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Isomorphism of commutative group algebras of closed $p$-groups and $p$-local algebraically compact groups


Author: Peter Danchev
Journal: Proc. Amer. Math. Soc. 130 (2002), 1937-1941
MSC (2000): Primary 20C07; Secondary 20K10, 20K20, 20K21
DOI: https://doi.org/10.1090/S0002-9939-02-06300-1
Published electronically: February 12, 2002
MathSciNet review: 1896025
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Abstract: Let $G$ be an abelian group and let $K$ be a field of $\mathrm{char} K=p>0$. It is shown via a universal algorithm that if the modified Direct-Factor Problem holds, then the $K$-isomorphism $KH\cong KG$ for some group $H$ yields $H\cong G$ provided $G$ is a closed $p$-group or a $p$-local algebraically compact group. In particular, this is the case when $G$ is closed $p$-primary of arbitrary power, or $G$ is $p$-local algebraically compact with cardinality at most $\aleph_1$ and $K$ is in cardinality not exceeding $\aleph_1$. The last claim completely settles a question raised by W. May in Proc. Amer. Math. Soc. (1979) and partially extends our results published in Rend. Sem. Mat. Univ. Padova (1999) and Southeast Asian Bull. Math. (2001).


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Additional Information

Peter Danchev
Affiliation: Department of Mathematics, Plovdiv State University, 4000 Plovdiv, Bulgaria – and – Insurance Supervision Directorate, Ministry of Finance, 1000 Sofia, Bulgaria
Address at time of publication: 13 General Kutuzov Street, bl. 7, floor 2, flat 4, 4003 Plovdiv, Bulgaria
Email: peter_v@bulstrad.bg, library@math.bas.bg

DOI: https://doi.org/10.1090/S0002-9939-02-06300-1
Keywords: Group algebras, isomorphism, closed $p$-groups, $p$-local algebraically compact groups
Received by editor(s): May 19, 2000
Received by editor(s) in revised form: February 5, 2001
Published electronically: February 12, 2002
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2002 American Mathematical Society

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