January 01, 1970
The class-breadth conjecture of Leedham-Green, Neumann and Wiegold states that the nilpotency class of any \(p\)-group is at most \(b(G) + 1\), where \(\displaystyle{b(G) = \max_{g\in G}\log_p[G:Z_G(g)]}\) denotes the breadth of \(G\). While several counter-examples to this conjecture have been found for \(p = 2\), it is still open in general for \(p>2\). This article is dedicated to the general case \(p>2\) of the conjecture. We propose a generalization for the case \(p>2\), which we prove under some additional conditions.
Let \(G\) be a finite \(p\)-group. The breadth \(b(x)\) of an element \(x\) of a group \(G\) is defined as \(b(x) := \log_p|G:Z_G(x)|\), where \(Z_G(x)\) denotes the centralizer of \(x\) in \(G\). The breadth \(b(G)\) of \(G\) is defined as \(\displaystyle{b(G) := \max_{g\in G}b(g)}\).
C. Leedham-Green, P. Neumann, and J. Wiegold in [1] stated the following conjecture: \(\operatorname{cl}(G)\leq b(G) + 1\) for all \(p\)-groups \(G\) (\(\operatorname{cl}(G)\) denotes the nilpotency class of \(G\)).
W. Felsch, J. Neubüser, and W. Plesken in [2] provided a series of counterexamples to the conjecture for \(p = 2\). Further counterexamples were found by B. Eick, M. Newman, and E. O’Brien in [3]. In [3], they also proved that the analogue of the conjecture for finite-dimensional nilpotent Lie algebras is false for all finite fields; i.e., for every finite field \(\mathbb{F}\), there exists a finite-dimensional nilpotent Lie algebra \(\mathfrak{g}\) over \(\mathbb{F}\) such that \(\operatorname{cl}(\mathfrak{g}) > b(\mathfrak{g}) + 1\), where \(b(\mathfrak{g}) := \max_{a\in\mathfrak{g}}(\dim \mathfrak{g} - \dim Z_{\mathfrak{g}}(a))\) and \(Z_{\mathfrak{g}}(a) := \{x\in\mathfrak{g} \mid [a, x] = 0\}\).
The conjecture has been proved in some special cases; see, for example: if \(b(G)\leq 4\) in [4], [5], if \(b(G)\leq p + 1\) in [6], if \(G\) is a \(p\)-group of maximal class in [7], if \(G\) is metabelian in [1], and if \(G\) is not covered by its two-step centralizers in [1]. See also [3] for more classes of \(p\)-groups satisfying the conjecture. A more detailed description of the results related to the class-breadth conjecture can be found in the introduction of [2].
In this paper, we study the general case \(p>2\) of the conjecture.
Notation and conventions. For any \(p\)-group \(G\), any of its subgroups \(H, H_1, H_2,\ldots, H_n\), any subset \(S\subseteq G\), and any elements \(a, b\in G\), we denote:
\([a, b] := aba^{-1}b^{-1}\),
\(a^b := bab^{-1}\),
\(H^a := aHa^{-1}\)
\(H_1H_2\ldots H_n := \langle \cup_{i=1}^nH_i\rangle\) (it should be noted that this notation differs from the standard one, in particular, \(H_1H_2\ldots H_n = H_{\sigma(1)}H_{\sigma(2)}\ldots H_{\sigma(n)}\) for each permutation \(\sigma\in S_n\)),
\([H_1, H_2,\ldots, H_n] := \left\langle\left\{ [[\ldots [[h_1, h_2], h_3],\ldots ], h_n] \mid h_i\in H_i, 1\leq i\leq n\right\}\right\rangle\),
\(\langle S\rangle\) – the subgroup of \(G\) generated by \(S\),
\(\langle\!\langle S\rangle\!\rangle^H := \langle\{s^h\;\vert\;h\in H, s\in S\}\rangle\) – the normal closure of \(S\) in \(\langle S\rangle H\),
\([a, S] := \{[a, s] \mid s\in S\}\),
\([S, a] := \{[s, a] \mid s\in S\}\),
\(\llbracket H_1, H_2,\ldots, H_n\rrbracket^H := \langle\!\langle\left\{ [[\ldots [[h_1, h_2], h_3],\ldots ], h_n] \mid h_i\in H_i, 1\leq i\leq n\right\}\rangle\!\rangle^H\),
\(N_G(H)\) – the normalizer of \(H\) in \(G\),
\(Z_G(H) := \cap_{h\in H}Z_G(h)\),
\(Z_i(G)\) – the \(i\)-th term of the upper central series of \(G\) (\(Z_0(G) = \{e\}, Z_1(G) = Z(G)\)),
\(\gamma_i(G)\) – the \(n\)-th term of the lower central series of \(G\) (\(\gamma_1(G) = G, \gamma_2(G) = G'\)),
\(G^{(i)}\) – the \(i\)-th derived subgroup of \(G\) (\(G^{(0)} = G\), \(G^{(1)} = G'\)),
\(d(G)\) – the derived length of \(G\),
for each element \(h\in H\), denote \(b_H(h) := \log_p[H : H\cap Z_G(h)]\),
by a maximal subgroup \(M\) of \(G\), we mean a proper subgroup not contained in any other proper subgroup of \(G\); such a subgroup is always normal in \(G\) and has index \(p\) in \(G\).
In Section 2, we introduce the necessary notions regarding \(G\)-groups.
In Section 3, we introduce a strengthened class-breadth conjecture for \(G\)-groups (see Conjecture 1). For this conjecture, we make some additional assumptions and state Theorem 1.
In Section 4, we prove Theorem 1.
I am grateful to Y. V. Novikova and Huawei Moscow Research Center, and to the Theoretical Physics and Mathematics Advancement Foundation “BASIS”.
Let \(p\geq 2\) be any prime number and \(G\) be any \(p\)-group. By a \(G\)-group, we will denote any \(p\)-group \(P\) such that \(G\subseteq Aut(P)\), endowed with the natural action of \(G\) via automorphisms (unless specified otherwise, when \(G\) is a \(p\)-group, any \(G\)-group is assumed to be a \(p\)-group for the same prime \(p\)).
Consider any \(G\)-group \(P\), let \(P_1 := G\ltimes P\) denotes the respective semidirect product.
