(a) Show that any rank 2 tensor $t_{i j}$ can be written uniquely as a sum of two rank 2 tensors $s_{i j}$ and $a_{i j}$ where $s_{i j}$ is symmetric and $a_{i j}$ is antisymmetric.

(b) Assume that the rank 2 tensor $t_{i j}$ is invariant under any rotation about the $z$-axis, as well as under a rotation of angle $\pi$ about any axis in the $(x, y)$-plane through the origin.

(i) Show that there exist $\alpha, \beta \in \mathbb{R}$ such that $t_{i j}$ can be written as

$$t_{i j}=\alpha \delta_{i j}+\beta \delta_{i 3} \delta_{j 3} .$$

(ii) Is there some proper subgroup of the rotations specified above for which the result $(*)$ still holds if the invariance of $t_{i j}$ is restricted to this subgroup? If so, specify the smallest such subgroup.

(c) The array of numbers $d_{i j k}$ is such that $d_{i j k} s_{i j}$ is a vector for any symmetric matrix $s_{i j}$.

(i) By writing $d_{i j k}$ as a sum of $d_{i j k}^{s}$ and $d_{i j k}^{a}$ with $d_{i j k}^{s}=d_{j i k}^{s}$ and $d_{i j k}^{a}=-d_{j i k}^{a}$, show that $d_{i j k}^{s}$ is a rank 3 tensor. [You may assume without proof the Quotient Theorem for tensors.]

(ii) Does $d_{i j k}^{a}$ necessarily have to be a tensor? Justify your answer.