# Algebraic Structures Test File. Test #1

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1 Algebraic Structures Test File Test #1 1.) True-False. Write out the word completely. Write "true" only if the statement is always true. If it is not always true, write "false" and give an example to show that it is false or a reason from a definition or theorem to show that it is false. a.) If a group <G, *> is abelian then x * x = e for all x in G. b.) 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. c.) The empty set can be considered a group. d.) There are no groups with 117 elements. e.) There is no binary operation on R 2 that gives a group structure. 2.) Prove the following. a.) In a group, the identity element is unique. b.) There is only one group with 3 elements. c.) Let G be a nonempty set together with an associative binary operation * such that a * x = b and y * a = b have unique solutions for every pair a, b of elements of G. Then <G, *> is a group. 3.) Determine whether or not the following are groups. If it is a group, determine whether or not it is abelian. a.) G = {non-negative real numbers}, a * b = a - b b.) G = {0, 1,..., 9}, a * b = ab (mod 10) c.) G = {1, -1, i, -i}, where i = (-1), a * b = ab d.) G = M 2x2, where A * B = AB 4.) Determine whether or not the following are equivalence relations. If it is not, then list all parts that fail. a.) S = Integers, arb iff a and b have the same number of digits in the usual base 10 notation b.) S = integers, arb iff a - b 7 5.) Solve for x. That is find an integer in {0, 1,..., 7} that makes the statement true. a.) x 39 (mod 7) b.) x -9 (mod 7) 6.) Write out the operation table for Z 5. 7.) Use mathematical induction to prove the following identity DOTSAXIS n = n + 1

2 8.) Define the following terms. a.) group b.) binary operation c.) identity element of a group d.) inverse of an element of a group e.) Z 7 f.) equivalence relation g.) abelian group 9.) True-False. Write out the word completely. Write "true" only if the statement is always true. If it is not always true, write "false" and give an example to show that it is false or a reason from a definition or theorem to show that it is false. a.) If x * x = e for all x in a group G, then the group is abelian. b.) The only groups that are abelian are ones in which x * x = e for all x in G. c.) If x * x = e for all x in a group G, then * is associative. d.) A group may have two distinct elements that are inverses. e.) Be sure to understand all of the true-false problems in the text. 10.) Prove the following. a.) In a group, inverses are unique. b.) In a group, the identity element is unique. c.) any of the problems in the book in the theory portions of the assignments 11.) Give examples of the following, if possible. If it is not possible, tell why not. a.) an operation and two sets so that the operation is a binary operation on one of the sets and is not on the other b.) a group with two distinct identities c.) a group that is not abelian d.) a group with 1 element e.) a group with an infinite number of elements f.) a relation on a set that is not reflexive and not symmetric but is transitive 12.) Determine whether or not the following are groups. If it is a group, determine whether or not it is abelian. a.) G = R, a * b = a - b b.) G = R +, a * b = a/b c.) G = {0, 1,..., 10}, a * b = 2a + b (mod 11) d.) G = {0, 1,..., 10}, a * b = a + b (mod 11) e.) G = C #, a * b = ab f.) G = C, a * b = a + b 13.) Determine whether or not the following are equivalence relations. If it is not, then list all parts that fail. a.) S = Integers, arb iff a b mod 7 b.) S = integers, arb iff a - b = 7 c.) S = {students at HSU}, arb iff both are under six feet tall d.) S = {students at HSU}, arb iff both are under six feet tall, both over six feet tall or both six feet tall e.) S = R #, arb iff a/b 1

