I am aware this has come up recently (Embedding of finite groups for example) but after searching I haven't found the particular answer I'm looking for. Suppose I know the character table and can figure out the minimal (degree) faithful representation of group G (such as a subgroup of a Frobenius group): can I then find/bound accurately the smallest m such that our group G embeds into S$_m$? It is merely my idle curiosity as to whether it is possible, but I would like to be able to give as accurate a bound on n as possible, or indeed find it. www.austms.org.au/Gazette/2008/Nov08/TechPaperSaunders.pdf seems to suggest in the first few lines that it simply is the degree of the min. faithful representation, but is it really that simple? Could anyone suggest anything? Thank you very much – Tom H
EDIT: Just to let everyone know I spoke further with a colleague and after reading a few books I circumvented the issue I was having, so problem solved: I now consider this question closed, in case it needs marking as such.
Best Answer
The paper you mention does not claim that the minimal degree of a faithful representation is your $n$, it says "of a faithful permutation representation." It is easy to see that the other statement would be too strong: $S_3$ has an irreducible faithful 2-dimensional representation, but you certainly cannot embed $S_3$ into $S_2$. Similarly, you cannot embed a cyclic group into $S_1$.
Your question has been asked on Mathoverflow, and as you can see from that discussion, it is a fairly delicate issue. You will not find a simple relationship between the minimal degree of a faithful representation and the smallest $n$, as the example of cyclic groups shows (the former is always 1, the latter can be arbitrarily large).