Visual Group Thoery, Lecture 5.5: p-groups

Visual Group Thoery, Lecture 5.5: p-groups

Before we can introduce the Sylow theorems, we need to develop some theory about groups of prime power order, which we call p-groups. In this lecture, we show that the number of fixed point of a p-group acting on a set S is congruent modulo p to |S|. Next, we state and prove the two-part "normalizer lemma". The first part says that for any p-group H, the index [N(H):H] is congruent to [G:H] modulo p. The second part says that if H is not a maximal p-group, then [N(H):H] is a multiple of p.

[sahhaf]

Thank you very much Prof Macauley, this was extremely helpful.

[BSplitt]

Dr. MaCauley, these videos are fantastic. Thank you so much.

[riccardoventrella]

I appreciated how you distinguished from the book uisng a right action instead of a left one, as done in the book. This of course works because the final goal is to reach the Normalizer, which makes right and left cosets equal in there.
Also your demo is by far more comprehensible than book one, which is somewhat too visual (even if I bought it exactly for that) talking about "unanimous arrows", which I honestly missed a bit (I'm referring to Fig. 9.10, page 207). Thank for your version of this. Really useful.

[Nico-vjjc]

There is a mistake at minute 11:58. The normalizer of H in G is the subgroup N_G(H) = {g in G | gHg^–1 = H }. In general xHx^-1 subset H does not imply xHx^-1 = H, there are examples of infinite subgroups H where this is false. It works in this case only because H is finite.


This means N_G(H) not equal to {g in G | gHg^–1 subset H } in general, even though they are equal for finite subgroups H.

[addemfrench]

Where you say "Cayley's Theorem" it should be "Cauchy's Theorem".

[dogbiscuituk]

Typo in the name of this lecture ("Thoery" should be "Theory")