Could you please provide me the details? step by step Recall that every finite g

Question

Could you please provide me the details? step by step

Recall that every finite group of even order 2n contains an element of order 2. Using the theorenm of Lagrange, show that if n is odd, then an abelian group of order 2n contains precisely one element of order 2. (Hint: Prove this by contradition, that is, assume there are two elements a and b of order 2, thern show ab is also an element of order 2, and check se, a, b, ab is a subgroup of order 4. Finally, use Lagrange to arrive at contradition.)

Explanation / Answer

Assume that there are at least two elements of order 2. Let a,b have order 2 with ab. Together they generate the four element subgroup H = { a, b, ab, e }. (If your group weren't abelian (say if it were dihedral), this group might have more than 4 elements.)

But Lagrange's theorem says that the order of H must divide the order of G. 4 doesn't divide 2n since n is odd, so we have a contradiction.

hence there are exactly one element of order 2

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