Hypersolvable Groups

Ahmet Arıkan Gazi Üniversitesi, Gazi Eǧitim Fakültesi, Matematik Eğitimi Anabilim Dalı 06500 Teknikokullar, Ankara, Turkey arikan@gazi.edu.tr

Ulrich Meierfrankenfeld Department of Mathematics Michigan State University East Lansing, MI 48824, USA meier@math.msu.edu

Abstract

Call a group G hypersolvable if it has an ascending series with G/C_{G}(A) solvable for each factor A of the series. In this paper we establish some basic facts about hypersolvable groups. We also prove that if G is a perfect Fitting p-group such that every proper subgroup is contained in a proper normal subgroup, then G has a proper non-hypersolvable subgroup.

1 Introduction

Let D be a class of pairs (B, A) such that B is a group acting faithfully on the group A. Let G be a group acting on a group N. A G-invariant normal series A of N is called a D-series for G on N if (G/C_{G}(A), A) ∈ D for all factors A of A.

An ascending D-series for G on N is called a hyper-D series. If such a series exists we say that G acts hyper-D on N. G is hyper-D means that G acts hyper-D on G. If G_{1}, G_{2} are classes of groups, then (G_{1}, G_{2}) denotes the class of pairs (B, A) with B ∈ G_{1}, A ∈ G_{2} and B acting faithfully on A. We denote the class of all groups with ∗. So (∗, ∗) denotes the class of all pairs of groups (B, A) with B acting faithfully on A.

Consider the case N = G. Observe that hyper-(∗,abelian) groups are the hyperabelian groups and hyper-(1,∗) groups are the hypercentral groups. We say that G is hypersolvable if G is hyper-(solvable,∗). This notation might be slightly misleading since one probably would be tempted to define a hypersolvable group to be a hyper-(∗,solvable) group. But as the hyper-(∗,solvable) groups are just the hyperabelian groups such a definition would not be of much use. Similarly we define a hypernilpotent group to be a hyper-(nilpotent,∗)-group.

Unwinding the definitions we see that a group G is hypersolvable if and only if G has a normal ascending series A such that G/C_{G}(A) is solvable for all factors A of A.

We say that G acts strongly hyper-D on N if for all G-invariant M ◁ N there exists a G-invariant M < M̃ ⊴ N with (G/C_{G}(M̃/M), M̃/M) ∈ D.

In section 2 we establish some basic facts about hyper-D groups. In particular, we show that if D is closed under quotients, then G acts hyper-D on N if and only if G acts strongly hyper-D on N.

In section 3 we investigate hyper-(G, ∗)-groups, where G is a countable union of group varieties.

In section 4 we apply Theorem 3.9 to obtain commutator conditions which characterize hypersolvable and hypernilpotent groups.

In section 5 it is shown that the class of hypersolvable groups lies strictly between the classes of hypercentral-by-solvable and hypercentral-by-(residually solvable) groups. Similarly we show that the class of hypernilpotent groups lies strictly between the classes of hypercentral-by-nilpotent and hypercentral-by-(residually nilpotent) groups.

Recall that a Fitting group is a locally (nilpotent and normal) group, that is a group in which every finitely generated subgroup lies in a nilpotent, normal subgroup. We say that a group G is NNC-proper if G is not the normal closure of a proper subgroup. NNC-proper Fitting p-groups are considered in [AÖ1] and given a criterion for these groups to be non-perfect. In Theorem 7.3 we prove that every NNC-proper, perfect, Fitting p-group has a proper non-hypersolvable subgroup.

As a supplement to Theorem 7.3, in section 8 we provide some conditions which ensure that a group is NNC-proper.

2 Basic Properties of hyper-D groups

Let D ⊆ (∗, ∗) (that is a class of pairs (A, B) of groups A and B with A acting faithfully on B, which is closed under isomorphism). We say that D is closed under subgroups if for all (A, B) ∈ D, all D ≤ A and all D-invariant E ≤ B we have (D/C_{D}(E), E) ∈ D. We say that D is closed under quotients if for all (A, B) ∈ D and all A-invariant E ⊴ B, (A/C_{A}(B/E), B/E) ∈ D. A group G is finitely hyper-D if it has a finite hyper-D-series.

Lemma 2.1 Let D ⊆ (∗, ∗) and let G be acting hyper-D on N.

(a) Suppose that D is closed under subgroups. Let H ≤ G and let M be an H-invariant subgroup of N. Then H acts hyper-D on M.

(b) Suppose that D is closed under quotients and that M is a G-invariant normal subgroup of N. Then G acts hyper-D on N/M.

Proof: Let (N_{α})_{α} be a hyper-D series for G on N.

(a) Just observe that (M ∩ N_{α})_{α} is a hyper-D series for H on M.

(b) Since quotients of ascending series are again ascending series, (N_{α}M/M)_{α} is a hyper-D series for G on N/M. □

Lemma 2.2 Let D ⊆ (∗, ∗) and let G be acting on N.

(a) If G acts strongly hyper-D on N, then G acts hyper-D on N.

(b) If D is closed under quotients, then G acts strongly hyper-D on N if and only if it acts hyper-D on N.

Proof: (a) Define N_{0} = 1. If α is a limit ordinal, put N_{α} = ⋃_{β<α} N_{β}. If α = β + 1 and N_{β} ≠ N, put N_{α} = Ñ_{β}. Then (N_{α})_{α} is a hyper-D series on N.

(b) Follows from (a) and 2.1(b). □

Lemma 2.3 Let D ⊆ (∗, ∗) and let G be acting on N. Suppose that there exists a G-invariant normal ascending series on N such that G acts hyper-D on each of the factors. Then G acts hyper-D on N. In particular, if (N_{i}, i ∈ I) is a family of groups with G acting hyper-D on each N_{i}, then G acts hyper-D on ⨁_{i∈I} N_{i}.

Proof: For the first statement use the series on the factors to refine the given series to a hyper-D series.

For the second statement well-order I such that I has a maximal element. For i ∈ I define N_{i}^{+} = ⨁_{j≤i} N_{j} and N_{i}^{−} = ⨁_{j<i} N_{j}. Then {N_{i}^{−}, N_{i}^{+} | i ∈ I} is Ginvariant normal ascending series on ⨁_{i∈I} N_{i} with factors N_{i}^{+}/N_{i}^{−} ≅ N_{i}. So the second statement follows from the first. □

Proposition 2.4 Let G be any class of groups.

