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For all finite groups $G$, define $S(G)$ to be the smallest $n\in\mathbb{Z}^+$ such that there exists an $H\leq S_n$ isomorphic to $G$ — i.e., $S(G)$ is the index of the first symmetric group in which $G$ can be embedded. Is there a standard (or at least somewhat common) name or notation for $S(G)$ or a similar concept?

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«Minimal degree of a faithful permutation representation» – Mariano Suárez-Alvarez May 23 '12 at 19:53
@MarianoSuárez-Alvarez: Why not make that an answer? – jwodder May 23 '12 at 20:14
up vote 5 down vote accepted

As Mariano mentioned, this is the minimal permutation representation. It is studied in the paper

David L. Johnson: Minimal permutation representations of finite groups. In: American Journal of Mathematics. 93, 1971, S. 857–866.

It is possible to classify the groups for which the regular representation (in Cayley's Theorem) is already minimal: These are the Klein Four Group, cyclic groups of prime power order and generalized Quaternion groups.

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