using lagrange's thm, subgroups of Q8 is of order 1, 2, 4 and 8.. get all subgroups then..

a) for ( ) is cyclic, then and . Let such that . then for some natural number . thus you can continue from here..

on the other direction, if , then for some integer . thus ...

i think, is non-abelian..

is non-abelian