Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Question

Let n be an integer greater than two. Show that no subgroup of order two is normal in Sn.

Expert Solution
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Step 1

To prove that no subgroup of order 2 in the symmetric group Sn (n >2) is normal.

Step 2

Statement of the problem. Note that n>2 ,as for n=2, all subgroups are normal.

G Spthe group of all permutations of n symbols; Let n> 2 Claim: No subgroup H of order 2 is normal in G.
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Step 3

. We may assume, (after reordering the symbols ) that we are dealing with the subgroup H = {e, (12)}. (any subgroup of order 2 is generated by a transposition)

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