Paper 3, Section II, D

Groups | Part IA, 2018

State and prove the Direct Product Theorem.

Is the group O(3)O(3) isomorphic to SO(3)×C2?S O(3) \times C_{2} ? Is O(2)O(2) isomorphic to SO(2)×C2S O(2) \times C_{2} ?

Let U(2)U(2) denote the group of all invertible 2×22 \times 2 complex matrices AA with AAˉT=IA \bar{A}^{\mathrm{T}}=I, and let SU(2)S U(2) be the subgroup of U(2)U(2) consisting of those matrices with determinant 1.1 .

Determine the centre of U(2)U(2).

Write down a surjective homomorphism from U(2)U(2) to the group TT of all unit-length complex numbers whose kernel is SU(2)S U(2). Is U(2)U(2) isomorphic to SU(2)×TS U(2) \times T ?

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