(a) Suppose G is closed under quotients. Then hypercentral-by-G groups are hyper- (G, ∗) and nilpotent-by-G groups are finitely hyper-(G, ∗).

(b) Hyper-(G, ∗) groups are hypercentral-by-(residually G). If G is closed under finite subdirect products then finitely hyper-(G, ∗)-groups are nilpotent-by-G.

(c) If G is closed under quotients and finite subdirect products, then the nilpotent-by-G-groups are exactly the finitely hyper-(G, ∗) groups.

Proof: (a) Let H ⊴ G such that H is hypercentral and G/H ∈ G. Let Z be the hypercentral series for H. Then Z is G-invariant. If Z is a factor of Z, then [Z, H] = 1 and so G/C_{G}(Z) is a quotient of G/H. Thus G/C_{G}(Z) ∈ G. Also G/C_{G}(G/H) is a quotient of G/H and so Z ∪ {G} is a hyper-(G, ∗) series for G. If H is nilpotent, Z is finite and (a) is proved.

(b) Let A = (A_{α})_{α} be a hyper-(G, ∗)-series for G and put

H = ⋂{C_{G}(A) | A a factor of A}.

Since G/C_{G}(A) ∈ G for all factors A, G/H is subdirect product of members of G and so residually-G. Moreover (A_{α} ∩ H)_{α} is a hypercentral series for H and so H is hypercentral. If A is finite and G is closed under finite subdirect products, then G/H ∈ G and H is nilpotent. So (b) holds.

(c) Follows from (a) and (b). □

3 Countable unions of group varieties

For n ∈ ℕ, F (n) denotes the free group on n-generators x_{1}, x_{2}, …, x_{n}. Let G be a group, m ∈ ℕ ∪ {∞} with m ≥ n and g = (g_{i})_{i=1}^{m} ∈ G^{m}. Then there exists a unique homomorphism ϕ_{g} : F (n) → G with x_{i} → g_{i} for all 1 ≤ i ≤ n. Given a word w ∈ F (n) we write w(g) for ϕ_{g}(w). So if w = x_{i_{1}}x_{i_{2}} … x_{i_{m}} with 1 ≤ i_{k} ≤ n, then w(g) = g_{i_{1}}g_{i_{2}} … g_{i_{m}}. If m ≤ n we view F (m) as a subgroup of F (n). Let m = m(w) ∈ ℕ be minimal with w ∈ F (m). Let F := ⋃_{n=1}^{∞} F (n) and let W be the set of subsets of F. So the elements of W are sets of words.

Put G^{w} := ⟨w(g) | g ∈ G^{n}⟩ and note that G^{w} is a normal subgroup of G. For a set W ∈ W let G^{W} = ⟨G^{w} | w ∈ W ⟩. Let G(W) be the class of groups G with G^{W} = 1, that is G(W) is the variety defined by W.

Proposition 3.1 Let W ∈ W and let G be a group. Then G is hyper-(G(W), ∗) if and only if G^{W} is hypercentral.

Proof: Let N ⊴ G. Then G/N ∈ G(W) if and only if G^{W} ≤ N if and only if G/N is residually G(W). Thus the proposition follows from 2.4. □

Definition 3.2 Let W = (W_{i})_{i=1}^{∞} ∈ W^{∞} be a sequence of sets of words.

(a) W is decreasing if F^{W_{i+1}} ≤ F^{W_{i}} for all i.

(b) W is almost decreasing if for all i, j ∈ ℤ^{+} there exists k ≥ j with F^{W_{k}} ≤ F^{W_{i}}.

(c) G(W) = ⋃_{i=1}^{∞} G(W_{i}).

Lemma 3.3 Let G be group.

(a) Let V, W ∈ W with F^{V} ≤ F^{W}. Then G^{V} ≤ G^{W}.

(b) Let W ∈ W^{∞} be almost decreasing. Then (G^{W_{i}})_{i=1}^{∞} is almost decreasing, that is for i, j ∈ ℤ^{+} there exists k ≥ j with G^{W_{k}} ≤ G^{W_{i}}.

Proof: (a) Let g ∈ G^{V}. Then g ∈ H^{V} for some finitely generated subgroup H of G. Let α : F → H be an onto homomorphism. Then H^{V} = α(F^{V}) ≤ α(F^{W}) = H^{W} and so g ∈ H^{W} ≤ G^{W}.

(b) follows from (a). □

Definition 3.4 Let G be a group acting on a group N, W ∈ W^{∞} and α an ordinal.

(a) Define H_{α} = Hyp_{α}^{W} (G, N) inductively as follows:

H_{α} =  1  if α = 0
H_{α} =  ⋃_{β<α} H_{β}  if 0 ≠ α is a limit ordinal
H_{α}/H_{α−1} = C_{N/H_{α−1}}([N, G^{W_{k}}]G^{W_{k}}) if α = β + k with
β a limit ordinal and k ∈ ℤ^{+}.

(b) δ = δ^{W} (G, N) is the least ordinal such that H_{δ} = H_{β} for all β ≥ δ. Moreover, Hyp^{W} (G, N) := H_{δ}

(c) A hyper-W series is a hyper-(G(W),∗) series and a hyper-W group is a hyper- (G(W),∗) group.

If α = β +k, β a limit ordinal and k ∈ ℤ^{+}, then H_{α}/H_{α−1} is the largest N-invariant subgroup of N/H_{α−1} centralized by G^{W_{k}}.

Define Hyp_{α}^{W} (G) = Hyp_{α}^{W} (G, G) and Hyp^{W} (G) = Hyp^{W} (G, G). If there is no doubt about the group G and the sequence W in question define H_{α} = Hyp_{α}^{W} (G).

Proposition 3.5 Let G be a group and W ∈ W^{∞}.

(a) (H_{α})_{α} is a hyper-W series for G on Hyp^{W} (G).

(b) Let A ⊴ G and (A_{α})_{α} be a hyper-W series for G on A.

