find a group of permutations isomorphic to the klein 4 group

find a group of permutations isomorphic to the klein 4 group

Solution

If you knew Cayley\'s theorem you could solve your problem by simply applying the proof of Cayley\'s theorem to the group V. I assume you don\'t know that theorem.

In the group V the elements a,b are elements of order two which commute, so let\'s start by finding two commuting permutations of order two. Permutations of order two are products of one or more disjoint transpositions, such as (1 2), (1 2)(3 4), (1 2)(3 4)(5,6) and like that. Multiplication of permutations is usually noncommutative, e.g. (1 2)(2 3)=(1 2 3)(1 3 2)=(2 3)(1 2),but multiplication of disjoint permutations is commutative. So let\'s try taking two disjoint transpositions: a=(1 2), b=(3 4), ab=ba=(1 2)(3 4),, and let\'s hope that

G={(1), (1 2), (3, 4), (1 2)(3 4)}