Wielandt subgroups of nilpotent residuals and Fitting-height bounds

Introduction

All groups in this paper are finite. We write $H \trianglelefteq\trianglelefteq G$ for subnormality; the remaining notation is collected in § 2.

Background: the Wielandt series

The Wielandt subgroup

was introduced by Wielandt [23], who proved that it contains every minimal normal subgroup of $G$; in particular, $\omega(G) \ne1$ whenever $G \ne1$. Consequently the upper Wielandt series

reaches $G$, and the least index $i$ with $\omega_i(G) = G$ is the Wielandt length $\operatorname{wl}(G)$. The groups of Wielandt length one are the $T$-groups, in which normality is transitive; Gaschütz [12] proved that soluble $T$-groups are metabelian, so they have Fitting height at most two.

If $H \trianglelefteq\trianglelefteq G$, then $H \trianglelefteq H\omega_1(G) \trianglelefteq\cdots\trianglelefteq H\omega_{\operatorname{wl}(G)}(G) = G$, so the subnormal defect of $H$ is at most $\operatorname{wl}(G)$. For soluble groups, Camina [6] showed that the derived length and the Fitting height are bounded in terms of the Wielandt length, and Bryce and Cossey [5] found the best possible bounds: if $G$ is soluble and $\operatorname{wl}(G) = n$, then

and the derived length of $G$ is at most $\lceil5n/3 \rceil$ (see [7 Theorems 1 and 2], where these bounds are stated and extended to infinite soluble groups). Casolo studied how the Wielandt length is controlled by the Fitting height together with the subnormal defects [8].

The residual-normalizer subgroup

The Wielandt subgroup is sensitive to the fine structure of nilpotent groups. Every subgroup of a nilpotent group $G$ is subnormal, so $\omega(G)$ is then the norm $\bigcap_{H\le G}N_G(H)$ introduced by Baer [1], which is contained in $Z_2(G)$ by a theorem of Schenkman [21] and is in general a proper subgroup. For instance, $\omega(D_8) = Z(D_8)$ and $\operatorname{wl}(D_8) = 2$. Since we are interested in the nonnilpotent part of a group, we replace each subnormal subgroup by its nilpotent residual. Let $\mathcal{N}$ be the class of nilpotent groups, and put

The upper W-series is

and its length is denoted $\ell_W(G)$, with $\ell_W(1)=0$. We call $X$ a W-group if $W(X)=X$, that is, if the nilpotent residual of every subnormal subgroup of $X$ is normal in $X$.

Since $H^{\mathcal{N}}$ is characteristic in $H$, we have $\omega(G)\leq W(G)$, and hence $\ell_W(G)\leq\operatorname{wl}(G)$ (Theorem 3.6). Every nilpotent group is a W-group, so $\ell_W(D_8)=1$. More generally, $W(G)$ contains the generalized Fitting subgroup $F^*(G)$ (Lemma 2.2). On the other hand, $W(G)$ can be a proper subgroup: in $G=S_3\wr C_2=(S_3\times S_3)\rtimes C_2$ the subgroup $S_3\times1$ is subnormal with residual $C_3\times1$, which is not normalized by the involution interchanging the factors; in fact $|G:W(G)|=4$ and $\ell_W(G)=2$ (Example 7.8).

The question

The upper W-series retains a direct connection with subnormality: for every $H\trianglelefteq\trianglelefteq G$, it gives a subnormal chain from $H^{\mathcal{N}}$ to $G$ of length at most $\ell_W(G)$ (Theorem 3.6). Thus $\ell_W(G)$ uniformly bounds the subnormal defects of these residuals. For soluble $G$ one also has $\ell_W(G)\leq h(G)$, where $h(G)$ is the Fitting height. To determine how this normalizer condition controls the global structure of $G$, we ask for the reverse comparison:

How much Fitting height can a single layer $W_{i+1}(G)/W_i(G)$ of the upper W-series carry?

Each layer is a W-group, so the question is to bound the Fitting height of soluble W-groups. The answer is four. The smallest example is the affine group

of order 432, whose Fitting layers are $\mathbb{F}_3^2$, $Q_8$, $C_3$ and $C_2$ (Proposition 6.2). Height four always arises in this way: from a primitive quotient whose point stabilizer has a quaternion subgroup $Q_8$ in its Fitting subgroup and induces on it the full group $\operatorname{Out}(Q_8)\cong S_3$.

Main results

The upper bounds are Theorem 5.8 and Corollary 5.9; the constructions are Theorems 8.1, 8.2 and 8.4. A group with $h(G)\geq2$ is not of prime-power order, so two primes are needed for every pair with $h\geq2$. Theorem 1.1 should be compared with the Bryce–Cossey bound $h(G)\leq\operatorname{wl}(G)+1$: for the $\{2,3\}$-groups $G_n$ with $\ell_W(G_n)=n$ and $h(G_n)=4n$ of Theorem 8.1 it gives

This is a lower bound for the ratio; we do not determine the exact Wielandt lengths of these groups.

The upper bound rests on a description of the soluble primitive W-groups. In it, the module enters only through faithfulness and irreducibility.

These assertions are Theorems 5.5 and 5.6, and Theorem 5.8. The implication from the structure of $\mathrm{F}(M)$ to $\mathrm{W}(M) = M$ is Proposition 4.3. For reducible modules we only use the sufficient condition of Lemma 5.3.

To show that the bounds are attained at every length, we stack W-groups by regular wreath products $B \wr E = B^E \rtimes E$. Ideally the base $B^E$ is exactly the first term of the W-series, so that the length increases by one and the heights add. For soluble $B$, let $\Sigma(B)$ be the set of primes $r$ such that $B$ has a primitive quotient $U \rtimes P$, with $P \ne1$ an $r$-group acting faithfully and irreducibly on $U$.