The quaternion group $Q_{8} := \{1,−1, i,−i, j,−j, k,−k \}$ of $\mathbb{H} $ is a group under multiplication.

Question

The quaternion group $Q_{8} := \{1,−1, i,−i, j,−j, k,−k \}$ of $\mathbb{H} $ is a group under multiplication.

Which of the following Wedderburn decomposition of $\mathbb{C} [Q_{8}]$ is correct? Why?Please help me.

$\mathbb{C} [Q_{8}] \cong M_{2}(\mathbb{C}) \times M_{2} ( \mathbb{C})$

$\mathbb{C} [Q_{8}] \cong M_{2}(\mathbb{C}) \times \mathbb{C} \times \mathbb{C} \times \mathbb{C} \times \mathbb{C} $

Answer 1

The Wedderburn decomposition of a group ring $\Bbb C G$ for a finite group $G$ always has a copy of $\Bbb C$ (corresponding to the trivial representation). That knocks out one of your possible structures...