Let g be a finite group with x,y ∈ g two elements of order 2. prove that ⟨x,y⟩ is either abelian or isomorphic to a dihedral group.

Step-by-step explanation:

We are given that G be  a finite group with have two elements of order two.

We have to prove that <x,y> is either abelian or isomorphic  to a dihedral group.

<x,y> means the group generated by two elements of order 2.

We know that   is a cyclic group and  number of elements of order 2 is always odd in number and generated by one element  .So , given group is not isomorphic to

But we are given that two elements of order 2 in given group

Therefore, group G can be or dihedral group

Because the groups generated by two elements of order 2 are and dihedral group.

We know that is abelian group of order 4 and every element of   is of order 2 except identity element and generated by 2 elements of order 2 and dihedral group can be  also generated by two elements of order 2

Hence, <x,y> is isomorphic to or .