(a) For every ordinal α there exists an ordinal α^{∗} with A_{α} ≤ H_{α^{∗}}. In particular, A ≤ Hyp^{W} (G).

(b) If W is almost decreasing we can choose α^{∗} such that α^{∗} = α + n_{α} for some n_{α} ∈ ℕ and n_{α} = 0 if α is a limit ordinal.

(c) G is hyper-W if and only if G = Hyp^{W} (G).

Proof: (a) Let α = β + k for some limit ordinal β and some k ∈ ℤ^{+}. Then G^{W_{k}} centralizes H_{α}/H_{α−1}. Hence G/C_{G}(H_{α}/H_{α−1}) ∈ G(W_{k}) ⊆ G(W) and (a) holds.

(b) By induction we may assume that for all β < α there exists β^{∗} with A_{β} ≤ H_{β^{∗}}. Suppose first that α is a limit ordinal. Let α^{∗} be the least ordinal with α ≤ α^{∗} and

H_{α^{∗}} = ⋃_{β<α} H_{β^{∗}}. Then

A_{α} = ⋃_{β<α} A_{β} ⊆ ⋃_{β<α} H_{β^{∗}} = H_{α^{∗}}

Moreover, if for all β < α, β^{∗} = β + n_{β} for some n_{β} ∈ ℕ then α^{∗} = α. So (b:a) and (b:b) hold for α.

Suppose next that α = β +k for some limit ordinal β and some k ∈ ℤ^{+}. Since (A_{α})_{α} is hyper-W there exists i ∈ ℤ^{+} with [A_{α}, G^{W^{i}}] ≤ A_{α−1}.

Assume that W is almost decreasing. By induction A_{α−1} ≤ H_{α−1+n_{α−1}} for some n_{α−1} ∈ ℤ^{+}. Since W is almost decreasing there exists n ∈ ℤ^{+} with n ≥ k + n_{α−1} and G^{W_{n}} ≤ G^{W_{i}}. Then

[A_{α}, G^{W_{n}}] ≤ [A_{α}, G^{W_{i}}] ≤ A_{α−1} ≤ H_{α−1+n_{α−1}} = H_{β+k−1+n_{α−1}} ≤ H_{β+n−1}.

Thus A_{α} ≤ H_{β+n} = H_{α+n−k} and (b:b) holds with n_{α} = n − k.

Assume next that W is not almost decreasing. Let γ be the smallest limit ordinal with (α − 1)^{∗} ≤ γ. Then

[A_{α}, G^{W_{i}}] ≤ A_{α−1} ≤ H_{(α−1)^{∗}} ≤ H_{γ} ≤ H_{γ+i−1}

and so A_{α} ≤ H_{γ+i}. Thus (b:a) holds.

(c) Follows from (a) and (b). □

Definition 3.6 (a) For i = 1, 2 let w_{i} be a word and m_{i} = m(w_{i}). Put

⌈w_{1}, w_{2}⌉ := [ w_{1}( (x_{i})_{i=1}^{m_{1}} ) , w_{2}( (x_{m_{1}+i})_{i=1}^{m_{2}} ) ] ∈ F (m_{1} + m_{2})

⌈w_{1}, w_{2}⌉ is called the outer commutator of w_{1} and w_{2}.

(b) Following Möhres [M3, (3) Definition], outer commutator words are inductively defined as follows:

(a) w = x_{1} is the only outer commutator word with m(w) = 1.

(b) If m(w) > 1 then w is an outer commutator word provided that there exist outer commutator words w_{1}, w_{2} with m(w_{i}) < m(w) and w = ⌈w_{1}, w_{2}⌉.

(c) Let w ∈ F^{n}, n ∈ ℕ ∪ {∞}. Then w̌ ∈ F^{n+1} is inductively defined as follows: w̌_{1} = x_{1} and w̌_{i+1} = ⌈w̌_{i}, w_{i}⌉.

(d) Let W ∈ W^{n}, n ∈ ℕ ∪ {∞}. Then W̌ ∈ W^{n+1} is inductively defined as follows: W̌_{1} = {x_{1}} and W̌_{i+1} = {⌈v, w⌉ | v ∈ W̌_{i}, w ∈ W_{i}}.

For example, ⌈x_{1}x_{2}^{3}, x_{1}x_{2}^{2}⌉ = [x_{1}x_{2}^{3}, x_{3}x_{4}^{2}]. Note that m(⌈w_{1}, w_{2}⌉) = m_{1} + m_{2}. Also W̌_{i+1} = {w̌_{i+1} | w ∈ \/_{j=1}^{i} W_{j}}. To improve readability we sometimes write wˇ for w̌.

Lemma 3.7 Let G be a group, w ∈ F^{∞}, g ∈ G^{∞} and i ∈ ℤ^{+}.

(a) Put n = m(w̌_{i}) and m = m(w_{i}). Then

w̌_{i+1}(g) = [w̌_{i}(g), w_{i}( (g_{n+j})_{j=1}^{m})].

(b) Let N ⊴ G. If w̌_{i}(g) ∈ N then also w̌_{j}(g) ∈ N for all j ≥ i.

(c) Let W ∈ W^{∞}. Then GW̌_{i+1} = [GW̌_{i}, G^{W_{i}}] ≤ GW̌_{i} ∩ G^{W_{i}}.

In particular, W̌ is decreasing.

Proof: (a) By definition w̌_{i+1} = ⌈w̌_{i}, w_{i}⌉. So (a) follows from the definition of the outer commutator.

(b) and (c) follow from (a). □

Definition 3.8 (a) Let W ∈ W^{∞}. Then H(W) is the class of groups G such that for all g ∈ G^{∞} and all w ∈ \/_{i=1}^{∞} W_{i} there exists n ∈ ℤ^{+} with w_{n}(g) = 1 (or equivalently for all g ∈ G^{∞}, there exists n ∈ ℤ^{+} with w_{n}(g) = 1 for all w_{n} ∈ W_{n}.)

(b) Let D ⊆ (∗, ∗). Then HD is the class of hyper-D-groups. FD is the class of finitely hyper-D-groups.

Observe that G(W) is the class of groups G for which there exists n ∈ ℤ^{+} with w_{n}(g) = 1 for all g ∈ G^{∞} and all w_{n} ∈ W_{n}. Thus G(W) ⊆ H(W).

