Mathematics · Group Theory

Let GL 2 (&Complexes;) denote the group of 2 × 2 invertible complex matrices with usual matrix multiplication. For S, T ∈ GL 2 (&Complexes;), ⟨S, T⟩ denotes the subgroup generated by S and T. Let S = ⌈ 0 -1 ⌉ ⌊ 1 0 ⌋ ∈ GL 2 (&Complexes;) and G 1 , G 2 , G 3 be three subgroups of GL 2 (&Complexes;) given by: G 1 = ⟨S, T 1 ⟩, where T 1 = ⌈ i 0 ⌉ ⌊ 0 i ⌋ , G 2 = ⟨S, T 2 ⟩, where T 2 = ⌈ i 0 ⌉ ⌊ 0 -i ⌋ , G 3 = ⟨S, T 3 ⟩, where T 3 = ⌈ 0 1 ⌉ ⌊ 1 0 ⌋ . Let Z(G i ) denote the center of G i for i = 1, 2, 3. Which of the following statements is correct?