[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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