Generalized flags in \(p\)-groups
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 closely related one where the subgroup lattices of Abelian \(p\)-groups are replaced by \(T\)-invariant subspace lattices of \(\mathbb{F}_p\) vector spaces with nilpotent linear transformations \(T\).
scientific article; zbMATH DE number 1471538