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Abstract Algebra One (Exam Three)

(1) Find the least positive such that

(2) (a) Write all the elements of both in two-row form and using cyclic notation. (b) Which elements of are their own inverses?

(3) Determine the group of symmetries for a rectangle including two of its half diagonals.

(4) Compute the right cosets of in

(5) Prove that if then

(6) Given groups and prove that

(7) Define a relation on by if an only if for some Prove that is an equivalence relation and give a complete set of equivalence class representatives.

(8) Prove or disprove that every subgroup of a finite cyclic group is cyclic.

(9) Give three examples of a group with order 6.

(10) Prove that if is a homomorphism and then

(11) Write the permutation associated with each element of (in ) by the isomorphism in the proof of Cayley's Theorem.

(12) Assume Prove that if is a prime, then is cyclic. Prove or disprove the converse.

(13) (a) Define a mapping such that for each the equation has exactly three solutions.
    (b) Define a mapping such that for each the equation has exactly solutions.
    (c) Define a mapping such that for each the equation has exactly infinitely many solutions.

(14) Assume that is a group with a subgroup such that and Find and

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