Abstract

We consider $p$-blocks with abelian defect groups and in the first part prove a relationship between its Loewy length and that for blocks of normal subgroups of index $p$. Using this, we show that if $B$ is a $2$-block of a finite group with abelian defect group $D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s$, where $a_i > 1$ for all $i$ and $r \geq 0$, then $d < \operatorname {LL}(B) \leq 2^{a_1}+\cdots +2^{a_r}+2s-r+1$, where $|D|=2^d$. When $s=1$ the upper bound can be improved to $2^{a_1}+\cdots +2^{a_r}+2-r$. Together these give sharp upper bounds for every isomorphism type of $D$. A consequence is that when $D$ is an abelian $2$-group the Loewy length is bounded above by $|D|$ except when $D$ is a Klein-four group and $B$ is Morita equivalent to the principal block of $A_5$. We conjecture similar bounds for arbitrary primes and give evidence that it holds for principal $3$-blocks.

1. Introduction

Let $G$ be a finite group and $k$ be an algebraically closed field of characteristic $p$. For a block $B$ of $kG$, write $\operatorname {LL}(B)$ for the Loewy length of $B$, that is, the smallest $n \in \mathbb{N}$ such that $\operatorname {Rad}(B)^n=0$ and write $e_B$ for the block idempotent corresponding to $B$. We are interested in upper and lower bounds for $\operatorname {LL}(B)$ in terms of the isomorphism type of the defect groups. For $p$-solvable groups it is proved in 10, p. 141 that the Loewy length is at most the order of a defect group and in 11 that it is strictly greater than $(p-1)d$, where $d$ is the defect of $B$. The upper bound does not hold when we remove the $p$-solvability hypothesis, as by 7, Theorem 2 the principal $2$-block $B_0(kA_5)$ of $A_5$ (with Klein-four defect groups) has Loewy length $5$, although it is tempting to think that blocks Morita equivalent to $B_0(kA_5)$ are the only counterexamples, as we will see is indeed the case for $p=2$. As remarked in 14 the lower bound $(p-1)d$ also does not hold when we remove the $p$-solvability hypothesis, although $d$ could still be a lower bound. In this paper we restrict our attention to blocks with abelian defect groups, but investigate bounds on the Loewy length of a block for arbitrary finite groups.

Using the results of 6, which depends on the classification of finite simple groups, we prove our main result (see Theorem 4.1).

Theorem.

Let $B$ be a $2$-block with abelian defect group $D$. Suppose $D \cong C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times (C_2)^s$, where $a_i > 1$ for all $i$ and $r \geq 0$. Write $|D|=2^d$.

Then $d < \operatorname {LL}(B) \leq 2^{a_1}+\cdots +2^{a_r}+2s-r+1$. If $s=1$, then $\operatorname {LL}(B) \leq 2^{a_1}+\cdots +2^{a_r}+2-r$.

The upper bounds in the above theorem are sharp for every isomorphism type of $D$: take $B$ to be the principal block of $k(C_{2^{a_1}} \times \cdots \times C_{2^{a_r}} \times SL_2(2^s))$ (see 1). The lower bound was suggested in 14, where there is an excellent discussion of lower bounds on Loewy lengths. Note that for blocks of $p$-solvable groups with abelian defect groups, we have $\operatorname {LL}(B) = \operatorname {LL}(kD)$. We also note that in 19 it is proved that for principal $2$-blocks with abelian defect groups, the Loewy length is bounded above by the maximum of $2d+1$ and $|D|$.

A crucial element in establishing the above bounds is the consideration of the case where there is $N \vartriangleleft G$ such that $G=ND$. For $D$ elementary abelian we may apply the main result of 13, which says that if $G$ is a split extension of $N$ (by a direct factor of $D$), then $B$ behaves as if $N$ were a direct factor of $G$. Theorem 2.1 generalizes this result and we use it to show how for arbitrary abelian $D$ we may compare the Loewy lengths of $B$ and the block of $N$ covered by $B$. This works for all primes, and we hope Theorem 2.1 is of wider interest.

The paper is structured as follows. In Section 2 we give the generalization of the theorem of 13 and the theorem concerning Loewy length of blocks when there is a normal subgroup of index $p$. In Section 3 we give some preliminary results needed for the proof of the bounds, which we give in Section 4. In Section 5 we consider similar bounds for odd primes and make a conjecture.

2. Normal subgroups of index $p$

We first prove a generalization of a theorem of Koshitani and Külshammer 13.

Theorem 2.1.

