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

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SUMMARY

A group of order 20 with 4 conjugacy classes does not necessarily have a trivial center. The Class Equation indicates that the order of the center Z(G) can be calculated as 20 = |Z(G)| + (20/4 + 20/4 + 20/5 + 20/5). This leads to |Z(G)| = 2, suggesting a non-trivial center. However, the existence of such a group must be demonstrated to validate this conclusion.

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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.
 

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