Abstract
In this paper, we provide a concrete and explicit decomposition of the quaternion group algebra through a suitable basis of the algebra. Thanks to this basis, the idempotent elements and the units of the algebra as well as the matrix representations of the group algebra are determined thereafter. The description of the quaternion group algebra through this basis then gives rise to a one-to-one correspondence between the representations of the quaternion group and the representations of the quaternion algebra.
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