John Stillwell - Naive Lie Theory.pdf - Index of

3.5 Maximal tori and centers 61
because in that case A is an isometry of the (e 1 ,e 2 )-plane that fixes O. The
only such isometries are rotations and reflections, and the only ones that
commute with all rotations are rotations themselves.
So, suppose that
A(e 1 )=a 1 e 1 + a 2 e 2 + a 3 e 3 .
By the hypothesis (*), A commutes with all R ′ θ
, and in particular with
⎛ ⎞
−1 0 0
R ′ π = ⎝ 0 −1 0⎠.
0 0 1
Now we have
AR ′ π (e 1)=A(−e 1 )=−a 1 e 1 − a 2 e 2 − a 3 e 3 ,
R ′ π A(e 1)=R ′ π (a 1e 1 + a 2 e 2 + a 3 e 3 )=−a 1 e 1 − a 2 e 2 + a 3 e 3 ,
so it follows from AR ′ π = R′ π A that a 3 = 0 and hence
A similar argument shows that
A(e 1 ) ∈ (e 1 ,e 2 )-plane.
A(e 2 ) ∈ (e 1 ,e 2 )-plane,
which completes the proof that T 1 is maximal in SO(3).
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An important substructure of G revealed by the maximal torus is the
center of G, a subgroup defined by
Z(G)={A ∈ G : AB = BA for all B ∈ G}.
(The letter Z stands for “Zentrum,” the German word for “center.”) It is
easy to check that Z(G) is closed under products and inverses, and hence
Z(G) is a group. We can illustrate how the maximal torus reveals the center
with the example of SO(3) again.
Center of SO(3)
An element A ∈ Z(SO(3)) commutes with all elements of SO(3), andin
particular with all elements of the maximal torus T 1 . The argument above
then shows that A fixes the basis vector e 3 . Interchanging basis vectors, we
likewise find that A fixes e 1 and e 2 . Hence A is the identity rotation 1.
Thus Z(SO(3)) = {1}.
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