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week09tutsols - THE UNIVERSITY OF SYDNEY Math2968...

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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. (a) Let T : V W be a linear transformation between vector spaces V and W both of the same finite dimension n . Use the Rank-Nullity Theorem to verify that T is a vector space isomorphism if and only if T is injective if and only if T is surjective.
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