Let $G$ be a finite group and $B$ a block of $kG$ with abelian defect group $D$ with primary decomposition $D=D_1\times \cdots \times D_t$. Let $N\vartriangleleft G$ such that $G=ND_r$ for some $r\leq t$, $D_j\leq N$ for all $j\neq r$ and $[G:N]=p$. Let $b$ be a block of $kN$ covered by $B$. Then $e_B=e_b$ and there exists an element $a\in Z(B)$ of multiplicative order dividing $|D_r|$ such that $B=\bigoplus _{j=0}^{p-1}a^jkNe_b$.

Proof.

We follow the proof of 13, where it is assumed that $D_r\cap N=1$. Let $\mathcal{A}$ be a root of $B$ in $C_G(D)$ and $N_G(D)_\mathcal{A}$ the stabilizer of $e_\mathcal{A}$ in $N_G(D)$. As in 13, $e_B=e_b$, $Z(B)$ is a crossed product of $Z(b)$ over $G/N$ and $N_G(D)_\mathcal{A}/C_G(D)$ is a $p'$-group.

Now $N_G(D)_\mathcal{A}/C_G(D)$ acts on both $D$ and $D\cap N$ by conjugation. Consider the subgroup $Q\leq D$ generated by $D_r$ and all its $N_G(D)_\mathcal{A}$-conjugates. This group has exponent $|D_r|$ and so by 9, $\S 5$, Theorem 2.2 $Q=Q_1\times Q_2$, where $Q_1$ and $Q_2$ are $N_G(D)_\mathcal{A}$-invariant, $Q_1$ is homocyclic of exponent $|D_r|$ and $Q_2$ has exponent strictly less than $|D_r|$. As $[Q_1:Q_1\cap N]=p$ we have $Q_1\cap N\cong R_1\times R_2$, where $R_1$ is cyclic of order $|D_r|/p$ and $R_2$ is homocyclic of exponent $|D_r|$. Again by 9, $\S 5$, Theorem 2.2 we can assume $R_1$ and $R_2$ are both $N_G(D)_\mathcal{A}$-invariant and therefore by 9, $\S 3$, Theorem 3.2 $R_2$ must have an $N_G(D)_\mathcal{A}$-invariant complement $R$ in $Q_1$. So $R$ is cyclic, $N_G(D)_\mathcal{A}$-invariant, $|R|=|D_r|$ and $[R:R\cap N]=p$.

Now $|\operatorname {Aut}(R)|_{p'}\!=\!p-1$ and the natural homomorphism $\operatorname {Aut}(R)\!\to \!\operatorname {Aut}(R/(R\cap N))$ is surjective and so any non-trivial $p'$-element of $\operatorname {Aut}(R)$ acts non-trivially on $R/(R\cap N)$. However, $[R,N_G(D)_\mathcal{A}]\leq R\cap N$ and so every element of $N_G(D)_\mathcal{A}$ acts trivially on $R/(R\cap N)$. Therefore, $N_G(D)_\mathcal{A}$ commutes with $R$.

Let $\alpha =\mathcal{A}^{C_G(C_D(N_G(D)_\mathcal{A}))}$. By Watanabe’s result 22, Theorem 2(ii), the map

$$\begin{align*} f:Z(B)&\to Z(\alpha ),\\ z&\mapsto \operatorname {Br}_{C_D(N_G(D)_\mathcal{A})}(z)e_\alpha \end{align*}$$

is an isomorphism of $k$-algebras. Note that since $R$ commutes with $N_G(D)_\mathcal{A}$ we have that $R\!\leq \! Z(C_G(C_D(N_G(D)_\mathcal{A})))$ and also $C_G(C_D(N_G(D)_\mathcal{A}))\!=\!RC_N(C_D(N_G(D)_\mathcal{A}))$. Therefore $\alpha$ and hence $Z(\alpha )$ are $G/N$-graded. Recall that $Z(B)$ is also $G/N$-graded and since $e_\alpha \in kN$, $f$ respects these gradings. Setting $a:=f^{-1}(e_\alpha g)$, where $R=\langle g\rangle$, proves the theorem.

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We now use Theorem 2.1 to prove upper and lower bounds on the Loewy length of $B$ in terms of the Loewy length of $b$. Note that using 5, Theorem 3.4(I) one can derive an upper bound of $p\operatorname {LL}(b)$. However, we require a much more precise bound.

Corollary 2.2.

Assume we are in the setting of Theorem 2.1. Then

(i) $\operatorname {Rad}(B)=\operatorname {Rad}(b)Z(B)+(1-a)B$;

(ii) $\operatorname {LL}(b)+(p-1)\leq \operatorname {LL}(B)\leq \operatorname {LL}(b)+|D_r|-|D_r|/p$.

Proof.

