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.*