Center of a Group and Abelian Groups
Not what you're looking for?
In any group G, some of the elements of G commute with all of the other elements in G.
The set of all such elements in G is called the centre of G, and is denoted by Z(G).
Hence Z(G) = {g|xg = gx x is a subset of G}.
For instance in any group the identity commutes with every element - so Z(G) is never empty. It should also be clear that:
Z(G) = G if and only if G is Abelian
Prove that this is always true. i.e Prove that in any group G, the centre Z (G) is a subgroup of G. In fact, Z (G) is always a normal subgroup of G (but you are not asked to prove this)
Purchase this Solution
Solution Summary
This solution shows how to calculate the center of a Group and Abelian Groups.
Solution Preview
Problem #4
(a) I show that Z(G)=G if and only if G is abelian.
"=>": If Z(G)=G, then for any a,b in G, a and b are also in Z(G).
By the definition of Z(G), we have ab=ba and thus G is abelian.
"<=": If G is ...
Purchase this Solution
Free BrainMass Quizzes
Probability Quiz
Some questions on probability
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts