Never mind, I think I managed to screw even this up. arrgh^(arrgh^arrgh) by forgetting that a prime's power can occur more than once. (The link is still valid, though.) —Dan
On Apr 24, 2016, at 10:03 PM, Dan Asimov <asimov@msri.org> wrote:
Inspired by that, here's an absurdly bold conjecture with remarkably little evidence: *Every* finite abelian group is isomorphic to some multiplicative group mod N. Someone prove me wrong.
Interesting suggestion.
After I get my program to find all such groups through at least N=1000 working, I'll check and see if it appears to be true. Of course whatever I find will constitute neither a proof or a disproof, but it may inspire insight. In support of that, is there any online database of small abstract abelian groups? I see little point in reinventing that wheel. Thanks.
There's a simple decomposition theorem for finite abelian groups.
Consider any choice
E: {2, 3, 5, 7,...} —> {0, 1, 2, 3, ...}
of a nonnegative integer e(p) for each prime p, *such that*
e(p) > 0
*for only finitely many primes* p.
Then such e are in one-to-one correspondence with finite abelian groups, up to group isomorphism.
Namely,
E <—> Z/2^e(2) + Z/3^e(3) + Z/5^e(5) + Z/7^e(7) + ...
where + denotes direct sum.
For more see
https://en.wikipedia.org/wiki/Finitely_generated_abelian_group#Classificatio...
(and ignore the Z^k direct summands, since that article more generally addresses the classification of finitely *generated* abelian groups, which is only infinitesimally more complicated).
—Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun