Abstract
Let [Formula: see text] denote the set of positive integers that may appear as the real genus of a finite abelian group. We obtain a set of (simple) necessary conditions for an integer [Formula: see text] to belong to [Formula: see text]. We also prove that the real genus of an abelian group is not congruent to 3 modulo 4 and that the genus of an abelian group of odd order is a multiple of 4. Finally, we obtain upper and lower bounds for the density of the set [Formula: see text].
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