Group Theory - 2018-07-30

That every finite group is the group of symmetries of a finite undirected graph. I don't know the proof and I suppose it's more a graph theory result but, still, it's very pretty.

ÍgjøgnumMeg

Ooooh I have a cool one

MatheinBoulomenos

@Shaun that's really cool

ÍgjøgnumMeg

I don't know the proof but you'll like it I think

and in fact it's related to the place I'll do my masters!

erm

Every finite group of $p$-power order is isomorphic to a closed subgroup of the Nottingham group

or something like this

I learned about it in the summer school I attended last year

MatheinBoulomenos

what is the Nottingham group?

ÍgjøgnumMeg

It's a group of formal power series with coefficients in $\Bbb F_p$

Nottingham group

In the mathematical field of infinite group theory, the Nottingham group is the group J(Fp) or N(Fp) consisting of formal power series t + a2t2+... with coefficients in Fp. The group multiplication is given by formal composition also called substitution. That is, if f = t + ∑ n = 2 ∞ a n t n ...

and there is a related group called the Fesenko group lol

or it's a subgroup

idk

MatheinBoulomenos

10:02 PM

wow, really nice

ÍgjøgnumMeg

Right, there was a lady who gave a seminar talk about it because she was one of the people who worked on this result

very interesting

MatheinBoulomenos

what I think is cool and also kind of funny about profinite groups is that you can define the index and the order of a profinite group using supernatural numbers and you can use that to transfer results that care for things being coprime etc.

for example the order of $\Bbb Z/3\Bbb Z \times \Bbb Z_2$ is $3 \cdot 2^\infty$ and that's "coprime" to $5^\infty$ which is the order of $\Bbb Z_5$

Shaun

Here's a link to the Fesenko group.

ÍgjøgnumMeg

@Mathein pretty cool

Shaun

Ditto.

ÍgjøgnumMeg

10:09 PM

unfortunately during the colloquium talk on the Nottingham group I didn't know what a profinite group was

so I didn't get much out of it

lol

MatheinBoulomenos

one cool result on profinite groups is that in a topologically finitely generated profinite group, every finite index subgroup is open

this means that the topology on a topologically finitely generated profinite group is completely determined by the algebra

interestingly, the proof uses the classification of finite simple groups

ÍgjøgnumMeg

Ah yes, now just to read the proof of the classification of finite simple groups

So there is a difference between topological generators and "regular" generators?

MatheinBoulomenos

yes

topological generators just mean that they generate a dense subgroup

ÍgjøgnumMeg

I see

makes sense; subset generates whole group and the closure is the whole group

Daminark

10:30 PM

Sylow theorems are currently my favorite, they kinda got me into group theory

Shaun

10:49 PM

I haven't so much as looked as Sylow's theorems since I did the groups, rings, and fields module at The University of York about four years ago. They're beautiful though.

Transcript for

$Mathematics$ Group Theory