From Littlewood-Richardson sequences to subgroup embeddings and back

Details From Littlewood-Richardson sequences to subgroup embeddings and back From Littlewood-Richardson sequences to subgroup embeddings and back

2007-09-18

arxiv.org/abs/0709.2920v1

Mathematics

Group Theory

5 pages

Scientific paper

Let $\alpha$, $\beta$, and $\gamma$ be partitions describing the isomorphism types of the finite abelian $p$-groups $A$, $B$, and $C$. From theorems by Green and Klein it is well-known that there is a short exact sequence $0\to A\to B\to C\to 0$ of abelian groups if and only if there is a Littlewood-Richardson sequence of type $(\alpha,\beta,\gamma)$. Starting from the observation that a sequence of partitions has the LR property if and only if every subsequence of length 2 does, we demonstrate how LR-sequences of length two correspond to embeddings of a $p^2$-bounded subgroup in a finite abelian $p$-group. Using the known classification of all such embeddings we derive short proofs of the theorems by Green and Klein.

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