Definition 1. For each subgroup \(H\) of \(P\), we say that \(H\) is a \(G\)-invariant subgroup of \(P\) if \(g\cdot h\in H\) for all \(g\in G\) and all \(h\in H\), which is equivalent to \([G, H]\subseteq H\) (\(G, H\subseteq P_1\)). Therefore, each \(G\)-invariant subgroup \(H\) of \(P\) is a \(G\)-group, equipped with the restricted action of \(G\) on \(H\).
Definition 2. For each \(G\)-group \(P\), define \(\gamma_2^G(P) := P'[G, P]\).
Definition 3. For each \(G\)-group \(P\), denote \(\gamma_1^G(P) := P\) and \(\gamma_i^G(P) := \gamma_2^G(\gamma_{i-1}^G(P))\) for each \(i\geq 2\). We will call \(\gamma_i^G(P)\) the \(G\)-lower central series of \(P\). Define \[\operatorname{cl}^G(P) := \min \{k \mid \gamma_{k + 1}^G(P) = \{e\}\}.\]
We will call \(\operatorname{cl}^G(P)\) the \(G\)-nilpotency class of \(P\).
Definition 4. Let \(P\) be a \(G\)-group. A finite sequence of subgroups \(C_i\), \(0\leq i\leq N\) of \(P\) such that \(C_0 = P\) and \(C_N = \{e\}\) is called a \(G\)-central series of \(P\) if, for each \(0 \le i \le N-1\), the following hold:
\(C_{i+1} \subseteq C_i\),
\([G, C_i] \subseteq C_{i+1}\).
The following two propositions are straightforward, and we omit their proofs.
Proposition 1. For each \(G\)-group \(P\) and each its \(G\)-central series \(C_i\), \(0\leq i\leq N\):
\(C_i\) are \(G\)-invariant subgroups of \(P\),
\(\gamma_i^G(P)\subseteq C_{i-1}\), \(\forall 1\leq i\leq N + 1\).
Proposition 2. In the case \(G = \{e\}\), we have that for each \(G\)-invariant subgroup \(H\) of \(P\), \(\gamma_2^G(H) = H'\), \(\gamma_i^G(H) = H^{(i-1)}\), and \(\operatorname{cl}^G(H) = d(H)\). In the case \(G = \operatorname{Int}(P)\trianglelefteq Aut(P)\), we have that for each \(G\)-invariant subgroup \(H\) of \(P\), \(\gamma_2^G(H) = [G, H]\), \(\gamma_i^G(P) = \gamma_i(P)\), and \(\operatorname{cl}^G(P) = \operatorname{cl}(P)\).
Definition 5. For each element \(x\in P\), define \(b^G(x) := b_{P_1}(x)\). We will call \(b^G(x)\) the \(G\)-breadth of \(x\in P\).
Definition 6. Define \(\displaystyle{b^G(P) := \max_{x\in P}b^G(x)}\). We will call \(b^G(P)\) the \(G\)-breadth of \(P\).
The following lemma is well-known, and we omit its proof.
Lemma 1. Consider any group \(A\) and any its subgroups \(B, C\) such that \(C\) is a normal subgroup of \(A\). Then \(BC = CB = \{bc \mid b\in B, c\in C\} = \{cb \mid b\in B, c\in C\}\).
The following lemma is straightforward to verify, and its proof is omitted:
Lemma 2. For each \(G\)-invariant subgroup \(H\) of \(P\), the group action of \(G\) on \(P\) induces an action of \(G\) on \(H\), and \(GH = G \ltimes H = \{gh \mid g\in G, h\in H\} \subseteq P_1\) is the respective inner semidirect product of \(G\) and \(H\).
Definition 7. For each \(G\)-invariant subgroup \(H\) of \(P\) and each element \(h\in H\), define \(b_H^G(h) := b_{GH}(h) \overset{\text{Lem. }\ref{lemma2}}{=} b_{G\ltimes H}(h)\).
Definition 8. Define \[\mathcal{M}^G(P) := \min \biggl\{ k \biggm| \begin{align} &\text{the set } \{x \in P \mid b^G(x) > k\} \text{ can be covered} \\ &\text{by two proper subgroups of } P \text{ containing } \gamma_2^G(P) \end{align} \biggr\}.\]
Remark. These two proper subgroups of \(P\) are \(G\)-invariant from the fact that they contain \([G, P]\subseteq \gamma_2^G(P)\).
The class-breadth conjecture for \(p > 2\) -groups can be generalized via the following statement that also generalizes [8]:
Conjecture 1. Consider any prime number \(p > 2\) and any \(p\)-group \(G\). Then for each \(G\)-group \(P\) there exists a \(G\)-central series \(C_i\), \(0\leq i\leq N\) of \(P\) such that:
\(C_0 = P\), \(C_1 = \gamma_2^G(P)\), \(C_N = \{e\}\),
\(\log_p[C_i : C_{i+1}]\leq \mathcal{M}^G(P) - i + 1\), \(\forall 1\leq i\leq N - 1\).
Proof. Without loss of generality, we may assume that \(C_{N-1}\not=\{e\}\). From Properties 1, 2, it follows that \(1\leq\log_p|C_{N-1}| = \log_p|C_{N-1}/C_N| \leq \mathcal{M}^G(P) - N + 2\), therefore, \(N\leq \mathcal{M}^G(P) + 1\) and, from Proposition 1, \(\gamma_{\mathcal{M}^G(P) + 2}^G(P)\subseteq\gamma_{N+1}^G(P)\subseteq C_N = \{e\}\), i.e. \(\operatorname{cl}^G(P)\leq \mathcal{M}^G(P) + 1\). In the case \(G = Int(P)\), from Proposition 2 we obtain the class-breadth conjecture for \(p>2\) -groups. To prove [8], notice that Properties 1, 2 imply \(\log_p|\gamma_2^G(P)| = \log_p|C_1| = \sum_{i=1}^{N-1}\log_p[C_i : C_{i+1}]\leq \sum_{i=1}^{N-1}\max (0, \mathcal{M}^G(P) - i + 1) \leq \mathcal{M}^G(P)(\mathcal{M}^G(P)+1)/2\). In the case \(G = Int(P)\), from Proposition 2 we obtain [8] for \(p>2\) -groups. ◻
In this paper, we prove the following special case of Conjecture 1:
Theorem 1. Consider any \(p>2\) -group \(G\) and any \(G\)-group \(P\), let \(P_1 := G\ltimes P\) denotes the respective semidirect product. Assume that there exists a normal and \(G\)-invariant subgroup \(M\) of \(P\) satisfying:
\(\log_p[P : M] \leq 1\),
\([G, M] = \{e\}\).
