Prove that Aut(S6) is semi-direct product of S6 by Z2.
1. Let ơ E Aut(R), where R denotes the field of real numbers. a) Prove that if a > b then σ(a) > σ(b) ( . (b) Prove that o is a continuous function. (c) Prove that ơ must be the identity function. Therefore Aut(R)-(1). (see problem 7 on pg. 567 for more details for each step).
1. Let ơ E Aut(R), where R denotes the field of real numbers. a) Prove that if a > b then σ(a) >...
Prove the Theorem:
Theorem: A graph G is a Cayley graph if and only if Aut(G) contains a subgroup that acts reg- ularly on G
Theorem: A graph G is a Cayley graph if and only if Aut(G) contains a subgroup that acts reg- ularly on G
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the line segment joining z1 to z2.)
12. Suppose that fis analytic on a convex domain D and that Re(f ,(z)) > 0 for all z E D. Show that f is one-to-one on D. (Hint: /(z2) - sz) J,f'(w) dw, where is the...
detailed solution for this one ?????
11. (a) Gi) If w=z+z-' prove that (i) z2 + z 2 = w2 -2 ; 24 +2° + z²+z+1 = z2 (W2 + w+1) = (z? +[1+V5]+1)(22 +[1–V5]+1). (b) Show that the roots of 24 +2+z2+z+1=0 are the four non-real roots of z' =1. (c) Deduce that cos 72° = +(15 – 1) and cos 36° = (15+1).
Find Aut(Z ) IS Aut (z is cyclic or not? " If not why 2
Prove that while the hyperbolic distance between two points zi-i and z2-ie in the upper half-plane H is k, their Euclidean distance in the unit disk D is tanh(k/2).
Prove that while the hyperbolic distance between two points zi-i and z2-ie in the upper half-plane H is k, their Euclidean distance in the unit disk D is tanh(k/2).
Prove that the limit does not exist: lim (x,y,z)--> (0,0,0) (xy +yz) / (x2+y2+z2)
3. (a) Let z1,z2, z3 € C, prove the following identity: (21 - 22)(22 – 23)(23 – £1) = (22 - 23)+23(23 – £1)+23(21 - 22). (b) In AABC, P is a point on the plane II containing A, B and C. Prove that aPA +bPB2 +cPC2 > abc.
Let z=5 where x, y, z E R. Prove that z? +z2+z?>