what are the possible orders of a subgroup of S4 explainSolu

what are the possible orders of a subgroup of S4? explain

Solution

the conjugacy classes are given by the “shapes” of the disjoint cycle decomposition of the elements. In the case of S₄ the conjugacy classes are as follows:

e (· ·) (· ·)(· ·) (· · ·) (· · · ·)

e (12) (12)(34) (123) (1234)

(13) (13)(24) (132) (1342)

(14) (14)(23) (124) (1423)

(23) (142) (1243)

(24) (134) (1432)

(34) (143) (1324)

(234)

(243)

The subgroups of S4 are the following:

n subgroups of S4 of order n =type #

1 {e} C₁ 1

2 {e,(12)}, {e,(13)}, {e,(14)}, {e,(23)}, {e,(24)}, {e,(34)} C₂ 6

{e,(12)(34)}, {e,(13)(24)}, {e,(14)(23)} C₂ 3

3. {e,(123),(132)}, {e,(124),(142)}, {e,(134),(143)}, {e,(234),(243)} C₃ 4

4 {e,(12),(34),(12)(34)}, {e,(13),(24),(13)(24)}, {e,(14),(23),(14)(23)} C₂ × C₂ 3

{e,(12)(34),(13)(24),(14)(23)} C₂ × C₂ 1

{e,(1324),(12)(34),(1423)}, {e,(1234),(13)(24),(1432)}, C₄   3

{e,(1243),(14)(23),(1342)}

6. {e,(123),(132),(12),(13),(23)}, {e,(124),(142),(12),(14),(24)}

, {e,(134),(143),(13),(14),(34)}, {e,(234),(243),(23),(24),(34)} S₃ 4

8 {e,(12),(34),(12)(34),(13)(24),(14)(23),(1324),(1423)},

{e,(13),(24),(13)(24),(12)(34),(14)(23),(1234),(1432)},

{e,(14),(23),(14)(23),(12)(34),(13)(24),(1243),(1342)} D₄ 3

12 {e,(12)(34),(13)(24),(14)(23),(123),(132),(124),(142),(134), A₄ 1

(143), (234),(243)}

24 S₄ S₄ 1

Each row of the table contains a conjugacy class of subgroups. The last column lists the number of subgroups in that conjugacy class. The second to the last column lists the isomorphism type, where C_k denotes the cyclic group of order k. In all we see that there are 30 different subgroups of S4 divided into 11 conjugacy classes and 9 isomorphism types.