Generalized flags in \(p\)-groups
type: Item
instance of: scholarly article
title: Generalized flags in \(p\)-groups
zbMATH Open document ID: 0962.20038
DOI: 10.1016/S0898-1221(00)00110-3
publication date: 2001-06-19T00:00:00Z
review text: Flags are defined as maximal sequences of subspaces of a finite-dimensional vector space. Taking a different point of view one can interpret flags as poset embeddings of the subgroup lattice of a cyclic \(p\)-group into the subgroup lattice of an elementary Abelian \(p\)-group (of the same order). This leads to the following definition: For a fixed prime \(p\), for a positive integer \(n\) and for Abelian \(p\)-groups \(G\), \(H\) of order \(p^n\), a generalized flag is a poset embedding of the subgroup lattice \(L(G)\) into the subgroup lattice \(L(H)\). This paper deals with the question for which groups \(G\), \(H\) such generalized flags exist. Note that, for \(G\) cyclic and \(H\) arbitrary, such an embedding clearly exists. In general, any Abelian group of order \(p^n\) is a direct sum of cyclic \(p\)-groups of orders \(p^{\mu_i}\) (\(i=1,\dots,t\)) with \(n=\mu_1+\cdots+\mu_t\) and hence one can associate such a group with the partition \(\mu=(\mu_1,\dots,\mu_t)\) of \(n\) and, conversely, any such partition \(\mu\) determines the Abelian \(p\)-group \(G_\mu=C_{p^{\mu_1}}\oplus\cdots\oplus C_{p^{\mu_t}}\) (assume \(\mu_1\geq\cdots\geq\mu_t\)). Put \(L_\mu=L(G_\mu)\). Then the question can be rephrased: When does \(L_\mu\) embed into \(L_\lambda\)? Some necessary conditions are proved in this article. For example, if \(L_\mu\) embeds into \(L_\lambda\) then \(\lambda\) refines \(\mu\), that is, \(\lambda\) is obtained from \(\mu\) by partitioning the \(\mu_i\)'s (and rearranging into the desired order). For \(\mu_2=1\) this condition is also sufficient. In the general case, an even stronger necessary condition holds. For another subclass a surprising characterization is shown: Assume \(\mu_3\geq 2\), then \(L_\mu\) embeds into \(L_\mu\) if and only if \(\lambda=\mu\). The remaining case (\(\mu_2\geq 2\), \(\mu_3=1\)) has been discussed but could not be solved completely. Some more results have been obtained concerning this problem and also concerning the close…
Mathematics Subject Classification ID: 20K01
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 51D25
Mathematics Subject Classification ID: 06B05
Mathematics Subject Classification ID: 05B35
zbMATH DE Number: 1471538
zbMATH Keywords: poset embeddings
zbMATH Keywords: finite Abelian \(p\)-groups
zbMATH Keywords: generalized flags
zbMATH Keywords: direct sums of cyclic \(p\)-groups
zbMATH Keywords: subgroup lattices
zbMATH Keywords: subspace lattices
author: Lynne M. Butler
author: Alfred W. Hales
reviewed by: Simone L. Wallutis
MaRDI profile type: Publication
cites work: Q5609457
cites work: Q5589390
cites work: Q3947818
cites work: Generalized flags in finite abelian \(p\)-groups
cites work: Q3663352
cites work: Nonnegative Hall polynomials
cites work: Q5534953
cites work: Order analogues and Betti polynomials
published in: Computers & Mathematics with Applications
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description: scientific article; zbMATH DE number 1471538