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

  • #1
1,462
44

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?
 

Answers and Replies

  • #2
14,587
11,945

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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