3 Test #2 1.) Define the following terms. a.) subgroup b.) order of a group c.) order of an element in a group d.) improper subgroup e.) trivial subgroup f.) cyclic group g.) subgroup generated by an element of a group h.) permutation i.) cycle j.) transposition k.) S A l.) S n m.) A n n.) symmetric group on n elements o.) alternating group on n elements p.) D 4 q.) orbit of an element of a group r.) generator of a cyclic group 2.) True-False. Write out the word completely. Write "true" only if the statement is always true. If it is not always true, write "false" and give an example to show that it is false or a reason from a definition or theorem to show that it is false. a.) Every group has at least one subgroup. b.) Every group has at least one proper subgroup. c.) Every group has at least one non-trivial subgroup. d.) Every group is cyclic. e.) Every subgroup of a group is cyclic. f.) Every group has at least one cyclic subgroup. g.) If a group is cyclic then every element in the group is a generator for that group. h.) If a group has no proper, non-trivial subgroups then it is cyclic. i.) Every quadratic equation has at most two solutions in a group. j.) Every quadratic equation has at least one solution in a group. k.) Cycles commute. l.) Cycles commute if and only if they are disjoint. m.) Review all true-false problems from the book. n.) Every non-cyclic group has at least three cyclic subgroups. 3.) Prove the following. a.) H is a subgroup of G if and only if whenever a, b are elements of H then ab -1 is in H. b.) All cyclic groups are abelian. c.) Subgroups of cyclic groups are cyclic. d.) Suppose H and K are subgroups of a group G. Then H K is also a subgroup of G. e.) Be able to do all of the theory problems from the book. 4.) Give examples of the following, if possible. If it is not possible, tell why not. a.) A non-abelian cyclic group. b.) A group with no non-trivial, proper subgroups. c.) An abelian group that is not cyclic. d.) A non-cyclic group of order 720. e.) A non-abelian group of order 360.

4 f.) An infinite group that is not cyclic. g.) A direct product of three groups, each with more than one element, that is cyclic. h.) Two finite groups with the same number of elements (more than six) that are not isomorphic. If such an example exists, give a reason they are not isomorphic. i.) A non-abelian group of order ) Determine whether the sets, H, are sub-groups of the group, G. a.) H = {differentiable 1-1 functions}, G = {1-1 functions from R to R} where our operation is function addition. b.) H = 3Z, G = Z c.) Others from the text 6.) Computational stuff. a.) Find the order of each of the following. (i) the element 4 in the group Z 14 (ii) the group S 6 (iii) the group A 6 (iv) the element in S 5 b.) Determine whether or not the following are cycles. (i) (ii) (iii) c.) Write the following permutations in cycle notation. Then write them as a product of transpositions. (i) (ii) d.) Perform the following multiplications and write the answer in cycle notation. (i) (ii) (iii) ( )(3 5 8) (iv) ( )( ) (v) ( )(2 8 9)( ) e.) Construct the group table for D n, for n = 3, 4, 5. f.) Find q and r as in the division algorithm when n is divided by m. (i) n = 28, m = 5 (ii) n = -37, m = 11 g.) Find the following. (i) (28, 36) (ii) (480, 560) (iii) (iv) (v) number of generators of Z 24 (vi) the subgroup lattice for Z 48 h.) Find the order of the element 18 in the group Z 44. i.) Find q and r as in the division algorithm when n is divided by m. n = 35, m = 6

5 j.) Find the number of generators of Z 77. k.) Find the order of each of the following. (i) the element 8 in the group Z 34 l.) Find the order of (4, 3) in Z 10 x Z 9. m.) Find all abelian groups, up to isomorphism, of order 1800 = ) True-False. Write out the word completely. Write "true" only if the statement is always true. If it is not always true, write "false" and give an example to show that it is false or a reason from a definition or theorem to show that it is false. a.) Every group has at least one proper subgroup. b.) Every element of a group generates a subgroup of the group that is cyclic. c.) If a group is cyclic then it has at least two generators. d.) If a group has no proper, non-trivial subgroups then it is cyclic. e.) A subgroup may be defined as a subset of a group. f.) If every proper subgroup of a group is cyclic, then the group is cyclic. g.) If every proper subgroup of a group is abelian, then the group is abelian. h.) A direct product of cyclic groups is cyclic. i.) G 1 x G 2 is abelian if and only if G 1 and G 2 are both abelian. j.) Suppose G = n, G is cyclic and k n. Then G has an element of order k. k.) Z 10 x Z 9 is a cyclic group. l.) The left and right cosets of a subgroup, H, in a group, G, are always the same. 8.) Prove the following. a.) H is a subgroup of G if and only if whenever a, b are elements of H then ab -1 is in H. b.) All cyclic groups are abelian. c.) Subgroups of cyclic groups are cyclic. 9.) Prove or disprove the following statement. Let H G. If ah = bh then b ε ah. 10.) Prove the following. a.) The direct product of two groups is abelian if and only if each group is abelian. b.) Let H be a subgroup of a group G such that g -1 hg ε H for every h ε H and every g ε G. Then for any g ε G, the left coset gh is the same as the right coset Hg. c.) If H G 1 and K G 2, then H x K G 1 x G 2. d.) If G = n, then g n = e for all g ε G. 11.) Give examples of the following, if possible. If it is not possible, tell why not. a.) An abelian group that is not cyclic. b.) A non-abelian group of order ) Determine whether or not the following sets, H, are sub-groups of the given group, G. a.) H = Q +, G = R + where the operation is multiplication.

