1. **Problem Statement:** Determine if the set of vectors of the form $nx + my$ where $n, m$ are integers and $x, y$ are vectors, forms a subgroup of the group of translations. 2. **Recall the definition of a subgroup:** A subset $H$ of a group $G$ is a subgroup if: - It contains the identity element of $G$. - It is closed under the group operation. - It is closed under taking inverses. 3. **Group of translations:** The group of translations in a vector space is $(\mathbb{R}^d, +)$, where the operation is vector addition. 4. **Check identity:** The zero vector $0$ is in the set since for $n=m=0$, $nx + my = 0$. 5. **Check closure under addition:** Take two elements $a = n_1x + m_1y$ and $b = n_2x + m_2y$ with $n_i, m_i \in \mathbb{Z}$. Then $$a + b = (n_1 + n_2)x + (m_1 + m_2)y$$ Since $n_1 + n_2$ and $m_1 + m_2$ are integers, $a + b$ is in the set. 6. **Check closure under inverses:** For $a = nx + my$, its inverse is $$-a = -nx - my = (-n)x + (-m)y$$ Since $-n, -m$ are integers, $-a$ is in the set. 7. **Conclusion:** The set $\{nx + my \mid n,m \in \mathbb{Z}\}$ is a subgroup of the group of translations. **Final answer:** Yes, the set of all integer linear combinations of vectors $x$ and $y$ forms a subgroup of the group of translations.