(i) Certainly $\operatorname {Rad}(b)Z(B)$ and $(1-a)B$ are nilpotent ideals of $B$ and so

$$\begin{align*} \operatorname {Rad}(B)\supseteq \operatorname {Rad}(b)Z(B)+(1-a)B. \end{align*}$$

Then

$$\begin{align*} B/(\operatorname {Rad}(b)Z(B))=B/(\operatorname {Rad}(b)B)\cong (b/\operatorname {Rad}(b))\otimes _{b}B\cong (b/\operatorname {Rad}(b))\otimes _kkC_p, \end{align*}$$

where the last isomorphism is given by $(x\otimes a)\mapsto (x\otimes g)$, where $C_p=\langle g\rangle$. Therefore

$$\begin{align*} \operatorname {Rad}(B)/(\operatorname {Rad}(b)Z(B)+(1-a)B) \end{align*}$$

is semisimple and so $\operatorname {Rad}(B)=\operatorname {Rad}(b)Z(B)+(1-a)B$.

(ii) If $\gamma _1,\dots ,\gamma _s \in \operatorname {Rad}(b)$ with $\gamma _1\dots \gamma _s\neq 0$, then $\gamma _1\dots \gamma _s (1-a)^{(p-1)}$ is a non-zero product of elements of $\operatorname {Rad}(B)$, and so $\operatorname {LL}(B)\geq \operatorname {LL}(b)+(p-1)$.

Next suppose $\beta _1,\ldots ,\beta _s \in \operatorname {Rad}(B)$ with $\beta _1\dots \beta _s\neq 0$ with $\beta _i=\operatorname {Rad}(B)$. Then by (i) there exist $\gamma _1,\dots ,\gamma _s$ such that $\gamma _1\dots \gamma _s\neq 0$ and each $\gamma _i$ is either $(1-a)$ or is an element of $\operatorname {Rad}(b)$. Say $t$ of the $\gamma _i$’s are $(1-a)$’s. Then clearly $t<|D_r|$ and since $(1-a)^p\in \operatorname {Rad}(b)$ we also have that $\lfloor t/p\rfloor +s-t<\operatorname {LL}(b)$. As increasing $t$ can only decrease the left hand side of this inequality, it is also true that $\lfloor (|D_r|-1)/p\rfloor +s-(|D_r|-1)<\operatorname {LL}(b)$ and so $|D_r|/p-1+s-(|D_r|-1)<\operatorname {LL}(b)$ giving that $s<\operatorname {LL}(b)+|D_r|-|D_r|/p$ and the theorem is proved.

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3. Further preliminary results

The following result is well known, but we are not aware of an explicit reference.

Lemma 3.1.

Let $G$ be a finite group and $B$ be a $p$-block of $G$ with defect group $D$. Let $N \vartriangleleft G$ and let $b$ be a block of $N$ covered by $B$. If $D \leq N$, then $\operatorname {LL}(B)=\operatorname {LL}(b)$.

Proof.

The proof given in 15, Lemma 4.1 for the case $G/N$ is a $p'$-group carries through in the general situation, but we include a simple argument suggested by Markus Linckelmann.

Let $I_G(b)$ be the stabilizer of $b$ in $G$. Since there is a block of $I_G(b)$ covering $b$ and Morita equivalent to $B$, we may assume that $G=I_G(b)$. Then $\operatorname {Rad}(b)B \subseteq \operatorname {Rad}(B)$. Since $D \leq N$, every $B$-module is relatively $N$-projective, and so in particular $B/\operatorname {Rad}(b)B$ and every quotient module is. Now the restriction of $B/\operatorname {Rad}(b)B$ to $kN$ is a semisimple $b$-module, so it follows that $B/\operatorname {Rad}(b)B$ is semisimple as a $B$-module. Hence $\operatorname {Rad}(B) \subseteq \operatorname {Rad}(b)B$, and so $\operatorname {Rad}(B)=\operatorname {Rad}(b)B$. Again since $b$ is $G$-stable, we have $\operatorname {LL}(B)=\operatorname {LL}(b)$.

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Lemma 3.2.

Let $G_1, \ldots , G_t$ be finite groups and $G=G_1 \times \cdots \times G_t$. For each $i$ let $B_i$ be a block of $G_i$ and let $B=B_1 \otimes \cdots \otimes B_s$ be the block of $G$ covering each $B_i$. Let $D_i$ be a defect group of $B_i$, so $D=D_1 \times \cdots \times D_t$ is a defect group of $B$. Then $\operatorname {LL}(B)=(\sum _{i=1}^t \operatorname {LL}(B_i))-t+1$.

