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 :

If you are (like me) in urgent need for a smile, try out

this newsvine article for more

bloopers.

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