• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.307-330
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• 2013

If $m$ and $n$ are non-negative integers, then three new classes of abelian $p$-groups are defined and studied: the $m$, $n$-simply presented groups, the $m$, $n$-balanced projective groups and the $m$, $n$-totally projective groups. These notions combine and generalize both the theories of simply presented groups and $p^{w+n}$-projective groups. If $m$, $n=0$, these all agree with the class of totally projective groups, but when $m+n{\geq}1$, they also include the $p^{w+m+n}$-projective groups. These classes are related to the (strongly) n-simply presented and (strongly) $n$-balanced projective groups considered in [15] and the n-summable groups considered in [2]. The groups in these classes whose lengths are less than ${\omega}^2$ are characterized, and if in addition we have $n=0$, they are determined by isometries of their $p^m$-socles.