Answer:
Step-by-step explanation:
We are given that G be a finite group with [tex]x,y\in G[/tex] 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 [tex]z_n[/tex] 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 [tex]Z_n[/tex]
But we are given that two elements of order 2 in given group
Therefore, group G can be [tex]K_4[/tex]or dihedral group
Because the groups generated by two elements of order 2 are [tex]K_4[/tex] and dihedral group.
We know that [tex]K_4[/tex] is abelian group of order 4 and every element of [tex]K_4[/tex] 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 [tex]K_4[/tex] or [tex]D_2[/tex].
