[FOM] New Proof of Fundamental Theorem of Arithmetic
Vaughan Pratt
pratt at cs.stanford.edu
Fri Sep 18 08:04:52 EDT 2009
Vaughan Pratt wrote:
> all paths in the modular
> lattice of normal subgroups of a finite group are isomorphic up to
> permutation. For abelian groups this lattice is distributive, but
Replace "abelian" by "cyclic" (the kind I was dealing with). Thanks to
Andreas Blass for drawing my attention to the Klein 4-group V4 as a
counterexample, sorry about that.
In general the lattice of subgroups of a group G is distributive if and
only if G is locally cyclic, meaning that any finite subset generates a
cyclic group (due to my great-grandadvisor Oystein Ore in 1938). V4
generates itself. The only locally cyclic groups are those isomorphic
to a subgroup of either the group Q of rationals or the rational circle
group Q/Z (the n-th roots of unity).
Vaughan Pratt
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