Abstract
If G is any finite Abelian group define γ ( G ) = ∑ i ( e i − 1 ) where e i are the canonic invariants of G ( e i+1| e i for all i). The primary result is the “super-additivity” of γ, i.e., γ ( G ) ⩾ γ ( G / H ) + γ ( H ) for all subgroups H of G . ( ∗ ) In the process of establishing (*) the structure of Abelian groups is studied in great detail and a technique is developed for proving inequalities analogous to (*) for other invariants than γ. We then apply (*) to obtain various further results of a combinatorial nature.
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