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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liingliing8
Answered 2021-02-09 Author has 95 answers

Suppose a and x are the elements of the group G.
Let a is the only element of order 2
a2=a
Since (xax1)2=xa2x1
=xax1 (Since a2=a)
Order of xax1 is 2
Which is a contradiction because Group G has only one element of order 2.
Hence, xax1=a
Now multiply both sides in xax1 on the right side by x
xax1x=ax
⇒=xa=ax (Since xx1=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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