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Totally Zippin $ p$-groups


Author: Charles Megibben
Journal: Proc. Amer. Math. Soc. 91 (1984), 15-18
MSC: Primary 20K10; Secondary 20K25, 20K35, 20K40
DOI: https://doi.org/10.1090/S0002-9939-1984-0735555-1
MathSciNet review: 735555
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Abstract: If $ G$ is a $ p$-group of limit length $ \lambda $, then it satisfies the $ \lambda $-Zippin property provided that whenever $ A/{p^\lambda }A \cong G \cong B/{p^\lambda }B$, every isomorphism between $ {p^\lambda }A$ and $ {p^\lambda }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 $ \lambda $-Zippin property. This leads to the existence of a great variety of $ G$'s that are totally Zippin in the sense that $ G/{p^\alpha }G$ satisfies the $ \alpha $-Zippin property for all limit ordinals $ \alpha \leqslant \lambda = {\text{length of }}G$. Hence totally Zippin $ p$-groups need not be $ S$-groups, although those of countable length turn out to be direct sums of countable groups.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0735555-1
Keywords: $ p$-groups, totally projective, $ \lambda $-elongation, totally Zippin, $ S$-groups, d.s.c, $ {C_\lambda }$-groups
Article copyright: © Copyright 1984 American Mathematical Society

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