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Integer sequences realized by the subgroup pattern of the symmetric group. (English) Zbl 1288.20002
From the introduction: The subgroup pattern of a finite group \(G\) is the table of marks of \(G\) together with a list of representatives of the conjugacy classes of subgroups of \(G\). In this article we describe a collection of sequences realized by the subgroup pattern of the symmetric group.
This paper is organized as follows. In Section 2 we study the conjugacy classes of subgroups of \(S_n\) for \(n\leq 13\). In Section 3 we examine the tables of marks of \(S_n\) for \(n\leq 13\) and describe how much more information regarding the subgroup structure of \(S_n\) can be obtained. In Section 4 we discuss the Euler transform and its applications in counting subgroups of \(S_n\).
MSC:
20B40 Computational methods (permutation groups) (MSC2010)
19A22 Frobenius induction, Burnside and representation rings
20C40 Computational methods (representations of groups) (MSC2010)
20D30 Series and lattices of subgroups
20B35 Subgroups of symmetric groups
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