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?
Name this property: First symmetric group in which a given group appears
1
$\begingroup$
abstract-algebra
group-theory
notation
terminology
-
2@MarianoSuárez-Alvarez: Why not make that an answer? – 2012-05-23
1 Answers
5
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.