Abstract Algebra One (Practice Exam One)

(1) Show that the following matrices form a group under matrix multiplication:

Is this group Abelian or not?

(2) Give an example of a mapping that is one-to-one and not onto.

(3) Give an example of a mapping that is onto and not one-to-one.

(4) Is it possible to have a mapping that is one-to-one and onto?

(5) If where and are subsets of then show is one-to-one if and only if

(6) Show that a mapping is invertible if and only if it is bijective.

(7) Prove that is a subgroup of a group if and only if for all Then use this property to show that the set of even integers is a subgroup of the integers under ordinary addition.

(8) If is a group, prove that where is the center of and is the centralizer for an element in

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