Let G be a finite group of order n. Let φ : G → G be the function given by φ(x) = z'n where rn E ...
(9) Let G be a group, and let x E G have finite order n. Let k and l be integers. Prove that xk = xl if and only if n divides l_ k.
(3) (7 points) Let G be a finite abelian group of order n. Let k be relatively prime to n. Prove the map : G G given by pla) = ak is an automor- phism of G
Let G be a finite group, and let H be a subgroup of order n. Suppose that H is the only subgroup of order n. Show that H is normal in G. [consider the subgroup of G] aha а
Let a ∈ G where G is a group. If X ⊆ G is a finite subset, write Xa = {xa | x ∈ X}. Show that X and Xa have the same number of elements.
2. Let G {g, g. . . , gn-le} be a cyclic group of order n, H a group, and h є H. Define a function φ : G → H by φ(gi-hi for all 0 < i n-1. Show that φ is a group homomor- phism if and only if o(h) divides o(g). Warning: mind your modular arithmetic! [10]
Only 2 and 3 1.) Let G be a finite G be a finite group of order 125, 1. e. 161-125 with the identity elemente. Assume that contains an element a with a 25 t e, Show that is cyclic 2. Solve the system of congruence.. 5x = 17 (mod 12) x = 13 mod 19) 3.) Let G be an abelian. group Let it be a subgroup o G. Show that alt -Ha for any a EG
Let G be a group and g E G such that g) is finite. Let og be the automorphism of G given by 09(x) = grg- (a) Prove that $, divides g. (b) Find an element b from a group for which 1 < 0) < 1b.
Theorem 4.27. Suppose G is a finite cyclic group of order n. Then G is isomorphic to Rn if n ≥ 3, S2 if n = 2, and the trivial group if n = 1. Most of the previous results have involved finite cyclic groups. What about infinite cyclic groups?
7. Let A be a Abelian group of finite order n and let m be a natural number. Define a map Om : A + A by Om(a) = a". Prove that Om is a homomorphism of A and identify the kernel of øm. Determine when om is an isomorphism.
16, Let x: U R2-, R, where x(8, φ) (sin θ cos φ, sin θ sin φ, cos θ), be a parametrization of the unit sphere S2. Let and show that a new parametrization of the coordinate neighborhood x(U) = V can be given by y(u, (sech u cos e, sech u sin e, tanh u Prove that in the parametrization y the coefficients of the first fundamental form are Thus, y-1: V : S2 → R2 is a conformal...