Abstract Algebra: Theory and Applications · Problem 15.8.4
Rings — Thomas Judson
Mathematics — Abstract Algebra · sylow-theorems
Problem
This exercise verifies Lemma 15.1.6. Let G be the dihedral group of order 36, D18. Let H be the one Sylow 3-subgroup. Let K be the subgroup of order 6 generated by the two permutations a and b given below. First, form a list of the distinct conjugates of K by the elements of H, and determine the number of subgroups in this list. Compare this with the index given in the statement of the lemma, employing a single (long) statement making use of the .order(), .normalizer() and .intersection() methods with G, H and K, only.
\begin{verbatim}
G = DihedralGroup(18)
a = G("(1,7,13)(2,8,14)(3,9,15)(4,10,16)(5,11,17)(6,12,18)")
b = G("(1,5)(2,4)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)")
\end{verbatim}