Abstract Algebra One (Practice 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 square including two of its half diagonals.

(4) Compute the right cosets of in

(5) Prove that if then

(6) Prove that if and are relatively prime then

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

(8) Find a necessary and sufficient condition on a subset of a group for

(9) Give three examples of a group with order 24.

(10) Define from to itself by for each where Prove that is an homomorphism and then find

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

(12) Prove that every element of has finite order. Also show that has an element of order for each positive integer

(13)     (a) If possible, define a mapping such that for each the equation has exactly three solutions.
    (b) If possible, define a mapping such that for each the equation has exactly solutions.
    (c) If possible, 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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