6 b.) H = 4Z, G = Z 13.) Computational stuff. a.) Find the order of the element 6 in the group Z 16. b.) Write the following permutation in cycle notation. Then write it as a product of transpositions c.) Perform the following multiplications and write the answer in cycle notation. Is it an even permutation or an odd permutation? ( )(3 5 2) d.) Find q and r as in the division algorithm when n is divided by m. (i) n = 37, m = 4 (ii) n = -29, m = 18 e.) Find the number of generators of Z 77.

7 Test #3 TRUE-FALSE - IF FALSE, GIVE AN EXAMPLE TO PROVE IT a.) Any group is isomorphic to a group of permutations on some set of integers. b.) There are no homomorphisms of Z 9 into Z 16. c.) There are no homomorphisms of Z 9 onto Z 3. d.) There are no homomorphisms of Z 9 onto Z 18. e.) If Φ is an isomorphism from G onto G' and σ is a homomorphism from G' onto G", then σφ is not an isomorphism of G onto G". f.) A n is a normal subgroup of S n. g.) There are no non-trivial homomorphisms of Z 9 onto Z 3. h.) Every factor group of an abelian group is abelian. i.) Every factor group of a non-abelian group is non-abelian. j.) There is a homomorphism of a group of 146 elements into a group of 13 elements. k.) If H is a subgroup of G and G/H is isomorphic to G then H = {e}. l.) If (m, n) = 1 then there are no non-trivial homomorphisms from Z m into Z n. m.) Any group is isomorphic to a group of permutations. n.) There are no non-trivial homomorphisms of Z 10 into Z 16. o.) There are no homomorphisms of Z 12 onto Z 18. p.) Every factor group of an abelian group is abelian. q.) Every factor group of a non-abelian group is non-abelian. r.) Every field is an integral domain. s.) A ring is finite if and only if its characteristic is non-zero. t.) Every field is a ring. u.) An integral domain can only be a field if it is finite. v.) Multiplication in a field is commutative. w.) There are exactly 6 different isomorphisms of Z 18 into Z 18. x.) There are exactly 3 homomorphisms of Z 6 into Z 9. y.) There are no non-trivial homomorphisms of Z 12 into Z. GIVE AN EXAMPLE IF POSSIBLE. IF NOT POSSIBLE, WHY NOT? a.) A homomorphism from a group, G, into a group, G', where G is abelian and G' is not. b.) An abelian group in which a subgroup is not normal. c.) A group that has a generating set containing only one element. d.) A group that has a generating set containing only two elements. e.) A group that has no generating set. f.) An isomorphism between two groups such that one group is a proper subgroup of the other. g.) Two finite groups with the same order that are not isomorphic. h.) A direct product of three groups, each with more than one element, that is cyclic. i.) Two groups with the same number of elements(more than six) that are not isomorphic. If such an example exists, give a reason they are not isomorphic. j.) A group with no non-trivial normal subgroups other than

