Compute the index of 〈3〉 in ℤ12. Then write down all of the cosets of 〈3〉 in ℤ12. Finally, write a Cayley table for ℤ12/〈3〉. Is this quotient an abelian?
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- Prove or disprove that the set of all diagonal matrices in Mn() forms a group with respect to addition.True or False Label each of the following statements as either true or false. 11. The invertible elements of form an abelian group with respect to matrix multiplication.Let G be an abelian group of order 2n, where n is odd. Use Lagranges Theorem to prove that G contains exactly one element of order 2.
- Let H1={ [ 0 ],[ 6 ] } and H2={ [ 0 ],[ 3 ],[ 6 ],[ 9 ] } be subgroups of the abelian group 12 under addition. Find H1+H2 and determine if the sum is direct.True or False Label each of the following statements as either true or false. 6. The set of all nonzero elements in is an abelian group with respect to multiplication.Exercises 11. According to Exercise of section, if is prime, the nonzero elements of form a group with respect to multiplication. For each of the following values of , show that this group is cyclic. (Sec. ) a. b. c. d. e. f. 33. a. Let . Show that is a group with respect to multiplication in if and only if is a prime. State the order of . This group is called the group of units in and designated by . b. Construct a multiplication table for the group of all nonzero elements in , and identify the inverse of each element.