Lecture 21, MTH 204A (Section B)

We talk about the automorphism groups of symmetric groups. First we see that, for any positive integer n not less than 3, if an automorphism of S_n takes transpositions to transpositions then it has to be inner. Next we show that, when n is not equal to 6, then every conjugacy class in S_n that consists of exactly nC2 elements of order 2 must contain a transposition. From this it follows that, every automorphism of S_n is inner, provided n is not equal to 6. When n=6, we see that the factor group Aut(S_n)/Inn (S_n) has order 2. Next we discuss about the simplicity of PSL_2 over any field having at least four elements. This is done in two steps, which are as follows: 1. The intersection of all conjugates of the standard Borel subgroup of GL_2 or SL_2 is the centre. 2. For SL_2, every normal subgroup is either contained in the centre or contains the commutator subgroup of SL_2, which is SL_2 itself.