Let G be a group and let C={g∈G|xg=gx for all x∈G} be the center of G....

Let G be a group and let C={g∈G|xg=gx for all x∈G} be the center of G. Prove that for any a ∈ G, aC = Ca.

Expert Solution

Colby Messinger answered 1 year ago

Let G be a Group. The center of, denoted by Z(G), is defined to be the...

Let G be a Group. The center of, denoted by Z(G), is defined to be the set of all elements of G that with every element of G. Symbolically, we have Z(G) = {x in G | ax=xa for all a in G}. (a) Prove that Z(G) is a subgroup of G. (b) Prove that Z(G) is an Abelian group.

Let G be a group. The center of G is the set Z(G) = {g∈G |gh...

Let G be a group. The center of G is the set Z(G) = {g∈G |gh = hg ∀h∈G}. For a∈G, the centralizer of a is the set C(a) ={g∈G |ga =ag } (a)Prove that Z(G) is an abelian subgroup of G. (b)Compute the center of D4. (c)Compute the center of the group G of the shuffles of three objects x1,x2,x3. ○n: no shuffling occurred ○s12: swap the first and second items ○s13: swap the first and third items ○s23:...

Let G be an abelian group. (a) If H = {x ∈ G| |x| is odd},...

Let G be an abelian group. (a) If H = {x ∈ G| |x| is odd}, prove that H is a subgroup of G. (b) If K = {x ∈ G| |x| = 1 or is even}, must K be a subgroup of G? (Give a proof or counterexample.)

Let G be a nontrivial nilpotent group. Prove that G has nontrivial center.

For X ≠ ∅, an action of G on X is transitive if and only if, given x and y in X, there is some g ∈ G such that y = gx.

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove...

Let G be a group. For each x ∈ G and a,b ∈ Z+ a) prove that xa+b = xaxb b) prove that (xa)-1 = x-a c) establish part a) for arbitrary integers a and b in Z (positive, negative or zero)

Let G be a group and a be an element of G. Let φ:Z→G be a...

Let G be a group and a be an element of G. Let φ:Z→G be a map defined by φ(n) =a^{n} for all n∈Z. a)Show that φ is a group homomorphism. b) Find the image ofφ, i.e.φ(Z), and prove that it is a subgroup ofG.

PLEASE SHOW ALL STEPS WITH EXPLANATIONS Let G be a group (not necessarily an Abelian group)...

***PLEASE SHOW ALL STEPS WITH EXPLANATIONS*** Let G be a group (not necessarily an Abelian group) of order 425. Prove that G must have an element of order 5

Let G be a finite group and H a subgroup of G. Let a be an...

Let G be a finite group and H a subgroup of G. Let a be an element of G and aH = {ah : h is an element of H} be a left coset of H. If b is an element of G as well and the intersection of aH bH is non-empty then aH and bH contain the same number of elements in G. Thus conclude that the number of elements in H, o(H), divides the number of elements...

Let G be a group, and let a ∈ G be a fixed element. Define a...

Let G be a group, and let a ∈ G be a fixed element. Define a function Φ : G → G by Φ(x) = ax−1a−1. Prove that Φ is an isomorphism is and only if the group G is abelian.

Question