In GL₂(ℂ), let S=pmatrix0&-1\\1&0pmatrix, T₁=iI,T₂=diag(i,−i), T 3=pmatrix0&1\\1&0pmatrix and G j=⟨S,T j⟩. Write Z(G) for the center. Which holds?

Z(G₁) is isomorphic to Z(G₂).

GATE 2023 MA – Question 36

Algebra · Groups · 2 marks · Multiple choice

In GL₂(ℂ), let $S=\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, T₁=iI,T₂=diag(i,−i), $T_3=\begin{pmatrix}0&1\\1&0\end{pmatrix}$ and G_j=⟨S,T_j⟩. Write Z(G) for the center. Which holds?

G₁ is isomorphic to G₃.
Z(G₁) is isomorphic to Z(G₂).
Z(G₃)={I}.
Z(G₂) is isomorphic to Z(G₃).

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Correct answer: (D) Z(G₂) is isomorphic to Z(G₃).

Explanation

S commutes with iI, so G₁ is abelian of order 8. S and T₂ square to −I and anticommute, giving the quaternion group with center {±I}. S and T₃ give the order-8 dihedral group, also with center {±I}. G₃ is nonabelian and its center is nontrivial. Thus **D**.

GATE 2023 MA – Question 36

Explanation

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