This question looks at two groups that arise from arithmetic modulo
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This question looks at two groups that arise from arithmetic modulo

Q 3. This question looks at two groups that arise from arithmetic modulo 14.

(a) Consider the cyclic group Z14 (which has group operation ⊕14 and identity element 0). Calculate 1 ⊕14 13. What is the inverse of 13?

(b) The set {3, 10} is a left coset of a subgroup H of Z14. Find H.

(c) Write down the other left cosets of H in Z14.

(d) Consider the multiplicative group U14 (which has group operation ⊗14 and identity element 1). Calculate 13 ⊗14 13. What is the inverse of 13?

(e) What is the order of the group U14?

(f) What is the order of the group U14×Z14? What is the identity element?

(g) What is the order of the element (5, 5) in U14 × Z14?

Hint
ComputerIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.The modulo (or "modulus" or "mod") is the remainder after dividing one number by another. Example: ...
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