Algebraic Structures Exam File Fall 2013 Exam #1


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1 Algebraic Structures Exam File Fall 2013 Exam #1 1.) Find all four solutions to the equation x = 0. Give your answers as complex numbers in standard form, a + bi. 2.) Do the following. a.) Write out the operation table for Z 5. b.) Construct the operation table for the set {1, 2, 3, 4} where the operation is multiplication modulo 5. Does this produce a group? Why or why not? 3.) Give examples of the following, if possible. If it is not possible, tell why not ("tell why not" does not mean prove, it can mean cite a result we proved in class). a.) A group with two distinct elements that are their own inverses. b.) A group that is not associative. c.) Two finite groups with the same order that are not isomorphic. 4.) Determine whether or not the following sets with the given operations are groups. If it is a group, determine whether or not it is abelian. "Determine" means "give reasons, not just saying 'yes' or 'no,' but give a reason." a.) G = R, a * b = a  b b.) G = {0, 1,..., 10}, a * b = a  b (mod 11) 5.) Determine if the given pairs of binary structures are isomorphic.
2 6.) a.) (G,*) (G', ) are groups. If there is an isomorphism from G onto G', and G is abelian then prove that G' is abelian. b.) If all elements of a group are their own inverses, prove that the group is abelian. c.) Prove or disprove: A group, <G, *>, is abelian if and only if, for every a, b G, (a * b) 1 = a 1 * b ) TrueFalse. Write out the word completely. Write "true" only if the statement is always true. If it is not always true, write "false." a.) A group may have more than one identity element. b.) If <G, *> is a finite group with identity e and a G, then there exists a positive integer, n, such that a n = e. c.) An abelian group can't be isomorphic to a nonabelian group. d.) Any two finite groups with the same number of elements are isomorphic. e.) Q with the usual multiplication is a group. f.) If x * x = e for all x in a group <G, *>, then G is abelian. g.) Z with the usual subtraction is a group. h.) If G is a set and * is a binary operation on G such that a * a = a for all a in G then <G, *> is not a group. i.) If a group <G, *> is abelian then x * x = e for all x in G. j.) If G is a group and a, b G then (ab) 1 = a 1 b 1. Exam #2 1.) Computational stuff. Show work. a.) Find the order of <6> in Z 16. b.) Perform the following multiplication and write the answer in cycle notation. Is it an even permutation or an odd permutation? ( )(3 5 2) c.) Find all cosets of 3Z in Z. d.) Find the index of <4> in Z 12. e.) Find the order of (2,3) in Z 4 x Z 9. 2.) Give examples of the following, if possible. If it is not possible, tell why not. a.) A noncyclic nonabelian group of order 720. b.) A noncyclic abelian group of order 720. c.) A nonabelian group of order 7. d.) e.) A nonabelian group of order ) Consider the following permutations in S 6. A group with no nontrivial, proper subgroups. ( ), ( ) Find each of the following. Show work. a.) 1 b.) 2 c.) order of d.) 347 e.) as a product of disjoint cycles f.) as a product of transpositions
3 Proofs. 1.) Prove that a nonempty subset, H, of a group, G, is a subgroup of G if and only if whenever a, b are elements of H then ab 1 is in H. 2.) Suppose H and K are subgroups of a group G. Prove that H K is also a 3.) a.) Suppose <G, *> is a group and H G. If a, b ε H and a * c = b then c ε H. b.) All cyclic groups are abelian. Exam #3 1.) For each of the following sets (with usual operations unless otherwise noted) write the letter corresponding to the correct algebraic structure. Be as restrictive as possible. (If something is a commutative ring with unity, then don't say it is a commutative ring.) Some answers may be used more than once. A ring B commutative ring C commutative ring with unity D ring with unity E division ring F none of the above Z 9 Z Z 5 Reals 2Z Irrationals {2 x 2 matrices} {invertible 2 x 2 matrices} R = {a, b, c, d} with operations as given below + a b c d a a b c d b b c d a c c d a b d d a b c * a b c d a a a a a b a a a a c a a a a d a a a a 2.) Determine the set of all units in the ring Z 8. In #3  #5, prove the statement. 3.) If a subgroup, H, of a group, G, has index 2 in G, then H is a normal of G. 4.) If G is abelian then any subgroup of G is normal. 5.) Let R be a commutative ring with unity. Let U(R) be the set of all units of R. Prove that U(R) is a group under the ring multiplication. In #6  #8, give an example if possible. If not possible, why not? 6.) A nontrivial group homomorphism from Z 4 into Z 12. You do NOT need to verify that it is a homomorphism. If there is not one, you must show why there can't be one. 7.) A group with no nontrivial normal subgroups other than the group itself. 8.) A group, G, where G = n and an integer, m, such m n and G has no subgroup of order m. In #9  #11, determine if the function is a ring isomorphism, a ring homomorphism or neither. 9.) Φ:Z 2Z where Φ(x) = 2x 10.) Φ:Z 8 Z 6 where Φ(x) = 0 if x is even and Φ(x) = 3 if x is odd. 11.) Φ:R R where Φ(x) = x 2
