Let H e 1234 1324 1432 Find the left cosets of H in S4Solut

Let H = {e, (1234), (13)(24), (1432)}. Find the left cosets of H in S₄.

Solution

ANSWER :-

Let H = {(1),(12)(34),(13)(24),(14)(23)}. Find the left cosets of H in S4. Because |S4| = 12 and |H| = 4, there are exactly 12/4 = 3 distinct cosets of H.

Given H = {e,(12)(34),(13)(24),(14)(23)}

(123) H = {(123)e, (123)(12)(34), (123)(13)(24), (123)(14)(23) }

                = {(123),(134),(243),(142)}

(124) H = { (124)e, (124)(12)(34), (124)(13)(24), (124)(14)(23)}

                = {(124),(143),(132),(234)}

So they are all of them. Indeed, H = (12)(34)H = (13)(24)H = (14)(23)H, (123)H = (134)H = (243)H = (142)H, (124)H = (143)H = (132)H = (234)H.