5. [2 marks] Write down the relation matrix of the abelian group Now reduce this matrix using elementary integer row and column operations, and hence write G as a direct sum of cyclic groups. 5. [2 marks] Write down the relation matrix of the abelian group Now reduce this matrix using elementary integer row and column operations, and hence write G as a direct sum of cyclic groups.
4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G 4 (a) Let G be an abelian group with identity e and let H- gEGI8-e. Prove that H is a subgroup of G
Let G = {1, 3, 5, 9, 11, 13} and let represent the binary operation of multiplication modulo 14. (a) Prove that (G, ) is a group. (You may assume that multiplication is associative.) (b) List the cyclic subgroups of (G, ). (c) Explain why (G, ) is not isomorphic to the symmetric group S3. (d) State an isomorphism between (G, ) and (Z6, +).
11. Let G = Z4 Z4, H = {0,0), (2,0), (0,2), (2,2)). Write the Cayley table for G/H. Is G/H isomorphic to Z4 or Z2 x Z ? Justify your answer. 12. Show that G = {1, 7, 17, 23, 49, 55, 65, 71} is a group under multiplication modulo 96. Then express G as an external and an internal direct product of cyclic groups.
22 Must the center of a group be Abelian? 23. Let G be an Abelian group with identity e and let n be some integer Prove that the set of all élements of G that satisfy the equation* - e is a subgroup of G. Give an example of a group G in which the set of all elements of G that satisfy the equation :2 -e does not form a subgroup of G. (This exercise is referred to in...
5. Let N be a normal subgroup of a group G and G/N be the quotient group of all right cosets of N in G. Prove each of the following: (a) (2 pts) If G is cyclic, then so is G/N. (b) (3 pts) G/N is Abelian if and only if aba-16-? E N Va, b E G. (c) (3 pts) If G is a finite group, then o(Na) in G/N is a divisor of (a) VA EG.
5. A field is a set F containing 0 and 1 that is an abelian group under addition, and (upon removing 0) Common examples of fields are abelian group under multiplication, for which the distributative law holds. an Q, R, and C. There is a unique finite field Fg of order q= p for every prime p and positive integer k. For all other q E N, there is no finite field of order g. For each of the fields...
(Abstract Algebra) Please answer a-d clearly. Show your work and explain your answer. (a) Let G be a group of order 4 with identity e. Show that G is either cyclic or a2-e for all (b) Does the result of part (a) generalize to groups of order p2 for any positive integer p? In other words, is it the case that if G is a group of order p2 with identity e, then is either cyclic or a- e for...
QUESTION 8 Let G is an abelian group with the additive operation. Define the operation of multiplication by the rule ab- a- b for all a,b of G. Is G a ring? QUESTION 8 Let G is an abelian group with the additive operation. Define the operation of multiplication by the rule ab- a- b for all a,b of G. Is G a ring?