Question

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.

Answer 1

Explanation:

We are given that G be a finite group with
`x,y\in G` 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
`z_n` 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
`Z_n`

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

Therefore, group G can be
`K_4`or dihedral group

Because the groups generated by two elements of order 2 are
`K_4` and dihedral group.

We know that
`K_4` is abelian group of order 4 and every element of
`K_4` 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
`K_4` or
`D_2`.