By Mark A. Armstrong

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This is a gradual creation to the vocabulary and lots of of the highlights of common staff concept. Written in an off-the-cuff type, the fabric is split into brief sections, each one of which offers with a major outcome or a brand new thought. contains greater than three hundred workouts and nearly 60 illustrations.

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**Additional info for Groups and Symmetry (Undergraduate Texts in Mathematics)**

**Example text**

Convert eaeh element IX of S" into an element IX. of S"+2 as folIows. The new permutation IX. behaves just like IX on the integers I, 2, ... , n. If IX is an even permutation then IX. fixes n + land n + 2, whereas if IX is odd IX* interehanges n + land n + 2. Verify that IX* is always an even permutation and that the eorrespondenee IX -+ IX* defines an isomorphism from Sn to a subgroup of A"+2. Work out this subgroup when n = 3. 9. If Gis a finite group of order n, prove that Gis isomorphie to a subgroup of the alternating group A,,+ 2.

Thus 123J [312 sends I to 3, 2 to 1, and 3 to 2. Remembering that aß meansfirst apply ß, then apply a, we calculate M. A. Armstrong, Groups and Symmetry © Springer Science+Business Media New York 1988 6. Permutations 27 123J [123J [123J [ 213 132 = 231 ' whereas [123J [123J [ 123J 132 213 = 312 . Therefore, S3 is not abelian. We can immediately say that S" is not abelian when n ~ 3. Why? When extended to higher values of n, this notation is too cumbersome to work with. For example, the element tX of S6 defined by tX(1) = 5, tX(2) = 4, tX(3) = 3, tX(4) = 6, tX(5) = 1, tX(6) = 2 becomes 123456J tX = [ 543612 .

Work out these elements when n = 4, a. = (2143), and ß = (423). 9. When n is odd show that (123) and (12 ... n) together generate An. Ifn is even show that (123) and (23 ... n) together generate An. 10. , prove that ß permutes those integers which are left fixed by a.. Show that ßmust be apower of a. when a. is an n-cycle. 11. 2. 12. Prove that the order of an element a. of Sn is the least common multiple of the lengths of the cycles which are obtained when a. is written as a product of disjoint cyclic permutations.