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

Answer:

Step-by-step 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 $D_2$.