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**Unformatted text preview: **Homework 2 for MATH 435 Solutions Problem 1 Book, p. 123, Exercise 2.21, (i)-(vi) ( The answers are in the back of the back, but give the reasons. ) Solution. (i) FALSE it is a set of n ! elements. (ii) TRUE in fact, any element of a finite group has finite order. (iii) TRUE (iv) FALSE For example, in S 3 , ( 12 )( 13 ) = ( 321 ) , ( 123 ) = ( 13 )( 12 ) . (v) FALSE see previous example. (vi) TRUE this follows from Proposition 2.33. Problem 2 Let T be a tetrahedron, and denote by T the tetrahedral group , i.e. the full symmetry group of T . Label the vertices of T as 1, 2, 3, 4. As discussed in class, every symmetry of T this way corresponds to a permutation in S 4 . Show that indeed every permutation in S 4 is realized this way. (I stated this in class but did not prove it.) Solution. Suppose that a permutation fixes a number, wlog ( 1 ) = 1. If fixes another element, say 2, then the permutation is represented either by the identity or a reflection along the plane through nodes 1, 2, and...

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