which of the following statements about a group are true?

Question

1.If a group has an odd number of elements then there exists no element which is inverse of itself. 2.If a group contains even no of elements then there exists only one element which is inverse of itself.

I am confused that if I have odd no of elements then in that case I will have one identity element and then I will be left with even no of elements,which can be paired up so then I will be left with atleast one element which is inverse of itself so then in the case where the group has even no of elements then can't we argue in the similar fashion .

Answer 1

"If a group contains even no of elements then there exists only one element which is inverse of itself."

What groups on four elements do yo know?

Answer 2

Both statements are false. The elements that are not their own inverses can be partitioned into (unordered) pairs of the form $\{a,a^{-1}\}$, hence the number of elements that are not selfinverse is even. Thus in an odd group there is at least one selfinverse element (indeed, the neutral element has this property); and in an even group, the number of such elements is even and - as the neutral element has this property - is at least $2$.