Theorem 3.9 Let W ∈ W^{∞}. Then

(a) G(W̌) ⊆ F(G(W), ∗) with equality if W is almost decreasing.

(b) H(W̌) ⊆ H(G(W), ∗) with equality if W is almost decreasing.

Proof: Suppose that GW̌_{n} = 1 for some n. Then by 3.7(c)

1 = GW̌_{n} ≤ GW̌_{n−1} ≤ … ≤ GW̌_{2} ≤ GW̌_{1} = G

is a finite hyper-W series on G. Thus G(W̌) ⊆ F(G(W), ∗).

Let G be a group which is not hyper-W. We will show that G is also not contained in H(W̌). By 2.2 there exists N ◁ G such that

(∗)  C_{G/N}(G^{W_{n}}) = 1 for all n ∈ ℤ^{+}.

Let g_{1} ∈ G \ N. Note that x_{1}(g_{1}) = g_{1} ∉ N. Suppose inductively that we already found (g_{i})_{i=1}^{n_{k}} ∈ G^{n_{k}} and w_{i} ∈ W_{i}, 1 ≤ i < k with w̌_{k}( (g_{i})_{i=1}^{n_{k}}) ∉ N. Then by (*)

[w̌_{k}( (g_{i})_{i=1}^{n_{k}}), G^{W_{k}}] ≰ N and there exist w_{k} ∈ W_{k} and (g_{n_{k}+j})_{j=1}^{m(w_{k})} ∈ G^{m(w_{k})} with [w̌_{k}(g_{i})_{i=1}^{n_{k}}, w_{k}( (g_{n_{k}+j})_{j=1}^{m(w_{k})} )] ∉ N. Put n_{k+1} = n_{k} + m(w_{k}). Then by 3.7(a),

w̌_{k+1}( (g_{i})_{i=1}^{n_{k+1}} ) ∉ N.

Put g = (g_{i})_{i=1}^{∞} and w = (w_{i})_{i=1}^{∞}. Then w̌_{k}(g) ≠ 1 for all k and so G ∉ H(W̌). Thus H(W̌) ⊆ H(G(W), ∗).

Suppose next that W is almost decreasing. We will prove the second assertions in (a) and (b) simultaneously. Let G be hyper-W and let (A_{α})_{α≤ρ} be any hyper-W series on G. Let i ∈ ℤ^{+}. If ρ is finite let V_{i} = W_{i} and H_{i} = G_{i}. If ρ is infinite pick w_{i} ∈ W_{i} and g_{i} ∈ G and put H_{i} = {g_{i}} and V_{i} = {w_{i}}

Let g ∈ \/_{i=1}^{∞} H_{i} and w ∈ \/_{i=1}^{∞} V_{i}. Then w̌_{1}(g_{1}) = g_{1} ∈ G = A_{ρ}. So we can choose an ordinal α minimal such that there exists n ∈ ℤ^{+} with w̌_{n}(g) ∈ G_{α} for all w ∈ \/_{i=1}^{∞} V_{i} and g ∈ \/_{i=1}^{∞} H_{i}.

We will show that α = 0. Suppose that α = β + 1 for some ordinal β. Since G/C_{G}(A_{α}/A_{β}) ∈ G(W), there exists m ∈ ℤ^{+} with [A_{α}, G^{W_{m}}] ≤ A_{β}. Since W is almost decreasing we may assume m ≥ n. Let w ∈ \/_{i=1}^{∞} V_{i}. Then w̌_{n}(g) ∈ A_{α} and m ≥ n. So by 3.7(b), w̌_{m}(g) ∈ A_{α}. Hence

w̌_{m+1}(g) ∈ [w̌_{m}(g), G^{W_{m}}] ≤ [A_{α}, G^{W_{m}}] ≤ A_{β}

for all w ∈ \/_{i=1}^{∞} V_{i} and g ∈ \/_{i=1}^{∞} H_{i}, a contradiction to the minimal choice of α. Thus α is a limit ordinal.

Suppose that α ≠ 0. Then ρ is infinite and so by our choice of V_{i}, |V_{i}| = 1 = |H_{i} and there exist a unique w ∈ \/_{i=1}^{∞} V_{i} and a unique g ∈ \/_{i∈I} H_{i}. Since A_{α} = ⋃_{β<α} A_{β} there exists β < α with w̌_{n}(g) ∈ A_{β}, a contradiction to the choice of α.

Thus α = 0 and so w̌_{n}(g) = 1 for all w ∈ \/_{i=1}^{∞} V_{i}.

If ρ is finite, V_{i} = W_{i} and H_{i} = G_{i}. Thus GW̌_{n} = 1 and G ∈ G(W̌). So (a) is proved. In any case, w_{n}(g) = 1 shows that G ∈ H(W̌) and (b) holds. □

The following example shows that the inclusions in 3.9 may be proper if W is not almost decreasing:

Let G = Sym(3), x = x_{1}, W_{1} = {x^{2}} and W_{i} = {x} for i ≥ 2. Then w = (x^{2}, x, x, x, …) is the unique element in \/_{i=1}^{∞} W_{i}. Also 1 ≤ Alt(3) ≤ Sym(3) is a finite hyper-(G(W), ∗) series. Thus Sym(3) ∈ F(G(W), ∗) ⊆ H(G(W), ∗).

Put g = ((12), (123), (12), (12), (12), …). Then w̌_{1}(g) = g_{1} = (12), w̌_{2}(g) = [(12), (123)^{2}] = (123), w̌_{3}(g) = [(123), (12)] = (123) and so for all n ≥ 2, w̌_{n}(g) = (123). Thus w_{n}(g) ≠ 1 for all n and Sym(3) ∉ H(W̌). Since G(W̌) ⊆ H(W̌) we see that G(W̌) ≠ F(G(W), ∗) and H(W̌) ≠ H(G(W, ∗).

On the other hand, given an arbitrary W ∈ W^{∞} define

V = (W_{1}, W_{1}, W_{2}, W_{1}, W_{2}, W_{3}, W_{1}, W_{2}, W_{3}, W_{4}, W_{1}, …).

