 # Prove that if "a" is the only elemnt of order 2 in a group, then "a" lies in the center of the group. Marvin Mccormick 2021-02-08 Answered
Prove that if "a" is the only elemnt of order 2 in a group, then "a" lies in the center of the group.
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Suppose a and x are the elements of the group G.
Let a is the only element of order 2
$⇒{a}^{2}=a$
Since ${\left(xa{x}^{-1}\right)}^{2}=x{a}^{2}{x}^{-1}$
$=xa{x}^{-1}$ (Since ${a}^{2}=a$)
$⇒$ Order of $xa{x}^{-1}$ is 2
Which is a contradiction because Group G has only one element of order 2.
Hence, $xa{x}^{-1}=a$
Now multiply both sides in $xa{x}^{-1}$ on the right side by x
$⇒xa{x}^{-1}x=ax$
$⇒=xa=ax$ (Since $x{x}^{-1}=e$)
This shows that element a commuts with every element of the group G.
Hence, By definition the leement a lies in the centre of the group G.