Let G be a cyclic group with generator a, and let G' be a group isomorphic to G. If</> : G -+ G' is an isomorphism, show that, for every x E G, (x) is completely determined by the value </>(a). That is, if</> : G -+ G' and 1fr : G-+ G' are two isomophisms such that </>(a)= lfr(a), then </>(x) = 1/r(x) for all x E G.

Let G be a cyclic group with generator a, and let G' be a group isomorphic to G. If</> : G -+ G' is an isomorphism, show that, for every x E G, (x) is completely determined by the value </>(a). That is, if</> : G -+ G' and 1fr : G-+ G' are two isomophisms such that </>(a)= lfr(a), then </>(x) = 1/r(x) for all x E G.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Similar questions

5. Let G be a group and define z" = z * z * *.. * z for n factors c, where z E G and n E Z+. (a) Suppose that G is abelian. Give a mathematical induction proof that (x * y)" = (x") * (y") for all x, y E G. (b) Now, for G not necessarily abelian, prove the following: if (x*y)² = (x²) * (y²) then the elements x and y commute, i.e. x * y = y * x.
Suppose that f: G G such that f(x) = and only if = axa. Then f is a group homomorphism if -> a = e a^3 = e a^4 = e a^2 = e Let (G1, -) and (G2, *) be two groups and p: G1- G2 be an isomorphism. Then " G2 miaht be abelian even if G1 is abelian ing TOSHIBA
Consider the group G = ℚ* × ℤ with operation * on G that can be expressed as: (w, x) * (y, z) = (wy + 1, xz - 1), for all (w, x), (y, z) ∈ ℚ* × ℤ. Is the group <G, *> abelian?
Show that x is a generator of the cyclic group (Z3[x]/<x3 + 2x + 1>*.
G is a finite group and be an automorphism of G satisfying the condition f(x)= x > x=e for xEG. Show that there can be 36EG for YafG such that a=6"f(6)
For an abelian group, the only inner automorphism is the identity mapping, whereas for non-abelian groups there exists non-trivial automorphism.
An element x of a group satisfies x2 = e precisely when x = x-1. Use this observation to show how that a group of even order must contain an odd numder of elements of order 2.
Show that for an abelian group, the only inner automorphism is the identity mapping, whereas for non-abelian groups there exists non-trivial automorphism.
Let p: G → G' be a group homomorphism. (a) If H < G, prove that o(H) is a subgroup of G' (b) If H
An endomorphism of the group G is a homomorphism : G → G. Let End (A) be the set of all endomorphisms of the abelian groups A, that is End(A) := {y: A → Alya homomorphism} Define binary operations + and * on End(A) by (+)(a) = y(a) + (a) and (*)(a) = ((a)) 4 for all a E A and p, & € End(A). Show that (End (A), +, *) is a ring.
If x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian.
Find the zeros of Q(x) = 5x - x² + 9 using the quadratic formula. Number of zeros: one X NOTE: Enter an exact value for the zero or round to three decimal places. be here to search X = -5+√11 2 O X