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

2it would be useful if you can say what else are you familiar with.... – Jun 30 '14 at 11:13

@Praphulla Mostly courses that involve a significant amount of algebra, such as group theory, group representations, Galois theory, some algebraic number theory. – user50229 Jun 30 '14 at 11:41

1If you prefer a topic that also has an application outside math I might suggest applications of fixedpointfree groups in multiantenna radio communication: B.M. Hochwald et al, *Systematic design of unitary spacetime constellations*, IEEE Trans. Inform. Theory, 46, 1962–1973, September 2000. For the group theoretic part you would need to learn about Zassenhaus' classification about such groups (needs some elementary rep theory also). If you feel like it, you can try and relax some of the constraints of those authors, but alas that well may have dried up researchwise. – Jyrki Lahtonen Jun 30 '14 at 12:44

1This is the sort of question you ought to ask whoever you would like to supervise said thesis. – Tobias Kildetoft Jul 01 '14 at 11:52
3 Answers
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 LeedhamGreen and Newman to classify finite $p$groups)
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If you're into programming, try implementing some algorithms in computational group theory.
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See also http://math.stackexchange.com/questions/416021/whatiscomputationalgrouptheory. – lhf Jun 30 '14 at 12:19