Then \(G\)-group \(P\) has a \(G\)-central series \(C_i\), \(0\leq i\leq N\) as in Conjecture 1.
Remark 1. When \(M = P\) and \(P\not= \{e\}\) (the statement is trivially true when \(P = \{e\}\)) in Theorem 1, we can replace \(M\) with any maximal subgroup of \(P\) that contains \(\gamma_2^G(P)\) (for example, \(M = M_0\cap P\), where \(M_0\) is any maximal subgroup of \(G\ltimes P\)), preserving Conditions 1 and 2 of Theorem 1 (\(M\) will be still \(G\)-invariant since \([G, M]\subseteq [G, P]\subseteq \gamma_2^G(P)\subseteq M\)). Therefore, without loss of generality, we will assume that \(\log_p[P : M] = 1\) in Theorem 1.
Remark 2. Theorem 1 in the case when \(M = P\) implies [8] by a similar argument to that in Proposition 3.
Lemma 3. Consider any group \(X\) and any its subgroups \(A, B, C, D\) such that either \(B\subseteq N_X(C)\) or \(C\subseteq N_X(B)\) holds, and \(D\trianglelefteq X\). Consider also any subset \(S\) of \(X\). Then
\([A, BC]\subseteq [A, B]\llbracket A, C\rrbracket^B\),
\([A, \langle\!\langle S\rangle\!\rangle^X]\subseteq \langle\!\langle \{[\langle\!\langle A\rangle\!\rangle^X, s]\;\vert\;s\in S\}\rangle\!\rangle^X\).
Proof. Part (a) follows from Lemma 1, the fact that either \(B\subseteq N_X(C)\) or \(C\subseteq N_X(B)\), and the fact that \([a, bc] = [a, b][a, c]^b\), \(\forall a, b, c\in A\).
Part (b) follows from \([a, bc] = [a, b][a, c]^b\), \(\forall a, b, c\in A\) and the induction on the length \(k\) of a word \(s = s_1^{x_1}\ldots s_k^{x_k}\in \langle\!\langle S\rangle\!\rangle^X\) (the base case \(k=1\) follows from \([a, s^x] = [a^{x^{-1}}, s]^x\in RHS\), \(\forall a\in A, \forall s\in S, \forall x\in X\)). ◻
Lemma 4. Consider any group \(X\) and any its subgroups \(A, B\), such that \(B\trianglelefteq X\). Then \([\langle\!\langle A\rangle\!\rangle^X, B] = \llbracket A, B\rrbracket^X\).
Proof. From \(B, \langle\!\langle A\rangle\!\rangle^X\trianglelefteq X\), we obtain \([\langle\!\langle A\rangle\!\rangle^X, B]\trianglelefteq X\) and, therefore, from \([A, B]\subseteq [\langle\!\langle A\rangle\!\rangle^X, B]\), that \(\llbracket A, B\rrbracket^X\subseteq [\langle\!\langle A\rangle\!\rangle^X, B]\). Inclusion \([\langle\!\langle A\rangle\!\rangle^X, B] \subseteq \llbracket A, B\rrbracket^X\) follows from Lemma 3(b). ◻
Lemma 5. For each \(p\)-group \(G\) and any its maximal subgroup \(H\) such that \(H' \subsetneq G'\), the set \(X := \{x\in H \mid [x, G]\subseteq H'\}\) is a proper subgroup of \(H\).
Proof. Note that from the maximality of \(H\) in \(G\), it follows that \(H\trianglelefteq G\) and \([G : H] = p\), which implies \(H' \trianglelefteq G\). Consider the natural homomorphism of factorization \(\pi : G \to G/H'\). It is straightforward to check that \(X = \pi^{-1}(Z(\pi(G))) \cap H\), which is clearly a subgroup of \(H\). If \(X = H\), then \(\pi(X) = \pi(H)\) is a central subgroup of index \(p\) in \(\pi(G)\). This implies that \(\pi(G)\) is abelian, and, consequently, \(G' = H'\), which is a contradiction. ◻
Lemma 6. If the conditions of Theorem 1 are satisfied, then the following hold:
\(M\trianglelefteq P_1\),
\(\llbracket G, M\rrbracket^{P_1} = \{e\}\),
\([\langle\!\langle G\rangle\!\rangle^{P_1}, M] = \{e\}\),
\([P_1, P]\subseteq M\),
\([G, P_1]\subseteq G'[G, P]\).
Proof. Proof of (a). It follows from:
the fact that \(P_1 = GP\),
the fact that the normalizer \(N_{P_1}(M)\) of \(M\) in \(P_1\) contains \(P\), because \(M\trianglelefteq P\), and
the fact that the normalizer \(N_{P_1}(M)\) of \(M\) in \(P_1\) contain \(G\), because \(M\) is \(G\)-invariant.
Proof of (b). It follows from the fact that \([G, M] = \{e\}\).
Proof of (c). It follows from Part (b) and Lemma 4.
Proof of (d). From the facts that \(M\) is a \(G\)-invariant subgroup of \(P\), \([P : M]\leq p\), and Lemma 2, we obtain \(GM = G\ltimes M\) and \([P_1 : GM] = [P : M]\leq p\), which implies \(P_1'\subseteq GM = G\ltimes M\). Therefore, from \(P_1'\subseteq G\ltimes M\) and \(P\trianglelefteq P_1\), it follows that \([P_1, P]\subseteq P\) and \[[P_1, P] = \underbrace{[P_1, P]}_{\subseteq P_1'\subseteq G\ltimes M}\cap P\subseteq (G\ltimes M)\cap P = M.\]
Proof of (e). It follows from Lemma 3(a), the fact that \(P_1 = GP\), and the fact that \([G, P]\) is a \(G\)-invariant subgroup of \(P\). ◻
Lemma 7. Consider any \(p\)-group \(G\) and any \(G\)-group \(P\), let \(P_1 := G \ltimes P\) denotes the respective semidirect product. Consider any \(G\)-invariant subgroup \(H\) of \(P\). Denote \(H_1 := GH \overset{\text{Lem. }\ref{lemma2}}{=}G\ltimes H\). Then:
\([G, H]\trianglelefteq H_1\),
\(\gamma_2^G(H)\) is a \(G\)-invariant subgroup of \(P\),
\(H_1' = G'\gamma_2^G(H) = G'\ltimes\gamma_2^G(H)\),
\(H_1' \cap P = \gamma_2^G(H) \subseteq H\).
