# Article

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Keywords:
global Warfield group; isotype subgroup; knice subgroup; \$k\$-subgroup; separable subgroup; compatible subgroups; Axiom 3; closed set method; global \$k\$-group; sequentially pure projective dimension
Summary:
If \$H\$ is an isotype knice subgroup of a global Warfield group \$G\$, we introduce the notion of a \$k\$-subgroup to obtain various necessary and sufficient conditions on the quotient group \$G/H\$ in order for \$H\$ itself to be a global Warfield group. Our main theorem is that \$H\$ is a global Warfield group if and only if \$G/H\$ possesses an \$H(\aleph _0)\$-family of almost strongly separable \$k\$-subgroups. By an \$H(\aleph _0)\$-family we mean an Axiom 3 family in the strong sense of P. Hill. As a corollary to the main theorem, we are able to characterize those global \$k\$-groups of sequentially pure projective dimension \$\le 1\$.
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