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Groups of matrices not isomorphical

Source: Romanian District Olympiad 2006, Grade 12, Problem 2

March 11, 2006

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Problem Statement

Let

G= \{ A \in \mathcal M_2 \left( \mathbb C \right) \mid |\det A| = 1 \}

and

H =\{A \in \mathcal M_2 \left( \mathbb C \right) \mid \det A = 1 \}

. Prove that

G

and

H

together with the operation of matrix multiplication are two non-isomorphical groups.