Proof. Proof of (a). It follows from [9] (see also [10], [11]).
Proof of (b). From \(H\trianglelefteq H_1\), we obtain \(H'\trianglelefteq H_1\) and \(G\subseteq N_{H_1}(H')\cap N_{H_1}(H)\). Therefore, for any element \(g\in G\), we have that \[(\gamma_2^G(H))^g = ([G, H]H')^g = [G^g, H^g](H')^g = [G, H]H' = \gamma_2^G(H).\]
Proof of (c). This follows by applying [12] to the semidirect product \(H_1 = G \ltimes P\).
Proof of (d). Applying Part (c), we get \(H_1'\cap P = \gamma_2^G(H)\). To prove the final inclusion, note that \(\gamma_2^G(H) \subseteq H\) holds from the fact that \(H\) is \(G\)-invariant. ◻
Lemma 8. In the situation of Theorem 1, consider any \(G\)-invariant subgroup \(H\) of \(P\) such that \([P : H] = p\). Denote \(H_1 := GH \overset{\text{Lem. }\ref{lemma2}}{=}G\ltimes H\) and \(X := \{x \in H_1 \mid [x, P_1] \subseteq H_1'\}\). Then:
The set \(X\) is a subgroup of \(H_1\) that is normal in \(P_1\).
In the case when \(H\not= M\), at least one of the following holds:
\(H_1' = P_1'\),
the set \(X\) is a proper subgroup of \(H_1\) that contains \(G \gamma_2^G(H) \overset{\text{Lems. }\ref{lemma2}\text{ and }\ref{lem9}\text{(b)}}{=} G\ltimes \gamma_2^G(H)\).
In the case when \(H_1'\subsetneq P_1'\) and \(X\cap H\subsetneq H\), we have that \(b_{H_1}(h) \leq b_{P_1}(h) - 1\) for all \(h \in H \setminus X\).
\(\gamma_2^G(P) = \gamma_2^G(H)\circ [x, P_1]\), \(\forall x\in P\setminus H\) (here \(A\circ B := \{ab\;\vert\;a\in A, b\in B\}\) for each subsets \(A, B\) of \(P_1\)).
Proof. Proof of (a). From \([P : H] = p\), we get \([P_1 : H_1] = [P : H] = p\). Therefore, \(H_1 \trianglelefteq P_1\), which implies \(H_1' \trianglelefteq P_1\). Consider the natural homomorphism of factorization \(\pi : P_1 \to P_1/H_1'\). Then we can express \(X\) as \[X = \pi^{-1}(Z(\pi(P_1))) \cap H_1,\] which is clearly a subgroup of \(H_1\). From \(H_1, \pi^{-1}(Z(\pi(P_1)))\trianglelefteq P_1\), we have that \(X\trianglelefteq P_1\).
Proof of (b). Assume that \(H_1' \subsetneq P_1'\), then Property (ii) holds due to the following arguments:
From (a), it follows that \(X\) is a subgroup of \(H_1\).
From Lemma 3(a), Lemma 6(b), and the facts that \(H\not= M\), \([P : H] = [P : M] = p\), it follows that \(P = HM\) and: \[[G, H] \subseteq [G, P] = [G, HM] \subseteq [G, H]\llbracket G, M\rrbracket^{P_1} = [G, H]\]
and, therefore, \[[G, P] = [G, H].\label{equationPH}\tag{1}\]
\(X\) is a proper subgroup of \(H_1\): this follows directly from Lemma 5.
\(X\) contains \(G\): from Lemma 6(e) and (4.1), we obtain \[\begin{align} [G, P_1] &\subseteq G'[G, P] \subseteq H_1'[G, P] =\\ &= [G, P]H_1' = [G, H]H_1' = H_1'. \end{align}\]
It contains \(\gamma_2^G(H)\): Since \([P_1 : H_1] = [P : H] = p\), we have \(H_1 \trianglelefteq P_1\), which implies \(H_1' \trianglelefteq P_1\). So \([H_1', P_1]\subseteq H_1'\), \(\gamma_2^G(H) = [G, H]H'\subseteq H_1'H' \subseteq H_1'\), and \[[\gamma_2^G(H), P_1] \subseteq [H_1', P_1] \subseteq H_1'.\]
Therefore, it contains the product \(G \gamma_2^G(H)\).
Proof of (c). Consider any element \(h\in H\) such that \(b_{H_1}(h) > b_{P_1}(h) - 1\). Since \(H_1 \subseteq P_1\), it is easy to see that then \(b_{H_1}(h) = b_{P_1}(h)\), which implies \([h, P_1] = [h, H_1]\subseteq [H, H_1]\subseteq H_1'\), i.e. that \(h\in X\).
Proof of (d). Consider any element \(x\in P\setminus H\). From the proof of [8] and the facts that \([P_1 : H_1] = [P : H] = p\) and \(x\in P\setminus H\subseteq P_1\setminus H_1\), it follows that \[P_1' = H_1'\circ [x, P_1],\text{ where }A\circ B := \{ab\;\vert\;a\in A, b\in B\}\text{ for each subsets }A, B\text{ of }P_1.\label{eqphx}\tag{2}\]
Therefore, from 2 , Lemma 7(c) (applied to \(P, H\subseteq P\)), Lemma 7(b) (applied to \(H\subseteq P\)), and Lemma 2, it follows that \[\begin{align}&\gamma_2^G(P) = P_1'\cap P = (H_1'\circ [x, P_1])\cap P = \\ &= ((P'\gamma_2^G(H))\circ [x, P_1])\cap P = ((P'\ltimes\gamma_2^G(H))\circ [x, P_1])\cap P = \gamma_2^G(H)\circ [x, P_1].\end{align}\] ◻
Lemma 9. Consider any \(p\)-group \(\widetilde{G}\) and any its normal subgroups \(G_1, G_2\) such that \(G_1G_2 = \widetilde{G}\). Let \(C\) be a maximal subgroup of \(G_2\). Assume that \([G_1, C] = \{e\}\) and \(C\) is a normal subgroup of \(\widetilde{G}\). Then \([G_1, x] = [G_1, G_2]\) for each element \(x\in G_2\setminus C\).
