Restrictions on the structure of subgroup lattices of finite alternating and symmetric groups. (English) Zbl 1185.20020
The paper deals with an important and well-known open question: which finite lattices can be represented as intervals in the subgroup lattice of a finite group? In particular the authors exhibit restrictions on the structure of the lattice \(\mathcal O_G(H)\) of subgroups of \(G\) containing \(H\leq G\) when \(G\) is a finite alternating or symmetric group.
For a finite lattice \(L\), let \(L'\) be the poset obtained by removing the minimum and maximum elements of \(L\) and, for \(n\in\mathbb{N}\), let \(\Delta(n)\) be the lattice of all subsets of \(\{1,\dots,n\}\), ordered by inclusion. For positive integers \(m_1,\dots,m_t\), the authors consider the lattice \(D\Delta(m_1,\dots,m_t)\) defined as follows: \(D\Delta(m_1,\dots,m_t)'\) is a disjoint union of posets \(\mathcal C_1,\dots,\mathcal C_t\) with \(C_i\cong\Delta(m_i)'\) for all \(i\) and if \(i\neq j\) then no element of \(C_i\) is comparable to any element of \(C_j\).
The authors prove that if \(t>1\) and \(m_1\geq m_2\geq\cdots\geq m_t\geq 3\), then \(G\in\{\text{Alt}(n),\text{Sym}(n)\}\) contains no subgroup \(H\) with \(\mathcal O_G(H)\cong D\Delta(m_1,\dots,m_t)\).
For a finite lattice \(L\), let \(L'\) be the poset obtained by removing the minimum and maximum elements of \(L\) and, for \(n\in\mathbb{N}\), let \(\Delta(n)\) be the lattice of all subsets of \(\{1,\dots,n\}\), ordered by inclusion. For positive integers \(m_1,\dots,m_t\), the authors consider the lattice \(D\Delta(m_1,\dots,m_t)\) defined as follows: \(D\Delta(m_1,\dots,m_t)'\) is a disjoint union of posets \(\mathcal C_1,\dots,\mathcal C_t\) with \(C_i\cong\Delta(m_i)'\) for all \(i\) and if \(i\neq j\) then no element of \(C_i\) is comparable to any element of \(C_j\).
The authors prove that if \(t>1\) and \(m_1\geq m_2\geq\cdots\geq m_t\geq 3\), then \(G\in\{\text{Alt}(n),\text{Sym}(n)\}\) contains no subgroup \(H\) with \(\mathcal O_G(H)\cong D\Delta(m_1,\dots,m_t)\).
Reviewer: Andrea Lucchini (Padova)
MSC:
| 20D30 | Series and lattices of subgroups |
| 20B35 | Subgroups of symmetric groups |
| 06B15 | Representation theory of lattices |
| 08A30 | Subalgebras, congruence relations |
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