How can I show that has a subgroup isomorphic to ? Can someone help please?
I wonder who set you this question, because it would seem impossible - indeed it can be shown that no subgroup of is isomorphic to
Note that , eight rotations and eight reflections, and that By Lagrange's theorem, if is a subgroup of then and it is very clear that isomorphic groups have the same number of elements. But is not divisible by , so no subgroup of is isomorphic to
Maybe it was a trick question!
Maybe he means : some authors write for because that shows the order of the group, while others find the notation more appropriate because of its geometric meaning (and would argue that you write and not !).
In this case it is true, the -sylow subgroups of are isomorphic to (or if its order is ).
Think of the symmetries of a square whose vertices are nammed , take a close look at the subgroup generated by for instance.