Let a 1 2 3 elementof Sym3 Determine all elements of Csym3 a

Let a:= (1, 2, 3) elementof Sym(3). Determine all elements of C_sym(3) (a).

Solution

Solution:-

Recall that S3 = {1,(12),(13),(23),(123),(132)}.
Checking one by one all the
Sub groups generated by a single element we get the following cyclic subgroups:
1 = <e>
{1,(12)} = <(12)>
{1,(13)} = <(13)>
{1,(23)} = <(23)>

{1,(123),(132)} = <(123)> = <(132)>