(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.
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 with subgroup H. Define E = { g^{-1} H g : g \in G }. Prove that |E| divides |G/H|.
Let G be a group, and let a ∈ G. Let φa : G −→ G be defined by
φa(g) = aga−1 for all g ∈ G. (a) Prove that φa is an automorphism
of G. (b) Let b ∈ G. What is the image of the element ba under the
automorphism φa? (c) Why does this imply that |ab| = |ba| for all
elements a, b ∈ G?
9. (5 points each) Let G be a group, and let...
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
the following questions are relative,please solve them,
thanks!
4. Let G be a group. An isomorphism : G G is called an automorphism of G. (a) Prove that the set, Aut(G), of all automorphisms of G forms a group under composition. (b) Let g E G. Show that the map ф9: G-+ G given by c%(z)-gZg", įs an automorphism. These are called the inner automorphisms of G (c) Show that the set of all g E G such that Og-Pe...
Let G be a group and let g G. Show that the mapping ф : G given by ф(x) gxg is an automorphism of G. Any such automorphism, obtained by conjugating by a fixed element g e G, is called an inner automorphism of G
(5 points each) Let G be a group, and let a € G. Let da: G+ G be defined by @a(g) = aga-l for all g E G. (a) Prove that Pa is an automorphism of G. (b) Let b E G. What is the image of the element ba under the automorphism ..? (c) Why does this imply that |ab| = |ba| for all elements a, b E G?
Let G be a finite group such that p is a prime and p divides
|G|. Let P be a p-Sylow subgroup of G such that P is cyclic and ?
. Let H be a subgroup of P . Prove
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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