week09tutsols - THE UNIVERSITY OF SYDNEY Math2968 Algebra...

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Unformatted text preview: THE UNIVERSITY OF SYDNEY Math2968 Algebra (Advanced) Semester 2 Tutorial Solutions Week 9 2008 1. (for general discussion) Let G , H and K be groups. In what sense are G ( H K ) , G H K and ( G H ) K the same? Are G H and H G the same in any meaningful sense? Is it possible to find a nontrivial finite group G such that G = G G ? Does your answer to the previous question alter if G is allowed to be infinite? Solution The groups G ( H K ), G H K and ( G H ) K are all isomorphic. The mapping ( g, ( h,k )) mapsto ( g,h,k ) for g G , h H , k K is easily seen to be an isomorphism from the first group to the second, and there is a similar isomorphism from the second to the third. The groups G H and H G are isomorphic under the map ( g,h ) mapsto ( h,g ) for g G and h H . If G is a nontrivial finite group then 2 | G | < | G | 2 = | G G | , so it is impossible even to find a bijection between G and G G . However, if G is infinite then it is possible for G to be isomorphic to G G . For example, let H be any nontrivial group and put G = { ( x 1 ,x 2 ,x 3 ,... ) | x i H for i = 1 , 2 ,... } with coordinatewise multiplication, which is easily seen to yield a group (a direct product of countably infinitely many copies of H ). Then the map ( x 1 ,x 2 ,x 3 ,... ) mapsto (( x 1 ,x 3 ,x 5 ,... ) , ( x 2 ,x 4 ,x 6 ,... )) for x 1 ,x 2 ,x 3 ,... C 2 is clearly an isomorphism from G to G G . 2. Verify that the direct product of abelian groups is abelian, but the direct product of cyclic groups need not be cyclic. Solution If G and H be abelian groups and g 1 ,g 2 G , h 1 ,h 2 H , then ( g 1 ,h 1 )( g 2 ,h 2 ) = ( g 1 g 2 ,h 1 h 2 ) = ( g 2 g 1 ,h 2 h 1 ) = ( g 2 ,h 2 )( g 1 ,h 1 ) , which verifies that G H is abelian. However a direct product of cyclic groups need not be cyclic. For example, if C 2 is a cyclic group of order 2, then C 2 C 2 is not cyclic, since its elements have order 1 or 2, so that there is no possibility of C 2 C 2 being generated by a single element, which would have to have order 4. 3. Write down the Rank-Nullity Theorem for linear transformations of vector spaces....
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This note was uploaded on 09/12/2009 for the course MATH 2968 taught by Professor Easdown during the One '09 term at University of Sydney.

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week09tutsols - THE UNIVERSITY OF SYDNEY Math2968 Algebra...

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