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Abstract Algebra One (Exam Two)

(1) Define (a) a subgroup of a group , (b) a partition of a set , and (c) an equivalence relation on a set .

Subgroup:

Partition:

Equivalence Relation:

(2) If possible, find a subgroup of with order 6.

(3) Determine the group of symmetries for a rectangle with its diagonals.

(4) For sets and let mean there is an invertible mapping from onto Show that is an equivalence relation.

(5) Solve the equations in and Find the order of in

(6) Prove that if is a fixed element of a group and is defined by for each then is one-to-one and onto.

(7) Determine the smallest subset of such that and is closed with respect to subtraction. Explain why.

(8) Prove that is abelian if and only if

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