(3) (7 points) Let G be a finite abelian group of order n. Let k be...
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.
2. Let G be an abelian group. Suppose that a and b are elements of G of finite order and that the order of a is relatively prime to the order of b. Prove that <a>n<b>= <1> and <a, b> = <ab> .
Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2 Let Ga finite abelian group. Prove that a)If pa primenumber divides G|, G has an element of order p b)If G2n with n odd, G has exactly oneelement with order 2
(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.
Let G be a abelian finite group.Prove that a)If pisa prime divisor of G, then G has an element with order p. b)If G 2n with n odd, G has eractly an element with order 2.
Problem 1. Let G be a finite group and f : G → G a group automorphism ( isomorphism for G to G) of order 2 (i.e. f(f(x)) = x), and f has no nontrivial fixed points (i.e. f(x) = x if and only if x = 1). Prove that G is an abelian group of odd order.
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 N. If gcd(rn, n) = 1, show that φ s an injective map. 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 N. If gcd(rn, n) = 1, show that φ s an injective map.
Define , a finite -group, such that isn't abelian. Let such that , where is abelian. Prove that there are either or such abelian subgroups, and if there are , then the index of in is T We were unable to transcribe this imageT K G:K=P We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageT We were unable...
You are given elements g, h, k of a finite abelian group G of orders 306030, 215447282, 116699, respectively. Use g, h, k to construct an element of G of order 108954317152067.
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