In particular, suppose $|D|=p^d$, $|D_i|=p^{d_i}$ and $D_i \cong C_{p^{a_{i,1}}} \times \cdots \times C_{p^{a_{i,r_i}}} \times (C_p)^{s_i}$, where each $a_{i,j}>1$. If $d_i<\operatorname {LL}(B_i) \leq p^{a_{i,1}}+\cdots +p^{a_{i,r_i}}+ps_i-r_i+1$ for each $i$, then $d<\operatorname {LL}(B) \leq \sum _{i=1}^t \sum _{j=1}^{r_i}p^{a_{i,j}}+p\sum _{i=1}^t s_i- (\sum _{i=1}^tr_i)+1$. If, in addition, $\operatorname {LL}(B_i) \leq p^{a_{i,1}}+\cdots +p^{a_{i,r_i}}+ps_i-r_i$ for at least one $i$, then $\operatorname {LL}(B) \leq \sum _{i=1}^t \sum _{j=1}^{r_i}p^{a_{i,j}}+p\sum _{i=1}^t s_i- (\sum _{i=1}^tr_i)$.

Proof.

The first part is 3, 1.1 and the second part is just a direct application of the first.

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Before proceeding we recall the definition and some properties of the generalized Fitting subgroup $F^*(G)$ of a finite group $G$. Details may be found in 4.

A component of $G$ is a subnormal quasisimple subgroup of $G$. The components of $G$ commute, and we define the layer $E(G)$ of $G$ to be the normal subgroup of $G$ generated by the components. It is a central product of the components. The Fitting subgroup $F(G)$ is the largest nilpotent normal subgroup of $G$, and this is the direct product of $O_r(G)$ for all primes $r$ dividing $|G|$. The generalized Fitting subgroup $F^*(G)$ is $E(G)F(G)$. A crucial property of $F^*(G)$ is that $C_G(F^*(G)) \leq F^*(G)$, so in particular $G/F^*(G)$ may be viewed as a subgroup of $\operatorname {Aut}(F^*(G))$.

Lemma 3.3.

Let $B$ be a block of a finite group $G$ with abelian defect group $D$. Then there is a group $H$ and a block $C$ of $H$ with defect group $D_C \cong D$ such that $\operatorname {LL}(C)=\operatorname {LL}(B)$ and the following are satisfied:

(i) $C$ is quasiprimitive, that is, for every normal subgroup $N$ of $H$, every block of $N$ covered by $C$ is $H$-stable.

(ii) $H$ is generated by the defect groups of $C$.

(iii) If $N \vartriangleleft H$ and $C$ covers a nilpotent block of $N$, then $N \leq Z(H)O_p(H)$. In particular, $O_{p'}(H) \leq Z(H)$.

(iv) $O_p(H) \leq Z(H)$.

(v) Every component of $H$ is normal in $H$.

(vi) If $B$ is the principal block of $G$, then $C$ is the principal block of $H$.

Proof.

Consider pairs $([G:O_{p'}(Z(G))],|G|)$ with the lexigraphic ordering. There are three processes, labelled (a), (b), (c), which will be applied repeatedly and in various combinations. We describe these processes and show that they strictly reduce $([G:O_{p'}(Z(G))],|G|)$ when applied non-trivially, so that repeated application of (a), (b) and (c) terminates.

(a) Let $N \vartriangleleft G$ and let $b$ be a block of $N$ covered by $B$. Write $I=I_G(b)$ for the stabilizer of $b$ in $G$, and $B_I$ for the Fong-Reynolds correspondent. Now $B_I$ is Morita equivalent to $B$ and they have isomorphic defect groups. Clearly $O_{p'}(Z(G)) \leq O_{p'}(Z(I))$, and if $I \neq G$, then $[I:O_{p'}(Z(I))]<[G:O_{p'}(Z(G))]$. Process (a) involves replacing $B$ by $B_I$.

(b) is the replacement of $G$ by $N=\langle D^g:g \in G \rangle$ and $B$ by any block $b$ of $N$ covered by $B$, as in Lemma 3.1. In this case $[N:O_{p'}(Z(N))] \leq [G:O_{p'}(Z(G))]$ and $|N| < |G|$ if $N \neq G$. Since defect groups of $b$ are intersections of defect groups of $B$ with $N$ (see 2, 15.1), it follows that $B$ and $b$ share a defect group.

(c) Let $N \vartriangleleft G$ and suppose that $B$ covers a nilpotent block $N$ of $G$ such that $N \not \leq Z(G)O_p(G)$. Let $b'$ be a block of $Z(G)N$ covered by $B$ and covering $b$. By performing (a) first we may assume that $b'$ is $G$-stable. Further $b'$ must also be nilpotent. Using the results of 17, as outlined in 6, Proposition 2.2, $B$ is Morita equivalent to a block $\tilde{B}$ of a central extension $\tilde{L}$ of a finite group $L$ by a $p'$-group such that there is an $M \vartriangleleft L$ with