Question

In: Advanced Math

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.

Expert Solution

Given:

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}

1 Let a, b ∈ Z(G).

Then ax = xa and bx = xb for all x ∈ G, and for any x ∈ G

we have (ab)x = a(bx) = a(xb) = (ax)b = (xa)b = x(ab), so ab ∈ Z(G).

Therefore, Z(G) is closed.

Also, we certainly have ex = xe = x for all x ∈ G, so e ∈ Z(G).

Finally, if a ∈ Z(G), then ,

since a commutes with every element of G.

Continuing, we have ,

so , and ∈ Z(G).

Therefore, Z(G) is a subgroup of G.

2. given definition of Z(G) is as follows

Z(G) = {x in G | ax=xa for all a in G}

this implies Z(G) is commutative.

by the definition and Part 1 we can conclude that Z(G) is an abelian group.[Z(G) is also a group]

Colby Messinger answered 2 years ago

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 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.

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 the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider...

Let the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider the following statement: for every positive integer n and every x in Z/nZ, there exists y ∈ Z/nZ such that xy = [1]. (a) Write the negation of this statement. (b) Is the original statement true or false? Justify your answer.

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

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove...

Let G be a group,a;b are elements of G and m;n are elements of Z. Prove (a). (a^m)(a^n)=a^(m+n) (b). (a^m)^n=a^(mn)

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.

Let f : Z × Z → Z be defined by f(n, m) = n −...

Let f : Z × Z → Z be defined by f(n, m) = n − m a. Is this function one to one? Prove your result. b. Is this function onto Z? Prove your result

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.