hw2sol - Math 113 Homework 2 – Solutions 1 As explained...

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Unformatted text preview: 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 ....
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This note was uploaded on 03/14/2012 for the course MATH 113 taught by Professor Ogus during the Spring '08 term at Berkeley.

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hw2sol - Math 113 Homework 2 – Solutions 1 As explained...

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