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**ux0** i) Prove that $\displaystyle Q/Z(Q) \cong V$, where $\displaystyle Q$ is the group of quaternions and $\displaystyle V$ is the four-group. Conclude that the quotient of a nonabelian group by its center can be abelian.

ii) Prove that $\displaystyle Q$ has no subgroup isomorphic to $\displaystyle V$. Conclude that the quotient $\displaystyle Q/Z(Q)$ is not isomorphic to a subgroup of $\displaystyle Q$.

For part two, is it enough to say

An isomorphism preserves the order of all the elements. Therefore $\displaystyle Q$ has only one element of order 2, which means every subgroup of $\displaystyle Q$ will have a maximum of 1 element of order 2, which is $\displaystyle A^2$ also known as $\displaystyle -I,$ because $\displaystyle V$ consist of 1 element of order 1,(the identity) and 3 elements of order 2, it is impossible to make a subgroup of $\displaystyle Q$ with 3 elements of order 2. Therefore $\displaystyle Q$ has no subgroup isomorphic to $\displaystyle V$.

If that is true, then it follows that $\displaystyle Q/Z(Q)$ is not isomorphic to a subgroup of $\displaystyle Q $ because, $\displaystyle Q/Z(Q) \cong V$, and we know that no subgroup of $\displaystyle Q$ is isomorphic to $\displaystyle V$, then it follows $\displaystyle Q/Z(G)$ is not isomorphic to a subgroup of $\displaystyle Q$.

*Side Note: I understand if this is the worst mathematical proof one has ever witness.*