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Fuchs' problem for 2-groups | | Eric Swartz
; Nicholas J. Werner
; | | Date: |
16 Apr 2019 | | Abstract: | Nearly $60$ years ago, L’{a}szl’{o} Fuchs posed the problem of determining
which groups can be realized as the group of units of a commutative ring. To
date, the question remains open, although significant progress has been made.
Along this line, one could also ask the more general question as to which
finite groups can be realized as the group of units of a finite ring. In this
paper, we consider the question of which $2$-groups are realizable as unit
groups of finite rings, a necessary step toward determining which nilpotent
groups are realizable. We prove that all $2$-groups of exponent $4$ are
realizable in characteristic $2$. Moreover, while some groups of exponent
greater than $4$ are realizable as unit groups of rings, we prove that any
$2$-group with a self-centralizing element of order $8$ or greater is never
realizable in characteristic $2^m$, and consequently any indecomposable,
nonabelian group with a self-centralizing element of order $8$ or greater
cannot be the group of units of a finite ring. | | Source: | arXiv, 1904.7901 | | Services: | Forum | Review | PDF | Favorites | |
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