Suggestions for research in Group Theory

Question

(This is about a help for not to lose interest from Group Theory. Dear Group Theorist or Algebraist please help; if question is not clear, give suggestions.)

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(1) Few days before, I came across a review of a book on $p$-groups by an expert in p-groups (C. R. Leedham-Green), some part of which is as below:

....The authors suggest no fewer than 1400 research problem......Take at random Problem 1200: Study the p-groups whose cyclic subgroups are characteristic in their centralisers. There is no objection to asking a rather imprecise question (“Study. . . ”), except that it could rise to a number of papers, but there is an objection to studying some oddly defined class of groups without knowing why. ......

Today, I was looking so many papers on the Research Topic $$\mbox{study of Frobenius groups $N\rtimes H$ acting on other group $G$ via automorphisms},$$ Concerning above review-comment, the first question came to mind was why to study such groups? I didn't find a good reason for their study in the papers. Introduction in many papers says (almost same statement):

many properties of $G$ are related with that of fixed points of $H$ in $G$.

I didn't find the reason interesting. Is there other motivation for study of such Frobenius actions?

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(2) After mental preparation that ''let's see these papers, without philosophical reason'', I went for reading the papers. But, I faced lot of problems in Symbols. It was not said in paper, what the symbol $G^{\mathfrak{A}(p-1)}$ denotes. But in online search, I found two different meanings of this:

• abelian radical (Subgroup Lattices of Groups, Volume 14 By Roland Schmidt)

• abelian residual (Products of Finite Groups by Ballester-Bolinches, ...)

And this pulled out my mind from the Research Topic!

What is a reasonable good way of research in Group Theory?

Answer 1

• 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.

Answer 2

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.