Dinner and math.

**EDIT**: Here’s an interesting sidenote. If you try to generalize your approach to PMATH 346 A5P4 (e.g., try to count how many distinct isomorphism classes of finite abelian groups of order there are — for arbitrary ) then you will find it is related to the number of integer partitions of the exponents of the primes appearing in the prime factorization of .

Some random problems I thought up:

What is the automorphism group of a dihedral group? (So meta…)

Is there a group such that the sequence defined by , and for all never becomes constant?

Does there exist an infinite group , a non-trivial inner automorphism (note immediately that this means must be non-abelian if it exists), and proper infinite subgroups and so that but ?

Are there two non-isomorphic groups and with injective homomorphisms and ?

(Feel free to submit responses).

campus bubble! 😛

I was pretty cynical of its ability to compare to Sweet Dreams… but I think I’m pretty satisfied after trying it (so begins the addiction) XD

Re: 3rd question. Take three nonabelian groups , , and look at their direct product . pick some (not equal to ) and set to be conjugation by . set , . which is a proper subset of .

Re: 4th question (this time for real)

joint soln with chenglong

let so countably many copies of , and let so countably many copies of with a stuck to the front. Clearly embeds in , but also embeds in by shifting everything over to the right. now we claim that there is no isomorphism from to . indeed, if there was such a map we just look at what it does to , say it gets mapped to . then the map is onto so there is something that gets mapped to which can’t come from because there’s no in . but now if maps to this thing then maps to the same thing as which violates injectivity.

Thanks! That’s insightful.

what does the arrow with the curved left end mean? Does it just mean the homomorphism is injective?

yeah, is usually used for “monic”, for “epi”