attention-span : one chat line

December 26, 2006 0

Never spend so much time on teaching than this semester and never felt so depressed afterwards. The final test for the first year course on grouptheory (60 hrs. going from nothing to Jordan-Holder and the Sylow theorems) included the following question :

Question : For a subgroup $H \subset G$ define the normalizer to be the subgroup $N_G(H) = \{ g \in G~:~gHg^{-1} = H \}$ . Complete the statement of the result for which the proof is given below.

theorem : Let P be a Sylow subgroup of a finite group G and suppose that H is a subgroup of G which contains the normalizer N_G(P) . Then …

proof : Let $u \in N_G(H)$ . Now, $P \subset N_G(P) \subset H$ whence $uPu^{-1} \subset uHu^{-1} = H$ . Thus, $uPu^{-1}$ , being of the same order as P is also a Sylow subgroup op H. Applying the Sylow theorems to H we infer that there exists an element $h \in H$ such that
$h(uPu^{-1})h^{-1} = P$ . This means that $hu \in N_G(P)$ . Since, by hypotheses, $N_G(P) \subset H$ , it follows that $hu \in H$ . As $h \in H$ it follows that $u \in H$ , finishing the proof.

A majority of the students was unable to do this… Sure, the result was not contained in their course-notes (if it were I\’m certain all of them would be able to give the correct statement as well as the full proof by heart. It makes me wonder how much they understood of the proof of the Sylow-theorems.) They (and others) blame it on the fact that not every triviality is spelled out in my notes or on my \’chaotic\’ teaching-style. I fear the real reason is contained in the post-title…

But, I\’m still lucky to be working with students who are interested in mathematics. I assume it can get a lot worse (but also a lot funnier)

and what about this one :