Show that given conditions, element is in center of group G

In summary: This means that for any ##a## and ##b##, ##a(ba) = (ba)a## which implies ##ba = ab## and thus ##b## is in the center of ##G##. In summary, we use the fact that ##m## and ##|G|## are coprime to show that any element ##b## in the group ##G## that satisfies ##a^m = ba^mb^{-1}## for all ##a## must be in the center of ##G##.
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Homework Statement


Let ##G## be a finite group and ##m## a positive integer which is relatively prime to ##|G|##. If ##b\in G## and ##a^mb=ba^m## for all ##a\in G##, show that ##b## is in the center of ##G##.

Homework Equations

The Attempt at a Solution


Let ##|G| = n## and ##b\in G##. Note that by Bézout 's identity ##nx + my = 1## for some ##x,y\in\mathbb{Z}##. Also, note that ##a^m = ba^mb^{-1}##. So

$$
\begin{align*}
a^m &= ba^mb^{-1}\\
(a^m)^y&= (ba^mb^{-1})^y\\
a^{my} &= ba^{my}b^{-1}\\
a^{1-nx} &= ba^{1-nx}b^{-1}\\
a(a^n)^{-x}&= ba(a^n)^{-x}b^{-1}\\
a(e)^{-x}&= ba(e)^{-x}b^{-1}\\
a&= bab^{-1}\\
\end{align*}
$$

Since ##a## is an arbitrary element of ##G##, we see that ##b\in Z(G)## ☐

Is this the correct argument?
 
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Mr Davis 97 said:

Homework Statement


Let ##G## be a finite group and ##m## a positive integer which is relatively prime to ##|G|##. If ##b\in G## and ##a^mb=ba^m## for all ##a\in G##, show that ##b## is in the center of ##G##.

Homework Equations

The Attempt at a Solution


Let ##|G| = n## and ##b\in G##. Note that by Bézout 's identity ##nx + my = 1## for some ##x,y\in\mathbb{Z}##. Also, note that ##a^m = ba^mb^{-1}##. So

$$
\begin{align*}
a^m &= ba^mb^{-1}\\
(a^m)^y&= (ba^mb^{-1})^y\\
a^{my} &= ba^{my}b^{-1}\\
a^{1-nx} &= ba^{1-nx}b^{-1}\\
a(a^n)^{-x}&= ba(a^n)^{-x}b^{-1}\\
a(e)^{-x}&= ba(e)^{-x}b^{-1}\\
a&= bab^{-1}\\
\end{align*}
$$

Since ##a## is an arbitrary element of ##G##, we see that ##b\in Z(G)## ☐

Is this the correct argument?
Yes. Because ##m## and ##n## are coprime, ##a## generates the same elements as ##a^m## so they can be interchanged.
 
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Likes Mr Davis 97

1. How do you determine if an element is in the center of a group?

To determine if an element is in the center of a group, you need to first identify the group's symmetry operations. Then, you can apply these operations to the element and see if it remains in the same position. If the element does not move, it is considered to be in the center of the group.

2. What are the conditions that need to be met for an element to be in the center of a group?

The conditions for an element to be in the center of a group are that it must not move when subjected to any of the group's symmetry operations and it must also be invariant under those operations. In other words, the element must have the same properties before and after each operation.

3. Can an element be in the center of more than one group?

Yes, an element can be in the center of more than one group. This is because there may be multiple groups with different symmetry operations that do not affect the position of the element.

4. How does an element being in the center of a group affect its properties?

An element being in the center of a group does not have a direct effect on its properties. However, it does indicate that the element has certain symmetry properties that are preserved under the group's operations.

5. How is the center of a group related to its overall structure?

The center of a group is an important aspect of its overall structure as it represents its symmetry properties. The elements in the center of a group are typically used as reference points to describe the group's structure and organization.

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