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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College
Algebra
Abstract algebra

Marvin Mccormick

Answered question

2021-02-08

Prove that if "a" is the only elemnt of order 2 in a group, then "a" lies in the center of the group.

Answer & Explanation

liingliing8

Skilled2021-02-09Added 95 answers

Suppose a and x are the elements of the group G.
Let a is the only element of order 2
$\Rightarrow a^{2} = a$
Since ${(x a x^{- 1})}^{2} = x a^{2} x^{- 1}$
$= x a x^{- 1}$ (Since $a^{2} = a$ )
$\Rightarrow$ Order of $x a x^{- 1}$ is 2
Which is a contradiction because Group G has only one element of order 2.
Hence, $x a x^{- 1} = a$
Now multiply both sides in $x a x^{- 1}$ on the right side by x
$\Rightarrow x a x^{- 1} x = a x$
$\Rightarrow= x a = a x$ (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.

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