i) Prove that , where is the group of quaternions and is the four-group. Conclude that the quotient of a nonabelian group by its center can be abelian.
ii) Prove that has no subgroup isomorphic to . Conclude that the quotient is not isomorphic to a subgroup of .
For part two, is it enough to say
An isomorphism preserves the order of all the elements. Therefore has only one element of order 2, which means every subgroup of will have a maximum of 1 element of order 2, which is also known as because consist of 1 element of order 1,(the identity) and 3 elements of order 2, it is impossible to make a subgroup of with 3 elements of order 2. Therefore has no subgroup isomorphic to .
If that is true, then it follows that is not isomorphic to a subgroup of because, , and we know that no subgroup of is isomorphic to , then it follows is not isomorphic to a subgroup of .
Side Note: I understand if this is the worst mathematical proof one has ever witness.