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
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.
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.
(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 with subgroup H. Define E = { g^{-1} H g : g \in G }. Prove that |E| divides |G/H|.
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
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.
Always give rigorous arguments I. (A) Let G be a group under * and let g E G with o(g) = n (finite) (i) Show that g can never go back to any previous positive power of g* (1k< n) when taking up to the nth power (cf. g), e., that there are no integers k and m such that 1< k<m<n and such that g*-gm (ii) How many elements of the set (e, g,g2.... .g"-) are actually distinct? (iii)...
let G be a finite group, prove that for every a in G there exists a positive integer n such that an=e, the identity.
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 We were unable to transcribe this imageWe were unable to transcribe this image