If L is a fully invariant
subgroup of the p-primary group G, and if G = B + L for all basic subgroups B of G,
then L is called a large subgroup of G; this definition is due to R. Pierce. In light of
K. Wallace’s generalization of the concept of basic subgroup to that of a λ-basic
subgroup, we extend Pierce’s definition by defining the fully invariant subgroup L to
be a λ-large subgroup of G if G = B + L for all λ-basic subgroups B of G. Our
main theorems are: (1) L is a λ-Iarge subgroup of the Cλ-group G if and
only if L = G(v) where v denotes an increasing sequence of ordinals less
than λ satisfying the gap condition. (2) If L is a λ-large subgroup of the
Cλ-group G, then G∕L is a totally projective group, and L is a Cμ-group
where μ denotes the length of L∕pλG. (3) If L is a λ-large subgroup of the
Cλ-group G, then L is a totally projective group only if G is a totally projective
group.
