Unitary group

(1)

The (central) factor acts in adjoint representation ("conjugation by ") by

(2)

— it does nothing to , only the part of actually moves .

Proof

(3)
(4)
(5)

Define the product map

(6)

  1. Surjectivity: Given , set

    (7)

    so .

  2. Kernel:

    (8)

    Hence

    (9)

By the First Isomorphism Theorem,

(10)

In words: every unitary matrix factors into an overall phase ( ) and a special–unitary part ( ), but since both share the th‐roots of unity in common one must mod out by that discrete subgroup.