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.

Solutions

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 1 year ago

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