4 12.) Below is the group table for S 3. H = {ρ 0, µ 1 } is a subgroup of S 3. Find ρ 1 H and Hρ 1. Is H a normal subgroup of S 3? Why or why not? ρ 0 ρ 1 ρ 2 µ 1 µ 2 µ 3 ρ 0 ρ 0 ρ 1 ρ 2 µ 1 µ 2 µ 3 ρ 1 ρ 1 ρ 2 ρ 0 µ 3 µ 1 µ 2 ρ 2 ρ 2 ρ 0 ρ 1 µ 2 µ 3 µ 1 µ 1 µ 1 µ 2 µ 3 ρ 0 ρ 1 ρ 2 µ 2 µ 2 µ 3 µ 1 ρ 2 ρ 0 ρ 1 µ 3 µ 3 µ 1 µ 2 ρ 1 ρ 2 ρ 0 Exam #4 1.) Use Fermat's Little Theorem to find mod ) Use Euler's Theorem to find mod ) a.) Solve the equation x 24x + 3 = 0 in Z 7. b.) Solve the equation x 24x + 3 = 0 in Z 6. 4.) Suppose R is a commutative ring with unity and characteristic 3. If a, b R, find (a + 2b) 4. Simplify your answer as much as possible. 5.) Let that p, q and r be distinct primes. Find (pqr). 6.) Consider D = {a + bi a, b Z, i = }, with the usual addition and multiplication over the complex numbers. Find the relation and equivalence classes produced by the construction of the field of quotients for D. 7.) An element, a, of a ring R is nilpotent if a n = 0 for some n Z +. Find the set of all nilpotent elements in Z 9. 8.) PROVE OR DISPROVE Every integral domain is a field. 9.) PROVE OR DISPROVE The direct product of two fields is again a field. 10.) Do one of the following. Circle the letter of the one you are doing. a.) An element, a, of a ring R is idempotent if a 2 = a. Show that a division ring contains exactly two idempotent elements. b.) Prove that every finite integral domain is a field. Final Exam 1.) For each of the following, give an example of the following, if possible. If it is not possible, tell why not ("tell why not" does not mean prove, it can mean cite a result we proved in class). a.) A group with two distinct elements that are their own inverses. b.) A group that is not associative. c.) d.) e.) f.) g.) Two finite groups with the same order that are not isomorphic. A nontrivial group homomorphism from Z 4 into Z 12. You do NOT need to verify that it is a homomorphism. If there is not one, you must show why there can't be one. A group with no nontrivial normal subgroups other than the group itself. A group, G, where G = n and an integer, m, such m n and G has no subgroup of order m. A finite integral domain that is not a field.
5 h.) A commutative ring that does not have unity. i.) A commutative ring with unity that is not an integral domain. j.) A nonabelian group. k.) A group with two distinct identities. l.) An abelian group that is not cyclic. 2.) An element, a, of a ring R is nilpotent if a n = 0 for some n Z +. An element, a, of a ring R is idempotent if a 2 = a. a.) Find the set of all nilpotent elements in Z 8. Be sure to show all work indicating whether elements are or are not nilpotent. b.) Find the set of idempotent elements in Z 8. Be sure to show all work indicating whether elements are or are not idempotent. 4.) Determine whether or not the following sets with the given operations are groups. If it is a group, determine whether or not it is abelian. a.) G = R, a * b = a  b b.) G = {0, 1,..., 10}, a * b = 2a + b (mod 11) c.) G = {0, 1,..., 10}, a * b = a + b (mod 11) The following are for #5 only. ( ) ( ) 5.) Do each of the following. a.) compute b.) compute c.) write in cycle notation The following are for #6 only = ( )( )( ) = ( )( ) 6.) Do each of the following. a.) compute < > b.) write as a product of transpositions c.) compute d.) compute ) List all possible orders for subgroups of G if G = ) a.) Solve the equation x 24x + 3 = 0 in Z 7. b.) Solve the equation x 24x + 3 = 0 in Z 8. 9.) Suppose H and K are subgroups of a group G. Prove that H K is also a 10.) Prove that all cyclic groups are abelian. 11.) An element, a, of a group (G, *) is idempotent if a * a = a. Prove that a group can have only one idempotent element. 12.) Suppose H is a subgroup of an abelian group, G. Prove that H is a normal 13.) If there is an isomorphism from G onto G', and G is abelian then prove that G' is abelian. 14.) Given a set, G, and an operation *, give the definition for (G, *) is a group. 15.) Given a set, R, and two operations, + and *, give the definition for (R, +, *) is a ring. 16.) Given a set, F, and two operations, + and *, give the definition for (F, +, *) is a field.
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