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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
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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