Then clearly V is almost decreasing. For any group G, G(W) only depends on {W_{i} | i ∈ ℤ^{+}} and so G(W) = G(V). Thus by 3.9

G(V̌) = F(G(W), ∗) and H(V̌) = H(G(W), ∗).

4 Hypersolvable and hypernilpotent groups

Definition 4.1 (a) τ(0) = (x_{1})_{i=1}^{∞} and inductively τ(i + 1) = τ(i)ˇ .

(b) ϕ is the unique sequence of words with ϕ = ϕ̌. So ϕ_{1} = x_{1} and inductively ϕ_{i+1} = ⌈ϕ_{i}, ϕ_{i}⌉.

It might be worthwhile to list the first few terms of the above sequence of words:

\begin{matrix}τ(0) : & x_{1} & x_{1} & x_{1} & x_{1} \\ τ(1) : & x_{1} & [x_{1}, x_{2}] & [[x_{1}, x_{2}], x_{3}] & [[[x_{1}, x_{2}], x_{3}], x_{4}] \\ τ(2) : & x_{1} & [x_{1}, x_{2}] & [[x_{1}, x_{2}], [x_{3}, x_{4}]] & [[[x_{1}, x_{2}], [x_{3}, x_{4}]], [[x_{5}, x_{6}], x_{7}]]] \\ ϕ : & x_{1} & [x_{1}, x_{2}] & [[x_{1}, x_{2}], [x_{3}, x_{4}]] & [[[x_{1}, x_{2}], [x_{3}, x_{4}]], [[x_{5}, x_{6}], [x_{7}, x_{8}]]] \end{matrix}

Lemma 4.2 (a) Let N (0) be the class of trivial groups and inductively let N (n + 1) be the class of nilpotent-by-N (n) groups. Then G(τ(n)) = N (n). In particular, G(τ(1)) the class of nilpotent groups.

(b) G(ϕ) is the class of solvable groups.

(c) H(τ(1)) is the class of hypercentral groups and H(τ(2)) is the class of hypernilpotent groups.

(d) H(ϕ) is the class of hypersolvable groups.

Proof: (a) Let w ∈ F^{∞} be decreasing. By 3.9(a), G(w̌) = F(G(w), ∗) and so by 2.4(c):

(∗)  G(w̌) is the class of nilpotent-by-G(w)groups.

Clearly G(τ(0))) is the class of trivial groups. Since τ(1) = τ(0)ˇ , (*) says that G(τ(1)) is the class of nilpotent-by-trivial groups and so the class of nilpotent groups. Hence G(τ(1)) = N (1). Inductively suppose that G(τ(n)) = N (n). So (*) implies that G(τ(n + 1)) is the class of nilpotent-by-N (n) groups. Thus G(τ(n + 1)) = N (n + 1) and (a) holds.

(b) We have G = G^{ϕ_{1}} = G^{(0)} and so inductively

G^{ϕ_{i+1}} = [G^{ϕ_{i}}, G^{ϕ_{i}}] = [G^{(i−1)}, G^{(i−1)}] = G^{(i)}.

Hence G(ϕ_{i}) is the class of solvable groups of derived length less than i and (b) holds. (c) and (d) follow from (a), (b) and 3.9(b). □

Lemma 4.3 Let G be a group and w an outer commutator word. Put m = m(w). Then G^{ϕ_{m}} ≤ G^{w}.

Proof: For m = 1 we have w = x_{1} = ϕ_{1}. If m > 1, then w = ⌈w_{1}, w_{2}⌉ where w_{1}, w_{2} are outer commutator words with m_{i} := m(w_{i}) < m. So

G^{ϕ_{m−1}} ≤ G^{ϕ_{m_{i}}} ≤ G^{w_{i}}

and thus

G^{ϕ_{m}} = [G^{ϕ_{m−1}}, G^{ϕ_{m−1}}] ≤ [G^{w_{1}}, G^{w_{2}}] = G^{w}.
□

Corollary 4.4 Let w be a sequence of outer commutator words. Then

H(w̌) ⊆ H(G(w), ∗) ⊆ H(ϕ).

Proof: The first statement follows from 3.9(b). Now let G be a group with a hyper- (G(w), ∗) series and T a factor of that series. Then [T, G^{w_{k}}] = 1 for some k. By 4.3 [T, G^{(m)}] = 1 for some m and so G is hypersolvable. Thus by 4.2(d), G ∈ H(ϕ). □

5 Examples

In this section we construct various examples of groups which are hyper-(G, ∗) for some class G of groups. By 2.4 we know that any such group is hypercentral-by-(residually G). The next proposition gives a partial converse:

Example 5.1 Let G be a class of groups, (H_{i}, i ∈ I) a family of members of G and H a subdirect product of (H_{i}, i ∈ I). For i ∈ I let A_{i} be a group with H_{i} acting on A_{i}. Suppose that

(i) H is hyper-(G, ∗).

(ii) For each i ∈ I, A_{i} is abelian and H_{i} acts faithfully on A_{i}.

(iii) For each 1 ≠ N ⊴ H, there exists i ∈ I such that N does not act hypercentrally on A_{i}.

Put A = ⨁ A_{i}. Note that H acts on A_{i} via its projection onto H_{i} and so also acts on A. Put G = AH. Then G is hyper-(G, ∗). Moreover, any hypercentral normal subgroup of G is contained in A.

Proof: Since G/C_{G}(A_{i}) ≅ H_{i} ∈ G, G acts hyper-(G, ∗) on A_{i}. So by 2.3, G is hyper- (G, ∗) on A. Also G/A ≅ H is hyper-(G, ∗) and hence by 2.3 G is hyper-(G, ∗).

Let M ⊴ G with M ≰ A. Then AM = AN for some 1 ≠ N ⊴ H. By (iii) there exists i ∈ I such that N does not act hypercentrally on A_{i}. So N also does not act hypercentrally on [A_{i}, N]. Since A is abelian, [A_{i}, N] = [A_{i}, M] ≤ M and M does not act hypercentrally on [A_{i}, M]. Thus M is not hypercentral. □

Example 5.2 Let G be a class of groups and H a group. Suppose H is residually-G and hyper-(G, ∗). Then there exists a hyper-(G, ∗) group G and an abelian normal subgroup A of G such that G/A ≅ H and such that every hypercentral normal subgroup of G is contained in A.

