homework2solutions_Math 113

homework2solutions_Math 113 - Math 113 Section 5 Homework...

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Unformatted text preview: Math 113 Section 5 Homework #2 Solutions Fall 2010 Due: Monday, September 13 1. Determine which of the following sets with 2-to- 1 operations form groups. For those that are groups, prove that they are groups. For those that are not groups, describe why they are not groups. (Complete solutions are not given here, but sketches/ideas are!) (a) Z with usual addition + Yes (b) Z with usual multipication No, is not invertible (c) Z /n Z where n is any positive integer, with multiplication Not a group in general, and a for ( a,n ) = 1 are not invertible (d) ( Z /n Z ) , the set of invertible elements of Z /n Z , with multiplication Yes (e) M n m , the set of n m matrices with entries in R , with matrix addition Yes (f) M n m , the set of n m matrices with entries in R , with matrix multiplication No, multiplication would only be defined in m = n and then, a matrix A would be invertible only if det ( A ) = 0 . (g) M n , the set of (square) n n matrices with entries in R , with matrix multiplication No, see above (h) G = { a + b 2 | a,b Q } R with multiplication No, G but is not invertible (i) G = { z C | z n = 1 } , where n Z > , with multiplication of complex numbers...
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homework2solutions_Math 113 - Math 113 Section 5 Homework...

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