Proof. Consider any element \(x\in G_2\setminus C\). Consider the natural homomorphism of factorization \(\pi : \widetilde{G}\to\widetilde{G}/C\) (\(C\trianglelefteq \widetilde{G}\) by the assumptions of the lemma). From \(G_2\trianglelefteq\widetilde{G}\) and \([G_2 : C] = p\), we obtain \(\pi(G_2)\trianglelefteq\pi(\widetilde{G})\) and \(|\pi(G_2)| = p\), therefore, \(\pi(G_2)\subseteq Z(\pi(\widetilde{G}))\) and \([\pi(G_2), \pi(\widetilde{G})] = \{e\}\). So, combining \([G_1, C] = \{e\}\) and \([\pi(G_2), \pi(\widetilde{G})] = \{e\}\), we obtain \[[\widetilde{G}, G_2]\subseteq\ker\pi = C\subseteq Z_{\widetilde{G}}(G_1).\label{eqgg2}\tag{3}\]
We have that \([G_1, x]\) is a normal subgroup of \(\widetilde{G}\), because:
\([G_1, x]\) is a subgroup of \(\widetilde{G}\), because: \[\forall a, b\in G_1,\quad [a, x][b, x] = (\underbrace{[a, x]}_{\mathclap{\in [\widetilde{G}, G_2]\underset{\eqref{eqgg2}}{\subseteq} Z_{\widetilde{G}}(G_1)}})^b[b, x] = [ba, x].\]
\([G_1, x]\trianglelefteq \widetilde{G}\), because:\[\begin{align} g\in G_1,\;\forall y\in \widetilde{G},\quad [g, x]^y &= [g^y, x^y] = [g^y, x[x^{-1}, y]] =\\ &= [g^y, x]\underbrace{[\overbrace{g^y}^{\in G_1}, \underbrace{[x^{-1}, y]}_{\mathclap{\in [G_2, \widetilde{G}]\underset{\eqref{eqgg2}}{\subseteq} Z_{\widetilde{G}}(G_1)}}]^x}_{\mathclap{\subseteq [G_1, Z_{\widetilde{G}}(G_1)]^x = \{e\}}} = [\underbrace{g^y}_{\in G_1}, x]\in [G_1, x]. \end{align}\]
Obviously, \([G_1, G_2]\supseteq [G_1, x]\), so after the factorization \(\widetilde{G}\to \widetilde{G}/[G_1, x]\), we may reduce this lemma to the case \([G_1, x] = \{e\}\). From \([G_1, x] = [G_1, C] = \{e\}\), the facts that \(C\trianglelefteq \widetilde{G}\), \(G_2 = \langle x\rangle C\), and Lemma 3(a), we obtain \(\langle x\rangle\subseteq Z_{\widetilde{G}}(G_1)\) and \[[G_1, G_2] = [G_1, \langle x\rangle C] \subseteq [G_1, \langle x\rangle]\llbracket G_1, C\rrbracket^{\langle x\rangle} = \{e\}\subseteq [G_1, x]\subseteq [G_1, G_2].\] ◻
Lemma 10. Consider any \(p\)-group \(A\) and any its normal subgroup \(N\). Then there exists a sequence of normal subgroups \(N_i\), \(0\leq i\leq \log_p|N|\) of \(A\) such that:
\(N_i\subseteq N\), \(\forall 0\leq i\leq \log_p|N|\),
\(N_i\subseteq N_{i+1}\), \(\forall 0\leq i\leq \log_p|N| - 1\),
\(N_0 = \{e\}\),
\(N_{\log_p|N|} = N\),
\([N_{i+1} : N_i] = p\), \(\forall 0\leq i\leq\log_p|N| - 1\),
\([A, N_{i+1}]\subseteq N_i\), \(\forall 0\leq i\leq\log_p|N| - 1\).
Proof. The proof is by induction on \(|N|\). The base case \(|N| = 1\) is trivial. Induction step: Assume that \(|N|>1\). From \(N\trianglelefteq A\) it follows that \(N\cap Z(A)\not=\{e\}\) and there exists an element \(e\not= x\in N\cap Z(G)\) such that \(x^p = e\). After the application of the induction hypothesis to \(A/\langle x\rangle\), we get that there exists a sequence \(K_i\), \(0\leq i\leq \log_p|N| - 1\) of normal subgroups of \(A/\langle x\rangle\) that satisfies Properties 1 – 6 for \(A/\langle x\rangle\) and \(N/\langle x\rangle\trianglelefteq A/\langle x\rangle\). Easy to check that \(N_0 := \{e\}\), \(N_i := \langle x\rangle K_{i-1}\), \(\forall 1\leq i\leq\log_p|N|\) satisfies Properties 1 – 6 of the lemma. ◻
We proceed by induction on the order \(|P|\) of the \(G\)-group \(P\). The base case \(|P| = 1\) is trivial. Consider any \(G\)-group \(P\) of order \(\geq p\) and assume that the statement is proved for all smaller \(G\)-groups \(P\).
Assume that \(P/\gamma_2^G(P)\) is cyclic (\(P'\subseteq \gamma_2^G(P)\subseteq P\), so \(\gamma_2^G(P)\trianglelefteq P\)).
