TITLE: Given a group $G$, can we determine the $n$ for which $G$ is a subgroup of $S_{n}$?
QUESTION [9 upvotes]: I was watching singingbananas video Introduction to Group Theory, and near the end, he says "all other groups can be found contained in some symmetric group", and I wondered "is it possible to find the $n$ for which, say, the monster $M$ is a subgroup of $S_{n}$? (or any other group)

REPLY [5 votes]: You might want to read http://www.math.uic.edu/~rtakloo/papers/group-paper/delta.12.pdf. There has been a lot of research on minimal S_n embeddings.