First week in PhD

I just officially started my PhD last week, but I already started 3 weeks ago. Here are some of the topics and cool facts I have been reading:

Frobenius Algebra (strongly separable condition, knowledgeable Frob. algebra)

Sheaf: not much apart from allowing me to make some jokes.

Geometric group theory

• Fix k. Let G be a finitely-generated group. There are finitely many subgroups index k.
• Every subgroup of finite index of a finitely-generated group is finitely generated.
• Let G be a free group of finite rank. Then a normal subgroup of G is of finite index if and only if it is finitely generated.
• Table-Tennis lemma: more jokes for me.
• Kurosh's theorem

Reidemeister-Schreier method:  I use this to prove the first point

1. Fix k. Let G be a finitely-generated group. Let H be a subgroup of index k. Then there exists a system of representatives such that the length of each representative is not exceeding k.
2. Let G be a finitely-presented group. Then every finite index subgroup is finitely presented.