hw2sol

# hw2sol - Math 113 Homework 2 Solutions 1 As explained on...

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Math 113 Homework 2 – Solutions 1. As explained on page 43 of the book, one can give a seemingly weaker definition of a group, which turns out to be equivalent to the usual one. Namely, consider a set G equipped with a binary operation such that • ∗ is associative, there exists a left identity element e G such that e x = x for all x G , for each a G there exists a left inverse a G such that a a = e . (a) Show that left cancellation holds, namely if a b = a c then b = c . Let a, b, c G be such that a b = a c . By the third axiom, there exists an element a G such that a a = e . Using associativity, we have a ( a b ) = ( a a ) b = e b , and by the second axiom, e b = b , therefore a ( a b ) = b . Similarly a ( a c ) = c . However a b = a c , therefore b = a ( a b ) = a ( a c ) = c . Thus a b = a c implies b = c . (b) Show that the left identity element e is also a right identity element, namely x e = x for all x G . (Hint: calculate x ( x e ) and use part (a)). Let x G , and let x G be such that x x = e (the existence of x is guaranteed by the third axiom). Using associativity, we calculate x ( x e ) = ( x x ) e = e e , however the second axiom implies that e e = e = x x . Therefore x ( x e ) = x x . Using the result of (a) (with a = x , b = x e , c = x ), we conclude that x e = x . (c) Show that the left inverse x is also a right inverse, namely x x = e . (Hint: calculate x ( x x ) ). Let x G , and let x G be the left inverse given by the third axiom. Then x ( x x ) = ( x x ) x = e x = x (using associativity and the fact that e is a left identity element). Since by (b) e is also a right identity, we have x ( x x ) = x = x e ; finally, using the result of (a) we deduce that x x = e . 2. (a) Show that multiplication is a well-defined binary operation on the set Z n of congruence classes of integers modulo n . As in HW1 Problem 3, we need to show that if a a mod n and b b mod n , then ab a b mod n (so that the product a b is the same congruence class independently of the choice of a and b in their congruence classes).

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