Relation With Alternating Group
For n≥5, the alternating group An is simple, and the induced quotient is the sign map: An → Sn → S2 which is split by taking a transposition of two elements. Thus Sn is the semidirect product An ⋊ S2, and has no other proper normal subgroups, as they would intersect An in either the identity (and thus themselves be the identity or a 2-element group, which is not normal), or in An (and thus themselves be An or Sn).
Sn acts on its subgroup An by conjugation, and for n ≠ 6, Sn is the full automorphism group of An: Aut(An) ≅ Sn. Conjugation by even elements are inner automorphisms of An while the outer automorphism of An of order 2 corresponds to conjugation by an odd element. For n = 6, there is an exceptional outer automorphism of An so Sn is not the full automorphism group of An.
Conversely, for n ≠ 6, Sn has no outer automorphisms, and for n ≠ 2 it has no center, so for n ≠ 2, 6 it is a complete group, as discussed in automorphism group, below.
For n ≥ 5, Sn is an almost simple group, as it lies between the simple group An and its group of automorphisms.
Read more about this topic: Symmetric Group
Famous quotes containing the words relation with, relation and/or group:
“[Man’s] life consists in a relation with all things: stone, earth, trees, flowers, water, insects, fishes, birds, creatures, sun, rainbow, children, women, other men. But his greatest and final relation is with the sun.”
—D.H. (David Herbert)
“Parents ought, through their own behavior and the values by which they live, to provide direction for their children. But they need to rid themselves of the idea that there are surefire methods which, when well applied, will produce certain predictable results. Whatever we do with and for our children ought to flow from our understanding of and our feelings for the particular situation and the relation we wish to exist between us and our child.”
—Bruno Bettelheim (20th century)
“Laughing at someone else is an excellent way of learning how to laugh at oneself; and questioning what seem to be the absurd beliefs of another group is a good way of recognizing the potential absurdity of many of one’s own cherished beliefs.”
—Gore Vidal (b. 1925)