Problem 4 15 points Prove that for every n 3 InnSn SnSoluti

Problem 4 (15 points). Prove that for every n 3, Inn(Sn) = Sn.

Solution

solution -:

proof -:

Sn is generated by transpositions. We proceed by induction.
The case where n = 1 or n = 2 is clear. Now suppose that Sn can be generated by
transpositions of the form (1 i). Then each of the transpositions (i j) for i; j n are
contained in the spam of (1 2); : : : ; (1 n+1), so we need only show that we can generate
(i n + 1) for each 1 < i < n + 1. But we have (1 i)(1 1n)(1 i) = (i ^1n)

so for every n 3,

we have Inn(Sn) = Sn. it is true hence proved