From Lemmas 2 and [lemy](b), it follows that \(\gamma_2^G(P)\) is a \(G\)-invariant subgroup of \(P\) and \(G\gamma_2^G(P) = G\ltimes \gamma_2^G(P)\). From Definitions 7 and 8, it follows that there exist two proper subgroups \(Q_1, Q_2\) of \(P\) such that \(\gamma_2^G(P) \subseteq Q_1 \cap Q_2\) and \[b_{P_1}(x) \leq \mathcal{M}^G(P), \quad \forall x \in P\setminus (Q_1 \cup Q_2).\]
Since \(\gamma_2^G(P) \subseteq Q_1 \cap Q_2\) and the quotient group \(P/\gamma_2^G(P)\) is cyclic, we have \(Q_1 \cup Q_2 \subseteq Q\), where \(Q\) is the unique maximal subgroup of \(P\) containing \(\gamma_2^G(P)\). Therefore, \[b_{P_1}(x) \leq \mathcal{M}^G(P), \quad \forall x\in P\setminus Q. \label{eq1}\tag{4}\]
Consider any element \[a\in P\setminus (Q\cup M)\label{eqelx}\tag{5}\] (such an element exists since a group \(P\) cannot be covered by its two proper subgroups \(Q\) and \(M\)). Combining the facts that \(a\in P\setminus Q\) and 4 , we obtain \[b_{P_1}(a)\leq\mathcal{M}^G(P).\label{eqclm}\tag{6}\]
From Lemma 7(a) applied to \(P\subseteq P\), we obtain \([G, P]\trianglelefteq P_1\). Consider the natural homomorphism of factorization \(\pi : P_1\to P_1/[G, P]\). From \(\pi(P)/\pi(P)' = \pi(P/P') = P/([G, P]P') = P/\gamma_2^G(P)\) – is a cyclic group, it follows that \(\pi(P)\) – is a cyclic group and, therefore, \(\pi(P') = \pi(P)' = \{e\}\), which implies \(P'\subseteq\ker\pi = [G, P]\) and \[\gamma_2^G(P) = [G, P]P' = [G, P].\label{eqpker}\tag{7}\]
From Lemma 6(c), we get \([\langle\!\langle G\rangle\!\rangle^{P_1}, M] = \{e\}\). Therefore, after the application of Lemma 9 to \(\widetilde{G} := P_1\), \(G_1 := \langle\!\langle G\rangle\!\rangle^{P_1}\), \(G_2 := P\), \(C := M\), \(x := a\), we get that \([\langle\!\langle G\rangle\!\rangle^{P_1}, a] = [\langle\!\langle G\rangle\!\rangle^{P_1}, P]\) and, therefore, \[\log_p|[\langle\!\langle G\rangle\!\rangle^{P_1}, P]| = \log_p|[\langle\!\langle G\rangle\!\rangle^{P_1}, a]|\leq \log_p|[P_1, a]| = b_{P_1}(a).\label{equationsublemma2}\tag{8}\]
Combining 6 , 7 , and 8 , we obtain \[\log_p|\gamma_2^G(P)| = \log_p|[G, P]|\leq \log_p|[\langle\!\langle G\rangle\!\rangle^{P_1}, P]| \leq b_{P_1}(a) \leq \mathcal{M}^G(P).\label{equationlemma}\tag{9}\]
From \(P'\subseteq \gamma_2^G(P)\subseteq P\) and Lemma 7(b) it follows that \(\gamma_2^G(P)\) is a normal and \(G\)-invariant subgroup of \(P\). So \(P, G\subseteq N_{P_1}(\gamma_2^G(P))\) and \(\gamma_2^G(P)\trianglelefteq GP = P_1\). Therefore, from Lemma 10 applied to \(\gamma_2^G(P)\trianglelefteq P_1\), it follows that there exists a sequence \(K_j\), \(0\leq j\leq \log_p|\gamma_2^G(P)|\) of normal subgroups of \(P_1\) such that:
\(K_j\subseteq \gamma_2^G(P)\), \(\forall 0\leq j\leq \log_p|\gamma_2^G(P)|\),
\(K_j\subseteq K_{j+1}\), \(\forall 0\leq j\leq \log_p|\gamma_2^G(P)| - 1\),
\(K_0 = \{e\}\),
\(K_{\log_p|\gamma_2^G(P)|} = \gamma_2^G(P)\),
\([K_{j+1} : K_j] = p\), \(\forall 0\leq j\leq \log_p|\gamma_2^G(P)| - 1\),
\([P_1, K_{j+1}]\subseteq K_j\), \(\forall 0\leq j\leq \log_p|\gamma_2^G(P)| - 1\).
We will prove that the sequence \(C_i\), \(0\leq i\leq \log_p|\gamma_2^G(P)| + 1\) of subgroups of \(P\), defined as \[\begin{align} C_0 &:= P\\ C_i &:= K_{\log_p|\gamma_2^G(P)|- i + 1},\quad \forall 1\leq i\leq \log_p|\gamma_2^G(P)| + 1 \end{align}\label{eqCK}\tag{10}\] is a \(G\)-central series of \(P\) that satisfies Properties 1 and 2 of Conjecture 1, which would complete the proof for the case of cyclic \(P/\gamma_2^G(P)\). This follows from:
\(C_i\), \(0\leq i\leq \log_p|\gamma_2^G(P)| + 1\) is a \(G\)-central series of \(P\) that satisfies Property 1 of Conjecture 1, because from 10 and Properties (2), (3), (4), (6) of \(K_j\) we obtain:
\(C_0 = P\),
\(C_{\log_p|\gamma_2^G(P)| + 1} = K_0 = \{e\}\),
\(C_1 \subseteq P = C_0\)
\(C_{i+1} = K_{\log_p|\gamma_2^G(P)| - i}\subseteq K_{\log_p|\gamma_2^G(P)| - i + 1} = C_i\), \(\forall 1\leq i\leq \log_p|\gamma_2^G(P)|\),
\([G, C_0] = [G, P]\subseteq \gamma_2^G(P) = K_{\log_p|\gamma_2^G(P)|} = C_1\),
\([G, C_i] = [G, K_{\log_p|\gamma_2^G(P)| - i + 1}]\subseteq [P_1, K_{\log_p|\gamma_2^G(P)| - i + 1}]\subseteq\)
\(\subseteq [P_1, K_{\log_p|\gamma_2^G(P)| - i}] = C_{i+1}\), \(\forall 1\leq i\leq \log_p|\gamma_2^G(P)|\),
\(C_0 = P\),
\(C_1 = K_{\log_p|\gamma_2^G(P)|} = \gamma_2^G(P)\),
\(C_{\log_p|\gamma_2^G(P)| + 1} = K_0 = \{e\}\).
