## SU(5) coupling question

I was reading a gauge field theory book and was told to refer to Quigg's Gauge Theories of the Strong, Weak, and Electromagnetic Interactions ch 9.2 The SU(5) Model, so I did not start from the first page of Quigg's book.

In page 277, he started from

Q = T3 + K To,
where T3 is a generator of SU(2) and To is a weak-isosinglet generator of SU(5).

Σ Q^2 = (1+K^2) Σ T3^2

working on 5* representation and get

K^2 = 5/3

So To differs by a factor of (3/5)^(1/2) from U(1) hypercharge operator Y, and

g' ^2 = (3/5) (g_SU(2))^2

My questions are,

1. Why is it that from K^2 = 5/3 we can get g' ^2 = (3/5) (g_SU(2))^2 ?

2. Since Q = I3 + (1/2)Y, why is it that To differs by a factor of (3/5)^(1/2) but not (1/2)*(3/5)^(1/2) from Y?

3. Is there any beginner friendly reference for the derivation of Σ Q^2 = (1+K^2) Σ T3^2 ?

Thanks!
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