Proof: Let M = {M ⊴ H | G/M ∈ G}. Since H is residually-G, ⋂ M = 1. In particular, H is a subdirect product of (G/M | M ∈ M). For M ∈ M put A_{M} = ℤ[G/M]. Then A_{M} is an abelian group with G/M acting faithfully on A_{M} by right multiplication. Let 1 ≠ N ⊴ H and choose M ∈ M with N ≰ M. Then N does not act hypercentrally on A_{M} (indeed if NM/M is infinite, C_{A_{M}} (N) = 0 and if NM/M is finite, choose a prime p with p ∤ |NM/M| and observe that N does not act hypercentrally on A_{M}/pA_{M}.)

So 5.1 completes the proof. □

Example 5.3 For each prime p there exists a locally finite, hypersolvable p-group which is not hypercentral-by-solvable.

Proof: For 1 < k ∈ ℕ let H_{k} be a solvable p-group of derived length k with Z(H_{k}) = 1. Let A_{k} = 𝔽_{p}H_{k} and H = ⨁_{k=2}^{∞} H_{k}. Let 1 ≠ N ⊴ H and choose k such that the projection N_{k} of N in H_{k} is not trivial. If N_{k} is finite, H_{k}/C_{N_{k}}(H_{k}) is a finite p-group acting on the finite p-group N_{k} and so C_{N_{k}}(H_{k}) ≠ 1, contrary to Z(H_{k}) = 1. So N_{k} is infinite. Hence C_{A_{k}}(N) = 1 and N does not act hypercentrally on A_{k}. Put A = ⨁ A_{k} and G = AH. 5.1 now completes the proof. □

Example 5.4 For each prime p there exists a hypernilpotent, 3-step elementary abelian, p-group G which is not hypercentral-by-hypercentral.

Proof: Let 𝔽 be an infinite field of characteristic p.

Let W be a vector space over 𝔽 with basis (w_{i}, i ∈ ℕ). For i ∈ ℕ let i = ∑_{j=0}^{∞} b_{ij}2^{j} with b_{ij} ∈ {0, 1}. For j ∈ ℕ and f ∈ 𝔽 define t_{jf} ∈ GL_{𝔽}(W) by t_{jf}(w_{i}) = w_{i} + fw_{i+2^{j}} if b_{ij} = 0 and t_{jf}(w_{i}) = w_{i} if b_{ij} = 1.

Let T_{j} = {t_{jf} | f ∈ 𝔽}. Then T_{j} is an infinite elementary abelian p-group isomorphic to (𝔽, +). Also [T_{j}, T_{k}] = 1 for all j, k and so also T =: ⟨T_{j} | j ∈ ℤ^{+}⟩ is an elementary abelian p-subgroup of GL(W).

Define W_{i} = ⟨𝔽w_{k} | k ≥ 2^{i}⟩. Then clearly W_{i} is an 𝔽T -submodule of W and so W_{i} is a normal subgroup of the semidirect product H = W T. Moreover, W/W_{i} is finite dimensional and H/W_{i} is nilpotent. Since ⋂_{i=1}^{∞} W_{i} = 1, H is residually nilpotent.

Let 1 ≠ N ⊴ H. We prove next that

(*) there exists k such that NW_{k}/W_{k} is infinite.

Since C_{H}(W) = W either [N, W] ≠ 1 or N ≤ W. In either case there exists 1 ≠ n ∈ N ∩ W. Let n = ∑_{i=0}^{l} k_{i}w_{i} with k_{i} ∈ F and pick j ∈ ℕ with 2^{j} > l. Put m = ∑_{i=0}^{l} k_{i}w_{i+2^{j}}. Then t_{jf}(n) = n + fm. Since F is infinite and fm ∉ W_{j+1} for all 0 ≠ f ∈ 𝔽 we conclude that (*) holds for k = j + 1.

Since H is a p-group, (*) implies Z(H) = 1 and so H is not hypercentral. Since H/C_{H}(W) ≅ T is abelian and H/C_{H}(H/W) = 1 we conclude that 1 ≤ W ≤ H is a hypernilpotent series on H. Therefore H is hypernilpotent.

Let A_{i} = 𝔽_{p}[H/W_{i}] and put A = ⨁_{i=1}^{∞} A_{i}. Then A is an elementary abelian p-group. Choose k as in (*). Then C_{A_{k}}(N) = 1 and so N does not act hypercentrally on A_{k}. Therefore the assumptions of 5.1 are fulfilled. Thus G = AH = AW T is hypernilpotent and every hypercentral normal subgroup of G is contained in A. Since G/A ≅ H is not hypercentral, G is not hypercentral-by-hypercentral. □

Many thanks to Jon Hall who simplified the description of the action of T on W in the preceding lemma.

6 Möhres’ Lemma

Fix a group G and let F be the set of finitely generated subgroups of G. For H, K ≤ G let F(H) = {E ∈ F | H ≤ E} and F(H, K) = {E ∈ F(H) | E ≰ K}. Put D(H, K) = G if F(H, K) = ∅, and D(H, K) = ⋂ F(H, K), otherwise. If the group G in question needs to be emphasized, we will also use the notations F_{G}, D_{G}(H, K), … in place of F, D(H, K), . . ..

For K ≤ G let K^{∘} ≤ G be such that ⟨K^{G}⟩ ≤ K^{∘} and K^{∘}/⟨K^{G}⟩ is the hypercenter of G/⟨K^{G}⟩.

Lemma 6.1 Let G be a group and K ≤ G.

(a) Let E ≤ G and put D = D(E, K). Then E ≤ D, F(E, K) = F(D, K) and D = D(D, K).

(b) K = K^{∘} if and only if K ⊴ G and Z(G/K) = 1.

(c) Let K ≤ L ≤ K^{∘}. Then L^{∘} = K^{∘}. In particular ⟨K^{G}⟩^{∘} = K^{∘} = (K^{∘})^{∘}.

(d) Suppose G is perfect. Then [K^{∘}, G] ≤ ⟨K^{G}⟩. Moreover, G = ⟨K^{G}⟩ if and only if G = K^{∘}.