\(C_i\), \(0\leq i\leq \log_p|\gamma_2^G(P)| + 1\) satisfies Property 2 of Conjecture 1, because:
in the case when \(\log_p|\gamma_2^G(P)| = 0\), there is no such \(1\leq i\leq \log_p|\gamma_2^G(P)|\) and Property 2 of Conjecture 1 is trivially satisfied,
in the case when \(\log_p|\gamma_2^G(P)| > 0\), from Property (5), 9 , and 10 it follows that \(\forall 1\leq i\leq \log_p|\gamma_2^G(P)|\),\[\begin{align} \log_p[C_i : C_{i+1}] &= \log_p[K_{\log_p|\gamma_2^G(P)|- i + 1} : K_{\log_p|\gamma_2^G(P)|- i}] = 1\leq\\ &\leq \log_p|\gamma_2^G(P)|- i + 1\leq \mathcal{M}^G(P) - i + 1.\end{align}\]
This concludes the case where \(P/\gamma_2^G(P)\) is cyclic.
Assume that \(P/\gamma_2^G(P)\) is non-cyclic (\(P'\subseteq \gamma_2^G(P)\subseteq P\), so \(\gamma_2^G(P)\trianglelefteq P\)).
Consider any two maximal subgroups \(C_1, C_2\) of \(P\) such that:
The following hold:
\(C_1, C_2\) are \(G\)-invariant,
\(\gamma_2^G(P)\subseteq C_1\cap C_2\),
\(\{x\in P \mid b_{P_1}(x) > \mathcal{M}^G(P)\}\subseteq C_1\cup C_2\).
Such subgroups exist by the definition of \(\mathcal{M}^G(P)\) and from the fact that each proper subgroup of \(P\) containing \(\gamma_2^G(P)\) is contained in a maximal subgroup of \(P\) which also contains \(\gamma_2^G(P)\) (which is therefore \(G\)-invariant).
\(C_1 \neq C_2\). These exist, because, if \(C_1 = C_2\) in Property i), then, from the fact that \(P\gamma_2^G(P)\) is non-cyclic, we can replace \(C_2\) with \(C_2^*\), where \(C_2^*\) is any maximal subgroup of \(P\) containing \(\gamma_2^G(P)\) (which is therefore \(G\)-invariant) and is different from \(C_1\).
Consider any maximal subgroup \(H\) of \(P\) such that:
\(C_1 \cap C_2 = H \cap C_1 = H \cap C_2 = H\cap (C_1\cup C_2)\),
\(H\not= M\).
Such a group exists since \(p>2\), \(P/(C_1\cap C_2)\cong \mathbb{Z}_p^2\), and \(M\subsetneq P\). From \(\gamma_2^G(P)\subseteq C_1\cap C_2 = H\cap C_1 \subseteq H\subseteq P\), it follows that \(H\) is a \(G\)-invariant subgroup of \(P\) and contains \(\gamma_2^G(P)\). Therefore, from Lemma 2, we get \(H_1 := GH \overset{\text{Lem. }\ref{lemma2}}{=} G\ltimes H\). From \([P : H] = p\), \([P : C_1\cap C_2] = p^2\), it follows that \([H : C_1\cap C_2] = p\) and \(C_1\cap C_2\) is a proper subgroup of \(H\).
We now prove the following two properties of \(H\):
\(\mathcal{M}^G(H) \leq \begin{cases} \mathcal{M}^G(P) & \text{if } H_1' = P_1', \\ \mathcal{M}^G(P) - 1 & \text{if } H_1' \subsetneq P_1', \end{cases}\)
\(\log_p|\gamma_2^G(P)|\leq\begin{cases} \log_p|\gamma_2^G(H)| & \text{if } H_1' = P_1', \\ \log_p|\gamma_2^G(H)| + \mathcal{M}^G(P) & \text{if } H_1' \subsetneq P_1'. \end{cases}\)
Proof of A1 and A2 in the case when \(H_1' = P_1'\).
Proof of A1. From the definition of \(C_1, C_2\) and the fact that \(H\setminus (C_1\cap C_2) = H\setminus (H\cap (C_1\cup C_2)) = H\setminus (C_1\cup C_2) \subseteq P\setminus (C_1\cup C_2)\) we obtain \[\begin{align} &\gamma_2^G(H)\subseteq \gamma_2^G(P)\subseteq C_1\cap C_2\text{ and} \\ &b_{H_1}(h) \leq b_{P_1}(h) \leq \mathcal{M}^G(P),\quad\forall h \in H \setminus (C_1 \cap C_2). \end{align}\]
Therefore, the set \(\{h \in H \mid b_{H_1}(h) > \mathcal{M}^G(P)\}\) can be covered by two coinciding proper subgroups \(A = B = C_1 \cap C_2\) of \(H\), which contain \(\gamma_2^G(H)\).
Item A2 follows from \(H_1' = P_1'\) and Lemma [lem10](d) applied to \(P\subseteq P\) and to \(H\subseteq P\): \[\log_p|\gamma_2^G(P)| = \log_p|P_1'\cap P| = \log_p|H_1'\cap P| = \log_p|\gamma_2^G(H)|.\]
Proof of A1 and A2 in the case when \(H_1' \subsetneq P_1'\). Denote \(X := \{x \in H_1 \mid [x, P_1] \subseteq H_1'\}\). By Lemma [lem4](a, b) \[G\gamma_2^G(H)\subseteq X\subsetneq H_1\text{ and }X\trianglelefteq P_1.\label{eqXH}\tag{11}\]
We have that \(H\nsubseteq X\), since otherwise, \(X\) would be a proper subgroup of \(H_1\) containing both \(G\) and \(H\), which contradicts to \(H_1 = GH\). From Lemma [lem5](c), we obtain \(b_{H_1}(h) \leq b_{P_1}(h) - 1\) for all \(h \in H\setminus X\). Thus, from the fact that \(H\setminus (X\cup (C_1\cap C_2))\subseteq H\setminus (\underbrace{C_1\cap C_2}_{\mathclap{ = H\cap (C_1\cup C_2)}}) = H\setminus (C_1\cup C_2)\subseteq P\setminus (C_1\cup C_2)\) and the definition of \(C_1, C_2\), it follows that \[b_{H_1}(h) \leq b_{P_1}(h) - 1 \leq \mathcal{M}^G(P) - 1, \quad \forall h \in H \setminus (X \cup (C_1 \cap C_2)).\]
Therefore, the set \(\{h \in H \mid b_{H_1}(h) > \mathcal{M}^G(G) - 1\}\) is covered by two subgroups \(X\cap H\) and \(C_1 \cap C_2\) of \(H\). From the definition of \(C_1, C_2\), we have \(\gamma_2^G(H)\subseteq \gamma_2^G(P)\subseteq C_1\cap C_2\), therefore, from \(H\nsubseteq X\), \([H : C_1\cap C_2] = p\), and 11 , it follows that \(X\cap H\) and \(C_1\cap C_2\) are proper subgroups of \(H\) that contain \(\gamma_2^G(H)\). Hence, \(\mathcal{M}^G(H) \leq \mathcal{M}^G(P) - 1\), and A1 is proved.
