Is the Centre of a Group with 4 Conjugacy Classes of Order 20 Trivial?

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In summary, the Conjugacy Class Problem is a mathematical problem in group theory that involves determining the number of conjugacy classes in a given group. It is important because it helps us understand the structure of a group and has applications in other areas of mathematics. The problem can be solved using various techniques such as the class equation, character theory, and representation theory. Examples of solving the problem include the dihedral group and the symmetric group. There are also open problems related to the Conjugacy Class Problem, including the existence of groups with a large number of conjugacy classes and the determination of the number of conjugacy classes in certain types of groups.
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Homework Statement


If a group has order 20 has 4 conjugacy classes, it must have a trivial centre. True or False?


Homework Equations


The Class Equation

The Attempt at a Solution


I believed the answer to be false with this following counterexample:
20 = lZ(G)l + (20/4 + 20/4 + 20/5 + 20/5) => order of Z(G) = 2 => the centre is non-trivial.
where 5, 5, 4, 4 is the size of the 4 conjugate classes respectively

Any feedback on my reasoning and any flaws are very much appreciated
 
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You haven't demonstrated that such a group exists.
 

1. What is the Conjugacy Class Problem?

The Conjugacy Class Problem is a mathematical problem in group theory that involves determining the number of conjugacy classes in a given group. A conjugacy class is a set of elements in a group that are equivalent under conjugation, meaning they can be transformed into one another by multiplying by a fixed element.

2. Why is the Conjugacy Class Problem important?

The Conjugacy Class Problem is important because it helps us understand the structure of a group by identifying the different ways its elements can be related through conjugation. It also has applications in other areas of mathematics, such as in the study of symmetry and in solving certain types of equations.

3. How is the Conjugacy Class Problem solved?

The Conjugacy Class Problem can be solved using various techniques, depending on the specific group being studied. One common method is to use the class equation, which relates the number of elements in a group to the number of conjugacy classes and the sizes of those classes. Other methods include using character theory and representation theory.

4. What are some examples of solving the Conjugacy Class Problem?

One example of solving the Conjugacy Class Problem is in the dihedral group Dn, which represents the symmetries of a regular n-gon. The number of conjugacy classes in Dn is equal to the number of divisors of n, and the size of each class can be determined using the class equation. Another example is in the symmetric group Sn, where the number of conjugacy classes is equal to the number of partitions of n.

5. Are there any open problems related to the Conjugacy Class Problem?

Yes, there are currently open problems related to the Conjugacy Class Problem. One such problem is the existence of groups with a large number of conjugacy classes. Another open problem is the determination of the number of conjugacy classes in certain types of groups, such as matrix groups and finite simple groups.

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