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MATH 504
Algebraic Structures and Functions
Richard Hammack
www.people.vcu.edu/rhammack/Math504/
Section 0: Cardinality of Innite Sets
Recall:
Sets X and Y have the same cardinality, written |X | = |Y |,
provided that there is a one-to-one and onto functi
Section 3 Solutions
8. Consider the binary structures M2 (R), and R, , and the map : M2 (R) R dened as (A) = det(A).
Is an isomorphism?
Notice that a property of determinants gives (A B) = det(A B) = det(A) det(B) = (A) (B), so does
satisfy the homomorphi
Section 4 Solutions
31. An element x of a group G is called idempotent if x x = x. Prove that any group G has exactly one
idempotent element.
Proof. Certainly e is idempotent, because e e = e, so G has at least one idempotent element, e. Could
there be ot
Section 0 Solutions
Use this solution key as a guide in resolving the problems (if any) you had on your homework. It also
gives an indication of the level of completeness and detail that Ill be looking for in your homework this
semester.
(2) cfw_m Z|m2 =
Section 2 Solutions
(2) (a b) c = b c = a
a (b c) = a a = a
Even though weve shown that (a b) c = a (b c), thats no guarantee that the opeation is
associative. We would have to show (x y) z = x (y z) for all possible values of x, y and z. In fact,
note th