8 the group itself. k.) Two nonisomorphic rings <R 1, +, > and <R 2, +, > such that <R 1, +> and <R 2, +> are isomorphic. Why do they fail to be isomorphic? l.) Let R = {-1, 1}. If possible, define two binary operations so that R will be a ring (proof is NOT necessary). If not possible, tell why not. PROVE THE FOLLOWING. 1.) An element a of a ring R is idempotent if a 2 = a. Show that a division ring contains exactly two idempotent elements. 2.) If there is a homomorphism from G onto G', and G is abelian then G' is abelian. Give an example to show that "onto" is necessary. 3.) Suppose Φ is a group homomorphism from a group G into a group G'. Let e be the identity element in G, e' the identity element in G' and a is an element in G. Prove that Φ(e) = e' and Φ(a -1 ) = Φ(a) ) In Z p, a p = a for all a. (Hint: Z p is a field) COMPUTATIONAL STUFF. 2.) Suppose R is a commutative ring with unity and characteristic 3. If a, b ε R, find (a + b) 4. Simplify your answer as much as possible. 3.) Solve the equation 3x 2 - x + 5 = 0 in Z 6. 4.) Determine the set of all units in the ring Z 10. PROVE OR DISPROVE a.) If there is a homomorphism from G onto G', and G' is abelian then G is abelian. b.) If there is a homomorphism, Φ, from G into G', and G is not abelian then Φ(G) is not abelian. c.) If ah = bh then b ε ah. d.) Two finite groups are isomorphic if and only if they have the same number of elements. e.) A homomorphism is 1-1 if and only the kernel contains the identity only. f.) A homomorphism is 1-1 if and only if it is an isomorphism. g.) If Φ is an isomorphism from G onto G' and σ is a homomorphism from G' onto G", then σφ is an isomorphism of G onto G". h.) If Φ is an isomorphism from G onto G' and σ is a homomorphism from G' onto G", then σφ is an homomorphism of G onto G". i.) The non-zero reals under multiplication is isomorphic to the non-zero complex numbers under multiplication. j.) Let Φ be a group homomorphism from a finite group G onto a group G'. Then Φ is an isomorphism if and only if G = G'. PROVE a.) If there is a homomorphism from G onto G', and G is abelian then G' is abelian.

9 b.) Give an outline of the proof of Cayley's theorem. Give the definition of the subgroup of S G and show it is a subgroup. c.) The direct product of two groups is abelian if and only if each group is abelian. d.) If a subgroup, H, of a group, G, has index 2 in G, then H is a normal of G. e.) Let H be a subgroup of a group G such that g -1 hg ε H for every h ε H and every g ε G. Then for any g ε G, the left coset gh is the same as the right coset Hg. f.) If g has order n in G and g' has order m in G' then (g, g') has order k in G x G' where k is the least common multiple of n and m. g.) If (m, n) = 1 then there are no non-trivial homomorphisms from Z m into Z n. h.) Show that the property of being cyclic is a structural property. i.) The composition of two isomorphisms is itself an isomorphism. j.) If G is abelian then any subgroup of G is normal. k.) Suppose Φ is a group homomorphism from a group G into a group G'. Let e be the identity element in G, e' the identity element in G' and a is an element in G. Prove that Φ(e) = e' and Φ(a -1 ) = Φ(a) -1. l.) Suppose Φ is a group homomorphism of G onto G' and ker(φ) = {e} where e is the identity element of G. Then Φ is an isomorphism. DEFINE a.) direct product of groups G and G' b.) homomorphism c.) isomorphism d.) surjection e.) direct sum of groups G and G' f.) normal subgroup g.) index h.) right coset of subgroup H i.) factor group j.) kernel of a homomorphism k.) image of A _ X under Φ where Φ is a mapping of X into Y l.) trivial homomorphism m.) structural property n.) automorphism o.) quotient group DETERMINE IF THE FOLLOWING FUNCTIONS ARE HOMOMORPHISMS a.) Φ:R R where Φ(x) = x b.) Φ:R R where Φ(x) = 2x c.) Φ:Z Z 2 where Φ(x) = x mod 2 d.) Φ:S n Z 2 where Φ(σ) = 0 if σ is odd 1 if σ is even e.) Φ:Z 16 Z 5 where Φ(x) = l(x mod 5) and l = 5/(16,5) f.) Φ:Z 6 Z 15 where Φ(x) = kl where k = x mod 3 and l = 5

