Thanks Josh. For some reason your comment is not showing up under “recent comments”, I really don’t know why.

And of course, no sooner have I written that, than it is there…

]]>1. I read Poincare’s essay on intuition. This convinced me that it was okay I struggled with certain technicalities, as Poincare made it clear that this has been a problem since antiquity.

2. A certain incident at the beginning of this year.

3. Really thinking hard about graphs, just drawing pictures, pulling out colored pencils, and trying to understand what is happening. It was fruitless at the time, but it has paid off when I was no longer doing pure graph theory.

4. A lot of encouragement from many people, who I am all grateful to. I think they all know I consider them fondly.

Compositionality, for instance, is a journal owned by a registered Charitable Incorporated Organisation in the UK solely dedicated to publishing the journal, and was set up by the people who started the journal (and the overlap with the current editors/steering committee is high)

]]>We use that fact that a finite group is nilpotent if and only if it is the direct product of its Sylow subgroups. With A and B as in your proof, write A = P_1 x … x P_r and B = Q_1 x … x Q_r, where (for each i) P_i and Q_i are Sylow p_i-subgroups of A and B respectively for the same prime p_i (some of these groups may be trivial). As you showed, each P_i and Q_i is normal in G and P_i Q_i is a p_i-group. Also, if i \not= j, then [P_i, Q,j] is a subgroup of P_i \cap Q_j = {1}; so P_i Q_j = P_i x Q_j. Now

AB = (P_1 x … x P_r)(Q_1 x … x Q_r) = P_1 Q_1 x … x P_r Q_r

is a direct product of its Sylow subgroups and we’re done.

]]>I don’t exactly know. This is what they claim. But I guess they will produce a constitution or something similar which will spell it out. Presumably the editorial board will also be the “board of directors”.

]]>I didn’t know that paper. But it was interesting reading Miller’s paper on the Frattini subgroup. He knew, for example, that a group which acts faithfully as a primitive permutation group has trivial Frattini subgroup.

]]>I can never resist a good pun.

Did you know that, if wearing a face mask makes your glasses steam up, you may be entitled to condensation?