then |Irr(H|λ)| = k(H/Z) by Gallagher's theorem).But the converse is not true.For instance, H := SmallGroup(128, 731) has a subgroup Z ≤ Z(H) ∩ H ′ of order 2 such that k(H) = 2k(H/Z).The non-trivial character of Z cannot extend to H since Z ≤ H ′ .Now H/Z acts non-trivially on V ∼ = C 6 3 .It turns out that the solvable group G := V ⋊ H has two 3-blocks B 0 and B 1 with defect group V and inertial quotient H/Z such that k(B 0 ) = k(B 1 ) = 84 and l(B 0 ) = l(B 1 ) = 16 (B 0 is the principal block of G).However, B 0 and B 1 are not Morita equivalent, because they have distinct decomposition matrices (the former matrix contains a 4, but the latter does not).

(
iv) |G|, |H| and |Z| have the same prime divisors apart from p. Proof.We follow closely Gagola's construction [6, Theorem 1.2].Since the trivial group is of p-central type, we may assume that G ̸ = 1.Let M ⊴ G such that q := |G : M | is a prime.By induction on |G|, there exists a solvable group K and a faithful µ ∈ IBr(K) with µ(1) 2 = |K : Z(K)| p ′ fulfilling the conclusion for M instead of G. Let

Proof of Theorem 4 .
Let b 0 be the principal p-block of Q.Then b ⊗ b 0 is a G-invariant block of H × Q.By [21, Corollary 9.21], B := (b ⊗ b 0 ) G is the unique block covering b ⊗ b 0 .Since A acts regularly on IBr(b ⊗ b 0 ), Clifford theory implies that l(B) = 1.Note that (Q, b ⊗ b 0 ) is a B-subpair.Since, G/HQ ∼ = A is not a p-group (as |A| = l(b) is not a p-power), B cannot be nilpotent.It is not so easy to find groups H with blocks b where Aut(H) acts transitively on IBr(b).The quasisimple groups H with the desired property were investigated and partially classified in[20].For alternating and sporadic groups, b must have defect 1.Here, l(b) divides p − 1 and one can choose A ∼ = C l(b) and Q ∼ = C p .The smallest case, p = 3 and H = SL(2,5), leads to the group H has a p-block B such that IBr(B) = {φ} and φ has a unique lift to Irr(B).