10 DETERMINE IF THE FOLLOWING FUNCTIONS ARE ISOMORPHISMS, HOMOMORPHISMS OR NEITHER. a.) Φ:Z 2Z where Φ(x) = 2x b.) Φ:Z 8 Z 6 where Φ(x) = 0 if x is even 3 if x is odd c.) Φ:R R where Φ(x) = x 2 CALCULATIONAL STUFF a.) Find all cosets of 3Z in Z. b.) Find all cosets of H in S 4 where H = <( )>. c.) Find the index of the subgroup in b.). d.) Find a smallest generating set for Z 4 x Z 6. Note: A smallest generating set is a generating set in which no proper subset is also a generating set. e.) Find a smallest generating set for Z 4 x Z 7. f.) Find the order of (2,3) in Z 4 x Z 9. g.) Find the order of (2,3) in Z 12 x Z 12. h.) How many homomorphisms are there from Z 12 into Z 12? i.) How many isomorphisms are there from Z 12 onto Z 12? j.) How many homomorphisms are there from Z 12 onto Z 9? k.) How many isomorphisms are there from Z 12 onto Z 9? l.) Find the order of the factor group Z 6 /<2>. m.) Find the order of the factor group (Z 6 x Z 9 )/<4,3>. n.) Find the order of the factor group Z/17Z. o.) Find all cosets of H in S 3 where H = <(1 2 3)>. p.) Find a smallest generating set for Z 4 x Z 8 and for Z 4 x Z 9. q.) Find the order of (2,3) in Z 10 x Z 12. r.) How many homomorphisms are there from Z 12 onto Z 9? Why? s.) Find the order of the factor group (Z 6 x Z 12 )/<4,3>. STATE THE FOLLOWING THEOREMS a.) Fundamental Theorem of Finite Abelian Groups b.) Cayley's Theorem c.) Theorem of Lagrange d.) Fundamental Homomorphism Theorem

11 Final Exam 1.) Define the following terms. a.) normal subgroup b.) group c.) ring d.) ring isomorphism e.) unit f.) field 2.) True-False. Write out the word completely. Write "true" only if the statement is always true. If it is not always true, write "false" and give an example to show that it is false or a reason from a definition or theorem to show that it is false. 1.) A ring is finite if and only if its characteristic is finite. 2.) A ring is finite if and only if its characteristic is non-zero. 3.) Addition in every ring is commutative. 4.) {a + b 2 : a, b ε Z} with the usual addition and multiplication is a ring. 5.) A Sylow p-subgroup of a finite group G is normal in G if and only if it is the only Sylow p-subgroup of G. 6.) An integral domain can only be a field if it is finite. 7.) C with the usual multiplication is a group. 8.) A n is a normal subgroup of S n. 9.) An equation of the form a * x * b = c always has a unique solution in a group. 10.) An abelian group can't be isomorphic to a nonabelian group. 11.) All cyclic groups are abelian. 12.) Any group is isomorphic to a group of permutations. 13.) Any group of order 16 is not simple. 14.) Any group of order 37 is simple. 15.) Any two Sylow p-subgroups of a finite group are conjugate. 16.) Any two groups of three elements are isomorphic. 17.) Every subgroup of a group is cyclic. 18.) Every group has at least one cyclic subgroup. 19.) Every group has at least one subgroup. 20.) Every factor group of an abelian group is abelian. 21.) Every factor group of a non-abelian group is non-abelian. 22.) Every field is a ring. 23.) Every ring has a multiplicative identity. 24.) Every ring with unity has at least two units. 25.) Every ring with unity has at most two units. 26.) Every element in a ring has an additive inverse. 27.) Every skew field is an integral domain. 28.) Every finite integral domain is a field. 29.) Every group is a subgroup of itself. 30.) Every subgroup of an abelian group G is a normal subgroup of G. 31.) If a group is cyclic then every element in the group is

12 a generator for that group. 32.) If a group has no proper, non-trivial subgroups then it is cyclic. 33.) If (m, n) = 1 then there are no non-trivial homomorphisms from Z m into Z n. 34.) If Φ is an isomorphism from G onto G' and σ is a homomorphism from G' onto G", then σφ is an isomorphism of G onto G". 35.) If p is prime, then Z p is a field. 36.) If p is prime, then Z p has no divisors of zero. 37.) If G is a group with identity element e, then it is impossible for there to be more than one solution in G to the quadratic equation x 2 = e. 38.) If G is a finite group with identity e and a ε G, then there exists a positive integer n such that a n = e. 39.) If R is a ring with unity then R is a division ring. 40.) If R is a ring with unity, then this unity 1 is the only multiplicative identity. 41.) Multiplication in a field is commutative. 42.) nz with the usual addition and multiplication is a ring. 43.) R + with the usual multiplication is a group. 44.) Q with the usual multiplication is a group. 45.) The set of all pure imaginary complex numbers ri for r ε R with the usual multiplication and addition is a ring. 46.) The nonzero elements of a field form a group under the multiplication in the field. 47.) The nonzero real numbers with the usual multiplication is a group. 48.) The set of all continuous functions from R to R, with composition as its operation, is a group. 49.) The empty set can be considered a group. 50.) The only groups that are abelian are ones in which x * x = e for all x in G. 51.) There are no non-trivial homomorphisms of Z 10 into Z ) There are no homomorphisms of Z 9 onto Z ) There are no homomorphisms of Z 12 onto Z ) There is a homomorphism of a group of 146 elements into a group of 13 elements. 55.) There are exactly 6 different isomorphisms of Z 18 into Z ) There are exactly 3 homomorphisms of Z 6 into Z ) There are no non-trivial homomorphisms of Z 12 into Z. 58.) Z + with the usual addition and multiplication is a ring. 59.) Z 5 with addition and multiplication modulo 5 is a ring. 60.) Z 6 with addition and multiplication modulo 6 is a field. 61.) Z is a subfield of Q. 62.) Z with the usual multiplication is a group. 63.) Z with the usual subtraction is a group. 3.) Prove the following. a.) The direct product of two integral domains is not an integral domain. b.) All cyclic groups are abelian.