Proof: (a) Clearly E ≤ D. Let E ≤ H ∈ F with H ≰ K. Then by definition of D, D ≤ H and so F(E, K) ⊆ F(D, K). Clearly F(D, K) ⊆ F(E, K) and so (a) holds.

(b) is obvious.

(c) Clearly ⟨K^{G}⟩^{∘} = K^{∘} and ⟨L^{G}⟩ ≤ K^{∘}. So we may assume that both K and L are normal in G. Since K^{∘}/K is hypercentral for G also K^{∘}/L is hypercentral for G. Thus K^{∘} ≤ L^{∘}. Since L^{∘}/K^{∘} and K^{∘}/K are hypercentral for G, L^{∘}/K is hypercentral for G and so L^{∘} ≤ K^{∘}.

(d) The first statement holds since the hypercenter of a perfect group is its center. The second follows from the first. □

The following lemma and its corollary have been abstracted from the proof of [M3, (4)Lemma].

Lemma 6.2 (Möhres’ Lemma) Let G be an NNC-proper, perfect group. Let U ∈ F and a ∈ G \ U. Then one of the following holds.

1. There exists N ◁ G and a ∉ V ∈ F(U) with a ∈ D(V, N).

2. Let α be any outer commutator word and a ∉ V ∈ F(U). Then

G = ⟨H^{α} | a ∉ H ∈ F(V)⟩.

Proof: We assume that (1) and (2) are both false. Since (1) is false:

(*) a ∉ D(V, N) for all N ◁ G and all a ∉ V ∈ F(U).

Since (2) is false, there exist an outer commutator word α with m(α) minimal and a ∉ V ∈ F(U) such that K := ⟨H^{α} | a ∉ H ∈ F(V)⟩ ≠ G. Let N = K^{∘}. Since G is NNC-proper, G ≠ ⟨K^{G}⟩ and so by 6.1(d), N ≠ G.

Suppose that m(α) = 1, that is α = x_{1}. From (*), a ∉ D(V, N) and so there exists H ∈ F(V) with a ∉ H and H ≰ N, a contradiction to H = H^{x_{1}} = H^{α} ≤ K ≤ N.

Thus m(α) ≠ 1 and so there exist outer commutator words β and γ with α = ⌈β, γ⌉. By the minimal choice of m(α), G = ⟨H^{β} | a ∉ H ∈ F(V)⟩ and so there exists a ∉ H ∈ F(V) with H^{β} ≰ N. Since Z(G/N) = 1, [H^{β}, G] ≰ N. Again by the minimal choice of m(α), G = ⟨R^{γ} | a ∉ R ∈ F(H)⟩ and thus there exists a ∉ R ∈ F(H) with [H^{β}, R^{γ}] ≰ N. Since H ≤ R, H^{β} ≤ R^{β} and so R^{α} = [R^{β}, R^{γ}] ≰ N, a contradiction to R^{α} ≤ K ≤ N. □

Corollary 6.3 Let G be an NNC-proper, perfect group. Let U ∈ F and a ∈ G \ U. Then one of the following holds:

1. There exist N ◁ G and a ∉ V ∈ F(U) with a ∈ D(V, N).

2. Let w = (w_{i})_{i=1}^{∞} be any sequence of outer commutator words. Then there exists g ∈ G^{∞} such that a ∉ ⟨U, g⟩ and w̌_{k}(g) ≠ 1 for all k ∈ ℤ^{+}. In particular, there exists a non-hypersolvable H ≤ G with a ∉ H and U ≤ H.

Proof: Suppose that (1) is false. Then 6.2(2) holds. In particular, there exists g_{1} ∈ G \ Z(G) with a ∉ ⟨U, g_{1}⟩. Put m_{k} = m(w_{k}) and n_{k} = m(w̌_{k}). Let k ∈ ℤ^{+} and suppose inductively that we have found

(*) g_{i} ∈ G_{i}, 1 ≤ i ≤ n_{k} such that a ∉ U_{k} := ⟨U, g_{i}, 1 ≤ i ≤ n_{k}⟩ and h_{k} := w̌_{k}( (g_{i})_{i=1}^{n_{k}}) ∉ Z(G).

Note that (*) holds for k = 1. Since G is perfect and h_{k} ∉ Z(G), [h_{k}, G] ≰ Z(G). So by 6.2(2), applied with α = w_{k} and V = U_{k} there exists H_{k} ∈ F(U_{k}) such that a ∉ H_{k} and [h_{k}, H_{k}^{w_{k}}] ≰ Z(G). Hence we can choose g_{n_{k}+i} ∈ H_{k}, 1 ≤ i ≤ m_{k} with with [h_{k}, w_{k}( (g_{n_{k}+i})_{i=1}^{m_{k}})] ∉ Z(G). Thus h_{k+1} ∉ Z(G). Moreover U_{k+1} ≤ H_{k} and so a ∉ U_{k+1}.

By induction (*) holds for all k ∈ ℕ. Put g = (g_{i})_{i=1}^{∞}. Then w̌_{k}(g) ≠ 1 for all k ∈ ℤ^{+}.

In the special case w_{i} = ϕ_{i}, 4.2(d) shows that ⟨U, g⟩ is not hypersolvable. □

7 Perfect NNC-proper Fitting p-groups

In this section we prove that every perfect, NNC-proper, Fitting p-group has a proper non-hypersolvable subgroup.

Lemma 7.1 Let G be a nilpotent p-group. Let H be a normal subgroup of G such that G/H is an infinite elementary abelian p-group. Let U be a finite subgroup of G and let a ∈ G \ U. Then there exists a subgroup V of G such that U ≤ V, a ∉ V and V/V ∩ H is infinite.

Proof: [M2, (6)Satz]. □

Corollary 7.2 Let G be a perfect Fitting p-group. Then U = D(U, N) for all finite subgroups U of G and all N ◁ G.

