Let G be a finite group, | G |= 24. Determine whether G must have an element of order 2.
Let G be a finite group, | G |= 24. Determine whether G must have an...
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
(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 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 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
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 A be a subset of a finite group G with |A| > |G|/2. Prove that every element of G can be written as the product of two elements of A. Is this also always true when |A| = |G|/2?
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.
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.
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.