Thread: quaternions and Diheral group

1. quaternions and Diheral group

Prove that the quaternions $\bold Q$ and the Dihedral group $D_8$ are non-isomorphic groups of order 8.

2. An isomorphism preserves the order of an element. Therefore if, for example, there are strictly more elements of order $2$ in $D_4$ than in $\bold Q,$ then theese groups are not isomorphic.

3. ok well i know that there is only one element in $\bold Q$ of order 2, which is $-I$, and i know it is normal, how do I figure out how many subgroups of order 2 are in $D_8$?

4. The dihedral group of order 8 can be seen as the group of symmetries of the square. There are two diagonal symetries, which are involutions, therefore $D_8$ (with your notation) contains at least 2 elements of order 2; that is sufficient to prove what you want.

$D_8$ has exactly 5 elements of order 2 (two diagonal symmetries, 2 lateral symmetries and one central symmetry) and 2 elements of order 4 (one rotation of $\frac{\pi}{2}$ and its inverse).

5. Originally Posted by clic-clac
There are two diagonal symetries, which are idempotent...
Maybe I am missing something here, but the only idempotent in a group is the identity - an idempotent is an element $a$ such that $a^2=a$. In a group we can cancel this to get that $a=1$...

6. You're not missing anything, I wrote idempotent thinking of involution... My bad
Thanks for pointing out the error.