Upper half-plane

H is the upper half-plane which contains the set of complex numbers with positive imaginary part.

This is modular group.

ad − bc = 1

This includes point at infinity.

Modular Group

a,b,c,d are integers.

ad-bc=1

This is modular group which is also isomorphic of PSL(2, Z).

a/b, a/c, c/d and b/d are irreducible.

(a,b,c,d)=(7,5,4,3),(4,11,17,47),(4,7,13,23),(2,5,17,43),(6,19,23,73),(8,13,19,31),(48,13,11,3)

You see 3 prime numbers and 1 even number in it.

Τ=3P+N

(P:prime number, N:even number)
2 is prime number, but it is also even number. This has special meaning in my blog.

This is also irreducible .

You see (7,5,4,3).

7p+5q

4p+3q

(p=2, q=3)

14+15=29

8+9=17

29/17=1.705882353

PSL(V)=SL(V)/SZ(V)

PSL is projective linear group, and V is vector space. Moreover, SZ(V) is the center, and SL(V) is factor group.

This is order.

d is the center.