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Math / Contemporary Abstract Algebra 8 / Chapter 5 / Problem 73E

Contemporary Abstract Algebra | 8th Edition | ISBN: 9781133599708 | Authors: Joseph Gallian

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Chapter 5, Problem 73E

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QUESTION:

Let H = {a2 | a ? S4} and K = {a2 | a ? S5}. Prove H = A4 and K = A5.

Questions & Answers

QUESTION:

Let H = {a2 | a ? S4} and K = {a2 | a ? S5}. Prove H = A4 and K = A5.

ANSWER:

Step 1 of 2

Given

To prove that

Since all the permutations of are even, therefore,

(i)

Now, we will show that .

We know that the elements of are of orders 1, 2 or 3.

Now, let .

If , then .

If , then, .

Now, suppose.

We know that for an element in a group of order , .

Since and , therefore,

Thus, (ii)

From (i) and (ii), we have .

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