Abstract Algebra: Theory and Applications · Problem 15.4.2

Rings — Thomas Judson

Mathematics — Abstract Algebra · sylow-theorems

Problem

Find all the Sylow 3-subgroups of

S_4

and show that they are all conjugate. Hint. The four Sylow 3-subgroups of

S_4

are

P_1 = \{(1),(1\,2\,3),(1\,3\,2)\}

P_2 = \{(1),(1\,2\,4),(1\,4\,2)\}

P_3 = \{(1),(1\,3\,4),(1\,4\,3)\}

P_4 = \{(1),(2\,3\,4),(2\,4\,3)\}