Can the center of a finite group determine the size of its conjugacy classes?

  • MHB
  • Thread starter Euge
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    2016
In summary, the size of the center of a finite group can determine the size of its conjugacy classes, as it is always a divisor of the group's order. The center restricts the possible sizes of the conjugacy classes, as it forms a unique conjugacy class of size 1 and the remaining elements must form conjugacy classes of equal size. A finite group with a non-trivial center cannot have conjugacy classes of different sizes, and the size of the center is always a divisor of the group's order. However, there are some exceptions to this rule, such as when a group has a non-trivial center and a normal subgroup that is not contained in the center, or in certain types of groups where the size of
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Euge
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Here is this week's POTW:

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Let $Z$ be the center of a finite group $G$. Prove that there are at most $(G : Z)$ elements in each conjugacy class of $G$.

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Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!
 
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  • #2
This week's problem was solved by johng. You can read his solution below.
Let $x\in G$. The number of conjugates of $x$ (the cardinality of the class of $x$) is $[G:C_G(x)]$ where $[G:C_G(x)]=\{g\in G\,:\,gx=xg\}$. Since obviously $Z\subseteq C_G(x),\,\,[G:Z]=[G:C_G(x)][C_G(x):Z]$ and so $[G:Z]\geq[G:C_G(x)]$.
 

1. Can the center of a finite group determine the size of its conjugacy classes?

Yes, the size of the center of a finite group can determine the size of its conjugacy classes. The order of the center (number of elements) is always a divisor of the order of the group, and the size of the conjugacy classes must also be a divisor of the order of the group. This means that the size of the conjugacy classes can only be equal to the size of the center or a multiple of it.

2. How does the center of a finite group affect the size of its conjugacy classes?

The center of a finite group affects the size of its conjugacy classes by restricting the possible sizes. As mentioned before, the size of the conjugacy classes can only be equal to the size of the center or a multiple of it. This is because the center elements form a unique conjugacy class of size 1, and the remaining elements must form conjugacy classes of equal size.

3. Can a finite group have conjugacy classes of different sizes if its center is non-trivial?

No, if a finite group has a non-trivial center, it cannot have conjugacy classes of different sizes. This is because the size of the center sets a limit on the possible sizes of the conjugacy classes, as explained earlier. Therefore, if the center is non-trivial, all conjugacy classes must be of equal size.

4. Is the size of the center of a finite group always a divisor of the order of the group?

Yes, the size of the center of a finite group is always a divisor of the order of the group. This is a fundamental property of groups, and it can be proven through group theory. The order of the center must divide the order of the group because all elements of the center commute with every element in the group, and the center is a subgroup of the group.

5. Are there any exceptions to the rule that the size of the center determines the size of conjugacy classes in a finite group?

Yes, there are some exceptions to this rule. In some cases, the center may not determine the exact size of the conjugacy classes. For example, if a group has a non-trivial center and a normal subgroup that is not contained in the center, the conjugacy classes of the group may be a multiple of the size of the center but not equal to it. Additionally, in certain types of groups such as symmetric groups, the size of the center may not be a divisor of the order of the group, resulting in some exceptions to this rule.

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