To prove A2, consider any element \(a\in P\setminus (H\cup C_1\cup C_2)\) (such an element exists from the fact that each non-trivial \(p>2\) -group cannot be covered by any its three proper subgroups). From \(a\in P\setminus (H\cup C_1\cup C_2)\subseteq P\setminus (C_1\cup C_2)\) and the definition of \(C_1, C_2\), we obtain \[b_{P_1}(a)\leq \mathcal{M}^G(P).\label{equationx2}\tag{12}\]
From Lemma [lemmaX](d) applied to \(H\subseteq P\), \(a\in P\setminus H\), and 12 , it follows that \[\begin{align} \log_p|\gamma_2^G(P)| &= \log_p|\gamma_2^G(H)\circ [a, P_1]|\leq\\&\leq \log_p|\gamma_2^G(H)| + \log_p|[a, P_1]| =\\&= \log_p|\gamma_2^G(H)| + b_{P_1}(a)\leq \log_p|\gamma_2^G(H)| + \mathcal{M}^G(P). \end{align}\label{equationx1}\tag{13}\]
Therefore, A1 and A2 are proved.
We are now ready to complete the proof in the case when \(P/\gamma_2^G(P)\) is non-cyclic.
From the induction hypothesis applied to the \(G\)-group \(H\) and its normal and \(G\)-invariant subgroup \(M\cap H\), it follows that there exists a \(G\)-central series \(K_i\), \(0\leq i\leq N\) of \(H\) such that: \(K_0 = H\), \(K_1 = \gamma_2^G(H)\), \(K_N = \{e\}\), and \[\log_p[K_i : K_{i+1}]\leq \mathcal{M}^G(H) - i + 1,\;\forall 1\leq i\leq N - 1.\label{eqK}\tag{14}\]
Let us analyze the two possible cases:
Case 1. Consider the case when \(H_1' = P_1'\). From Lemma 7(d) and \(P_1' = H_1'\), we obtain \[\gamma_2^G(P) = P_1'\cap P = H_1'\cap P = \gamma_2^G(H).\label{equationcpk}\tag{15}\]
We will prove that the sequence \(C_i\), \(0\leq i\leq N\) of subgroups of \(P\), defined as \[\begin{align} C_0 &:= P\\ C_i &:= K_i,\quad \forall 1\leq i\leq N \end{align}\label{equationcpn}\tag{16}\] is a \(G\)-central series of \(P\) that satisfies Properties 1 and 2 of Conjecture 1, which would complete the proof for the case of non-cyclic \(P/\gamma_2^G(P)\). This is because from 14 , 15 , 16 , and A1, we obtain:
\(P = C_0\supseteq \gamma_2^G(P) = \gamma_2^G(H) = K_1 \supseteq K_2 = C_2\supseteq\ldots\supseteq K_N = C_N = \{e\}\),
\([G, C_0] = [G, P]\subseteq \gamma_2^G(P) = \gamma_2^G(H) = K_1 = C_1\),
\([G, C_i] = [G, K_i]\subseteq K_{i+1} = C_{i+1}\), \(\forall 1\leq i\leq N - 1\),
\(C_0 = P\),
\(C_1 = K_1 = \gamma_2^G(H) = \gamma_2^G(P)\),
\(C_N = K_N = \{e\}\).
\(\log_p[C_i : C_{i+1}] = \log_p[K_i : K_{i+1}]\leq \mathcal{M}^G(H) - i + 1 \leq \mathcal{M}^G(P) - i + 1\), \(\forall 1\leq i\leq N\).
Case 2. Consider the case when \(H_1' \subsetneq P_1'\). We will prove that the sequence \(C_i\), \(0\leq i\leq N + 1\) of subgroups of \(P\), defined as \[\begin{align} C_0 &:= P\\ C_1 &:= \gamma_2^G(P)\\ C_i &:= K_{i-1},\quad \forall 2\leq i\leq N + 1 \end{align}\label{equationcpm}\tag{17}\] is a \(G\)-central series of \(P\) that satisfies Properties 1 and 2 of Conjecture 1, which would complete the proof for the case of cyclic \(P/\gamma_2^G(P)\). This is because from 14 , 17 , A1, and A2, we obtain:
\(P = C_0\supseteq \gamma_2^G(P) = C_1\supseteq\gamma_2^G(H) = K_1 = C_2\supseteq K_2 = C_3\supseteq\ldots\supseteq K_N = C_{N+1} = \{e\}\),
\([G, C_0] = [G, P]\subseteq \gamma_2^G(P) = C_1\),
\([G, C_1] = [G, \underbrace{\gamma_2^G(P)}_{\subseteq H}] \subseteq [G, H]\subseteq \gamma_2^G(H) = K_1 = C_2\),
\([G, C_i] = [G, K_{i-1}]\subseteq K_i = C_{i+1}\), \(\forall 2\leq i\leq N\),
\(C_0 = P\),
\(C_1 = \gamma_2^G(P)\),
\(C_{N + 1} = K_N = \{e\}\),
\(\log_p[C_1 : C_2] = \log_p[\gamma_2^G(P) : K_1] = \log_p[\gamma_2^G(P) : \gamma_2^G(H)]\leq \mathcal{M}^G(P)\),
\(\log_p[C_i : C_{i+1}] = \log_p[K_{i-1} : K_i]\leq \mathcal{M}^G(H) - (i-1) + 1\leq \mathcal{M}^G(P) - i + 1\), \(\forall 2\leq i\leq N\).
This finishes the proof of Theorem 1. \(\square\)