On the \(p\)-length of a finite group with conditionally permutable factors. (English) Zbl 1538.20016
We start with some notions/definitions as needed in the formulation of the two main theorems of the paper. All groups are finite here. A subgroup \(H\) of a group \(U\) is called permutable in \(U\) if \(HK = KH\) for all subgroups \(K\) of \(U\).
Subgroups \(A\) and \(B\) of a group \(G\) are called mutually permutable in \(G\) if \(UB = BU\) and \(VA = AV\) for all subgroups \(U\) of \(A\) and all subgroups \(V\) of \(B\).
The subgroup \(T\) of a group \(G\) is called a tcc-subgroup in \(G\) if \(G = TS\) for some subgroup \(S\) of \(G\) in which for any subgroup \(X\) of \(T\) and for any subgroup \(Y\) of \(S\) there exists an element \(u\) in the subgroup \(\langle X,Y\rangle\) such that \(XuYu^{-1}\) is a subgroup of \(G\).
The normal series \(1 = H_0\le H_1\le \dots \le H_n = N\) is called a permutable series in the nilpotent group \(N\), if for any index \(1\le i\le n\) and for any cyclic subgroup \(K_i\) of \(H_i / H_{i-1}\) it holds that \(K_i\) is permutable in \(N/H_{i-1}\). That character \(n\) is called the length of such a permutable series. Define \(P(N) = \min \{ n \mid n\) is the length of some permutable series of \(N\}\).
The \(p\)-length \(l_p(G)\) of a group \(G\) is the usual one as defined in [B. Huppert, Endliche Gruppen. I. Berlin: Springer-Verlag (1967; Zbl 0217.07201)].
Now we state the first main result of the paper.
Theorem 1. Let \(G = AB\) be the product of tcc-subgroups \(A\) and \(B\). Suppose that \(A\) and \(B\) are \(p\)-soluble subgroups of \(G\). Then \(G\) is \(p\)-soluble and \(l_p(G)\le \max \{l_p(A), l_p(B)\}\).
And here is the second main result of the paper.
Theorem 2. Let \(G = AB\) be the product of \(p\)-soluble mutually permutable subgroups \(A\) and \(B\). Let \(Ap\) be a Sylow \(p\)-subgroup of \(A\) and likewise \(Bp\) a Sylow \(p\)-subgroup of \(B\). Then \(l_p(G)\le \max \{P(Ap), P(Bp)\}\).
The max-bound stated in Theorem 2 is precisely 1 lower (whence better) than the max-bound stated in the paper by J. Cossey and Y. Li [Arch. Math. 110, No. 6, 533–537 (2018; Zbl 1494.20030)]. View more details about it in the present paper.
There is a lot to be learned in the proofs of the theorems, the argumentations are well done.
Subgroups \(A\) and \(B\) of a group \(G\) are called mutually permutable in \(G\) if \(UB = BU\) and \(VA = AV\) for all subgroups \(U\) of \(A\) and all subgroups \(V\) of \(B\).
The subgroup \(T\) of a group \(G\) is called a tcc-subgroup in \(G\) if \(G = TS\) for some subgroup \(S\) of \(G\) in which for any subgroup \(X\) of \(T\) and for any subgroup \(Y\) of \(S\) there exists an element \(u\) in the subgroup \(\langle X,Y\rangle\) such that \(XuYu^{-1}\) is a subgroup of \(G\).
The normal series \(1 = H_0\le H_1\le \dots \le H_n = N\) is called a permutable series in the nilpotent group \(N\), if for any index \(1\le i\le n\) and for any cyclic subgroup \(K_i\) of \(H_i / H_{i-1}\) it holds that \(K_i\) is permutable in \(N/H_{i-1}\). That character \(n\) is called the length of such a permutable series. Define \(P(N) = \min \{ n \mid n\) is the length of some permutable series of \(N\}\).
The \(p\)-length \(l_p(G)\) of a group \(G\) is the usual one as defined in [B. Huppert, Endliche Gruppen. I. Berlin: Springer-Verlag (1967; Zbl 0217.07201)].
Now we state the first main result of the paper.
Theorem 1. Let \(G = AB\) be the product of tcc-subgroups \(A\) and \(B\). Suppose that \(A\) and \(B\) are \(p\)-soluble subgroups of \(G\). Then \(G\) is \(p\)-soluble and \(l_p(G)\le \max \{l_p(A), l_p(B)\}\).
And here is the second main result of the paper.
Theorem 2. Let \(G = AB\) be the product of \(p\)-soluble mutually permutable subgroups \(A\) and \(B\). Let \(Ap\) be a Sylow \(p\)-subgroup of \(A\) and likewise \(Bp\) a Sylow \(p\)-subgroup of \(B\). Then \(l_p(G)\le \max \{P(Ap), P(Bp)\}\).
The max-bound stated in Theorem 2 is precisely 1 lower (whence better) than the max-bound stated in the paper by J. Cossey and Y. Li [Arch. Math. 110, No. 6, 533–537 (2018; Zbl 1494.20030)]. View more details about it in the present paper.
There is a lot to be learned in the proofs of the theorems, the argumentations are well done.
Reviewer: Robert W. van der Waall (Huizen)
MSC:
| 20D40 | Products of subgroups of abstract finite groups |
| 20D10 | Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks |
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