
Isomorphic?
How do I show whether the pair of groups $\displaystyle S_{5},D_{60}$ is isomorphic?
I know that the order of the two groups is 120, they are both noncommutative, they are both noncyclic. But I believe they are not isomorphic since I can't really find a function that does the trick.
Please help, thanks!
K

The only element $\displaystyle \sigma \in S_n$ with $\displaystyle n\geq 3$ that has a property that $\displaystyle \sigma \tau = \tau \sigma$ i.e. it commutes with everything, is $\displaystyle \sigma = i$, i.e. the identity element.*
Now is that true for $\displaystyle D_n$, i.e. it has a trivial center? I think not. So they are nonisomorphic.
*)If you every learned the meaning of "center" we will say the center of $\displaystyle S_n$ is trivial for $\displaystyle n\geq 3$.