Suggestions for research in Group Theory

• Research means finding your own problems and finding solutions to them.
• Research does not mean that you will have to find a very hard problem and then start solving it.It arises when you read a particular topic and then find something striking about it.
• It is always better to find problems on your own and may be asking on MSE or somewhere else about the originality of the result.
• You will definitely lose interest about the topic if your focus is only on doing research in that topic.You should concentrate more on enjoying the topic and motivating yourself to learn further step by step .
• Research can start at any elementary level.

Hope this helps.

Let $G$ be a finite group , by Jordan Hölder we know that

$1=G_0\leq G_1 \leq G_2...\leq G_n=G$

Such that $G_i$ is maximal normal in $G_{i+1}$. That means that $G_{i+1}/G_i$ is a simple group.

Thus, we have two main problem for understanding the group $G$ ?

$1)$ What are all finite simple groups ?

$2)$ If we know $G/M$ and $M$, can we know $G$ ? (extension problems)

(These can be seen as main problems in finite group theory)

The fisrt problem is finished at $2004$. (all papers related to this problems is about $10000$ pages).

The second one is not finished yet.(seems to be far away to finish)

The Second problem is very diffucult even $G=M\ltimes H$. In that case: $H$ acts on $M$ by automorphism.

There are many known result if $(|M|,|H|)=1$ called as coprime action. More specificly, Frobenius action. (it is also one of the coprime action.)

All such problems are the part of cases of second question.

Besdie these, Some people study on very specific groups like extra special groups. At first they can be seen as very specific and useless but when you notice that trying to solve many problems by induction force many groups to reduct to some special cases, you see that they are important indeed. Among them, extraspecial groups, frobenius groups, supersolvable groups ... Thus, these are not very "special case", are "general case".

As an example, assume that you want to solve $$x^2-bx+c =0$$

Some people say that I solved this when $b=0$. At first it can be seen that it is very specific case but

$$(x-\dfrac{b}{2})^2+c-\dfrac{b^2}{4}=0$$ set $t=(x-\dfrac{b}{2})^2$

$$t^2-C =0$$

Actually, you solved the problem !

I hope what I mean is clear.