13 c.) Every field is an integral domain. 4.) Determine whether the indicated operations of addition and multiplication are defined on the set and give a ring structure. If a ring is not formed, tell why not. If a ring is formed, state whether the ring is commutative, has unity, or is a field. R = {a + b 2 a, b ε Q} with the usual addition and multiplication 5.) Determine the set of all units in the ring Z 8. 6.) Determine if the following function is a group homomorphism, group isomorphism or neither. Φ:R R where Φ(x) = x ) Determine whether or not the following are groups. If it is a group, determine whether or not it is abelian. G = {1, -1, i, -i}, where i = (-1), a * b = ab 8.) Computational stuff. a.) An element a of a ring R is nilpotent if a n = 0 for some n ε Z +. Find the set of all nilpotent elements in Z 9. b.) Suppose R is a commutative ring with unity and characteristic 4. If a, b ε R, find (a + b) 4. c.) Solve the equation 2x 2-3x + 5 = 0 in Z 8. d.) Find the order of (6,2) in Z 8 x Z 12. e.) Perform the following multiplication and write the answer in cycle notation. ( )(3 5 8) 9.) Give examples of the following, if possible. If it is not possible, tell why not. a.) Two nonisomorphic rings <R 1, +, > and <R 2, +, > such that <R 1, +> and <R 2, +> are isomorphic. Why do they fail to be isomorphic? b.) A homomorphism from a group, G, into a group, G', where G is abelian and G' is not. c.) Two finite groups with the same order that are not isomorphic. d.) A non-abelian cyclic group. 10.) State the following theorems. a.) Cayley's Theorem b.) Theorem of Lagrange 11.) PROVE OR DISPROVE Every integral domain is a field. 12.) We know that if p and q are primes then the cyclic group Z pq has pq - p - q + 1 generators. State and prove a similar result for Z pqr where p, q and r are primes. 13.) If m n then there exists a homomorphism from Z n onto Z m. 14.) Every factor group of a cyclic group is cyclic. 15.) Every group has at least one cyclic subgroup. 16.) If x * x = e for all x in a group G, then the group is abelian. 17.) If there is a homomorphism from G onto G', and G is abelian then G' is abelian. 18.) If a subgroup, H, of a group, G, has index 2 in G, then H is a normal of G. 19.) Find all abelian groups, up to isomorphism, of order 3600.

14 20.) Find the subgroup lattice for Z ) An element a of a ring R is idempotent if a 2 = a. Show that a division ring contains exactly two idempotent elements. 22.) Prove that no group of order n is simple for each of the following integers. a.) n = 32 b.) n = ) Prove that if there is a homomorphism from G onto G', and G is abelian then G' is abelian. 24.) Every field is an integral domain. 25.) Determine whether or not the following is a group. If it is a group, determine whether or not it is abelian. G = {1, -1, i, -i}, where i = (-1), a * b = ab 26.) Find the order of (4,8) in Z 6 x Z ) An element a of a ring R is nilpotent if a n = 0 for some n ε Z +. Find the set of all nilpotent elements in Z ) Suppose R is a commutative ring with unity and characteristic 2. If a, b ε R, find (a + b) ) Find all abelian groups, up to isomorphism, of order 360.

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