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- W2021579635 abstract "If G is a p-group of limit length A, then it satisfies the A-Zippin property provided that whenever A/pxA = G = B/pxB, every isomorphism between px A and px B extends to an isomorphism between A and B. We show that if G is almost balanced in a totally projective group, then G does satisfy the A-Zippin property. This leads to the existence of a great variety of G's that are totally Zippin in the sense that G/paG satisfies the a-Zippin property for all limit ordinals a u), however, examples in (8) suggest that nothing very definitive can be established about the structure of p-groups with the A-Zippin property. Consequently, Warfield (13) proposed as a more tractable class the G's that are totally Zippin in the sense that G/paG has the a-Zippin property for all limit ordinals a < A. He moreover suggested that a reasonable conjecture would be that the totally Zippin p-groups are precisely the S-groups. This turns out not to be so and we shall, in fact, establish the existence of a considerable variety of totally Zippin p-groups of length H = u). On a more positive note, we show that when A is a countable limit ordinal, the only totally Zippin p-groups of length A are the d.s.c.'s (direct sums of countable reduced p- groups); that is, Warfield's conjecture does hold for groups of countable length. Call a subgroup H oi G almost balanced if it is isotype (i.e., paG H H = paH for all ordinals a) and pa(G/H) = paG + H/H for all a < A = length of G. The isotype and X-dense (G = paG + H for all a < A) subgroups H oi G are just the almost balanced subgroups with G/H divisible. As the following technical result indicates, this latter class of subgroups plays a crucial role in the study of A-elongations by G." @default.
- W2021579635 created "2016-06-24" @default.
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- W2021579635 date "1984-01-01" @default.
- W2021579635 modified "2023-10-17" @default.
- W2021579635 title "Totally Zippin $p$-groups" @default.
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- W2021579635 doi "https://doi.org/10.1090/s0002-9939-1984-0735555-1" @default.
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