Proof: Suppose U ≠ D(U, N) for some finite U ≤ G and some N ◁ G. Let U < D ≤ D(U, N) with D finite. Since G is perfect, G/N is not nilpotent. As G is a Fitting group, ⟨D^{G}⟩N/N is nilpotent. Thus G ≠ ⟨D^{G}⟩N and we may assume that D ≤ N. Also G ≠ N^{∘} and so we may assume N = N^{∘}. Since G/N is a Fitting group, there exists a non-trivial abelian normal subgroup E/N in G/N. Choose g ∈ E \ N with g^{p} ∈ N and put M = ⟨D^{G}, g^{G}⟩. Then M is nilpotent, M/M ∩ N ≅ MN/N is elementary abelian and U < D ≤ D_{M}(U, M ∩ N).

Suppose that M/M ∩N is infinite. Pick a ∈ D\U. Then by 7.1 (applied with G = M and H = N ∩ M) there exists U ≤ V ≤ M with a ∉ V and V/V ∩ (M ∩ N) infinite.

Pick v ∈ V \ (M ∩ N). Then ⟨U, v⟩ ≤ V and so a ∉ ⟨U, v⟩. Hence a ∉ D(U, M ∩ N), a contradiction to a ∈ D.

Thus M/M ∩ N is finite. So also MN/N is finite. Since G is perfect, we get [M, G] ≤ N and M ≤ N^{∘} = N, a contradiction to g ∈ M \ N. □

Proposition 7.3 Let G be a NNC-proper, perfect, Fitting p-group. Let U be a finite subgroup of G and a ∈ G \ U. Then there exists a non-hypersolvable subgroup H of G with a ∉ H and U ≤ H.

Proof: From 7.2 V = D(V, N) for all finite subgroups V of G. Thus 6.3(1) does not hold and so 6.3(2) does. □

8 Normal closure of subgroups

Let S be a set of subgroups of a group G. We say that G is NNC-S if G ≠ ⟨S^{G}⟩ for all S ∈ S (here NNC stands for “not normal closure”). Note that this is the case if and only if every member of S lies in a proper normal subgroup of G. If G is a class of groups, we say that G is NNC-G if G is NNC-S, where S = {S ≤ G | S ∈ G}. So G is NNC-abelian if G is not the normal closure of an abelian subgroup. G is strongly NNC-G if each non-trivial quotient of G is NNC-G. We say that G is NNC-proper if G is NNC-P where P is the set of proper subgroups of G. We say G is NNC-centralizers if G is NNC-C where C = {C_{G}(x) | 1 ≠ x ∈ G}. Note that G is NNC-centralizer if and only if G ≠ ⟨H^{G}⟩ for all H ≤ G with Z(H) ≠ 1.

The goal of this section is to prove Proposition 8.4, which provides conditions which imply that G is NNC-proper.

Lemma 8.1 Let G be a group and i ∈ ℤ^{+}. Then the following are equivalent:

(a) G is strongly NNC-abelian.

(b) G is strongly NNC-solvable.

(c) Let K ≤ G. Then G = ⟨K^{G}⟩ if and only if G = ⟨(K^{(i)})^{G}⟩.

Proof:

(a) implies (b): Let H be a non-trivial quotient of G and let S be a solvable subgroup of H. By induction on the derived length of S, N := ⟨S^{′H}⟩ ≠ H. Since SN/N is abelian, H/N ≠ ⟨SN/N^{H}⟩ and so also H ≠ ⟨S^{H}⟩.

(b) implies (c): Put N := ⟨(K^{(i)})^{G}⟩. Clearly G ≠ ⟨K^{G}⟩ implies N ≠ G. Now suppose N ≠ G. Since KN/N is solvable, (b) implies G/N ≠ ⟨KN/N^{G}⟩ and so ⟨K^{G}⟩ ≠ G.

(c) implies (a): Let H = G/N be a non-trivial quotient of G and A = K/N an abelian subgroup of G/N. Then K^{(i)} ≤ N < G and so G ≠ ⟨(K^{(i)})^{G}⟩. Thus by (c) G ≠ ⟨K^{G}⟩ and so also H ≠ ⟨A^{H}⟩. □

Definition 8.2 Let G be a group. Then Sol^{∗}(H) = Hyp^{ϕ}(G).

Observe that by 4.2 and 3.5 Sol^{∗}(H) is the largest normal subgroup of G on which G acts hypersolvablely.

Lemma 8.3 Let G be an NNC-centralizer and strongly NNC-abelian group. Then G ≠ ⟨K^{G}⟩ for all K ≤ H such that Sol^{∗}(K) ≠ 1. In particular, G is NNC-hypersolvable.

Proof: Let K ≤ G with Sol^{∗}(K) ≠ 1. Then there exists a non-trivial normal subgroup A of K such that K/C_{K}(A) is solvable. Let 1 ≠ x ∈ A. Then C_{K}(A) ≤ C_{G}(x) and since G is NNC-centralizer we get N := ⟨C_{K}(A)^{G}⟩ ≠ G. Then KN/N is solvable. Since G is strongly NNC-abelian, 8.1 implies that G is strongly NNC-solvable. Thus G/N ≠ ⟨KN/N^{G}⟩ and G ≠ ⟨K^{G}⟩. □

Proposition 8.4 Suppose G is NNC-centralizer and that one of the following holds:

(i) G is minimal non-hypercentral.

(ii) G is minimal non-hypersolvable and strongly NNC-abelian.

Then G is NNC-proper.

Proof: Let K be a proper subgroup of G.

If (i) holds, K is hypercentral. Hence Z(K) ≠ 1 and since G is NNC, G ≠ ⟨K^{G}⟩. If (ii) holds, then K is hypersolvable and so Sol^{∗}(K) = K ≠ 1. Thus by 8.3,

G ≠ ⟨K^{G}⟩. □

Corollary 8.5 Every non-trivial, NNC-centralizer, strongly NNC-abelian, perfect Fitting p-group has a proper non-hypersolvable subgroup.

Proof: Suppose G is a counterexample. Since G is non-trivial and perfect, G is not hypersolvable. So G is minimal non-hypersolvable. Thus 8.4 implies that G is NNC-proper. But then the assumption but not the conclusion of 7.3 are fulfilled, contradiction. □

Acknowledgments. This paper has been written while the first author was a visiting in Michigan State University. He is grateful to Professor Jonathan I. Hall and Professor Ulrich Meierfrankenfeld for their warm hospitality and many helpful discussions. He is also grateful to Gazi University for the financial support and the nice organization.

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