Consider the subgroup of S4 generated by the cycle 1234 Writ

Consider the subgroup of S_4 generated by the cycle (1234). Write down the elements of H = ((1234)) and find all (six) rowels off S_4. Show that cosset multiplication is not well defined in this ease. That is choose two aH and bH and locate two different representatives in each cosset a, a\' Element aH and b, b\' Element bH so that ab and a\' b\' land in different cossets. What can you say about the normality of H? This must mean that there is another subgroup of S_4 which is isomorphic. Use inner automorphisms to find this group.

Solution

(a) The elements of H are given by :

{ (1234), (1243), (1324), (1432), (1342), (1423)}

The left cosets are :

<(1234)>

(12)<(1234)> = {(12)(1234), (12)(1243), (12)(1324), (12)(1432),(12)(1342), (12)(1423)} =

{(234), (243), (13)(24), (143),(134),(142)}

(13)<(1234)> = {(12)(34),(124),(24),(14),(341),(142)}

(23)<(1234)> = {(134), (13)(24), (124),(142),(12)(34),(143)}

(34)<(1234)> = {(124),(123),(14)(23),(132),(142),(13)(24)}

(412)<(1234)> = {(1423),(1432),(1342),(34),(13),(23)}

(421)<(1234)> = {(23),(34),(13),(1243),(1324),(1234)}

These are possible cosets

(b) As H doesn\'t partition all elelments and so H is not normal