[Math] Finding Sylow p-subgroups

Question

I am trying some examples of finding Sylow p-subgroups in specific groups and looking for the most efficient way to do so. For example, lets say we need to find Sylow-3 subgroups in $A_4$ and $D_6$, dihedral group of order 12. Using sylow theorems I found there can be only $1,4$ sylow 3-subgroups in group of order 12. But how do we actually find these subgroups? Since these groups are of small order maybe check the orders of each element might work but this is not the best way. Are there any tricks or theorems out there to find the subgroups?

Answer 1

In both $A_4$ and $D_6$, the Sylow 3-subgroups are of order $3$ (the largest power of $3$ that divides the group order $12$), hence cyclic. So finding one such subgroup amounts to finding an element of order $3$ in the group, which you can do by inspection. And once you've got one Sylow 3-subgroup, you can find all the others (if there are others) because they're conjugates of the one you've got (or you can just check for more elements of order $3$).