Lesson 28 Solutions

Exercise 1) What is the smallest subgroup in $S_4$ containing the element $(1, 2, 3, 4)$ ?

Solution: We use the procedure that we developed in the lesson, i.e., just taking products of this element with itself over and over until we get back to the identity, placing each new product in our set as we go. Let’s start by just calculating:

$(1, 2, 3, 4, )^2=(1, 2, 3, 4)\cdot (1, 2, 3, 4)=(1, 3)(2, 4)$ ,

$(1, 2, 3, 4)^3=(1, 2, 3, 4)^2\cdot (1, 2, 3, 4)=(1, 3)(2, 4)\cdot (1, 2, 3, 4)=(1, 4, 3, 2)$ ,

$(1, 2, 3, 4)^4= (1, 2, 3, 4)^3\cdot (1, 2, 3, 4)=(1, 4, 3, 2)\cdot (1, 2, 3, 4)=e$ .

Thus the subgroup that we’re after is $\{e, (1, 2, 3, 4), (1, 3)(2, 4), (1, 4, 3, 2)\}$ .

Exercise 2) What is the smallest subgroup in $S_5$ (the group of permutations of 5 objects) containing the element $(1, 4)(2,5)$ ?

Solution: We take the same approach, only this time it’s easy since $(1,4)(2,5)$ is its own inverse (as can be easily checked). Thus, our subgroup is simply $\{e, (1, 4)(2,5)\}$ .