I would like some ideas on topics in group theory which would be suitable for a masters' thesis. What sort of problems would be suitable for this level? Because it is at masters' level, no original work is necessarily needed but it can instead be an original comprehensive survey of some area. Thank you
I once wrote a detailed answer of some topics in geometric and combinatorial group theory which would be suitable for a talk or a master's thesis. This is the branch of group theory which deals with (loosely) actions of groups, presentations, a bit of algebraic geometry, etc. The post can be found here. Special mention goes to Dehn's problems (decision problems in groups) and to Burnside's problem (does there exist an infinite, finitely generated group where every element has finite order?).
I should also point out that you should speak to your (potential) supervisor about this. When I wrote my masters thesis I spend the first 6 months meeting with my supervisor to discuss possible topics, and the second 6 months writing...
I have no idea what constitutes the appropriate level, but here some interesting topics:
- The Monster group (its construction, history, representations and moonshine, ...)
- Higman's PORC conjectures (results on the number of groups of order $p^n$ for $n\le7$, the counting techniques involved, reasons for current suspicion of likely falsehood, ...)
- Coclass theory (started by Leedham-Green and Newman to classify finite $p$-groups)
If you're into programming, try implementing some algorithms in computational group theory.