Vol. 75, No. 2, 1978

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λ-large subgroups of Cλ-groups

Ronald Charles Linton

Vol. 75 (1978), No. 2, 477–485

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.

Mathematical Subject Classification 2000
Primary: 20K40
Received: 11 November 1974
Revised: 4 May 1976
Published: 1 April 1978
Ronald Charles Linton