Problem 1

Consider the Klein four-group $\mathrm{V}$ with presentation

Recall that $\mathrm{V}\cong(\mathbb{Z}_{2})^{2}$.

1. Find the unique composition series for $\mathrm{V}$ up to isomorphism.

2. Let $\mathrm{Q}$ denote the quaternion group. Find the two distinct composition series for $\mathrm{Q}$ up to isomorphism.(Hint: consider the alternate definition of the quaternion group.)

3. Let $H$ be the subgroup of $\mathrm{Q}$ that is isomorphic to $\mathrm{C}_{4}$, and $K$ be a subgroup of $\mathrm{Q}$ that is isomorphic to $\mathrm{V}$. Prove that

   $$\mathrm{Q}/H\cong\mathrm{Q}/K.$$

   (Hint: consider 1.4.)