5. Show that the following code is a group code. (00000) (00101) (10110) (10011)
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- In Exercises 114, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition 3.1 that fails to hold. The set of all multiples of a positive integer n is group with operation multiplication.Let n be appositive integer, n1. Prove by induction that the set of transpositions (1,2),(1,3),...,(1,n) generates the entire group Sn.In Exercises 114, decide whether each of the given sets is a group with respect to the indicated operation. If it is not a group, state a condition in Definition 3.1 that fails to hold. The set of all positive irrational numbers with operation multiplication.
- Show that the set {5, 15, 25, 35} is a group under multiplication modulo 40. What is the identity element of this group? Can you see any relationship between this group and U(8)?Construct the complete group table for K.Create steps along with justifications to verify that in a group system (G, +) the following property holds:For any two elements from G, ‘a’ and ‘b’, -(a + b) = (-b) + (-a). In other words, the claim is that (-b) + (-a) plays the role of the inverse of a + b. Create steps to show that the element (-b) + (-a) does, in fact, play the role of an inverse to the element a + b, i.e., show that:i. (a + b) + ( (-b) + (-a) ) = e, where e represents the identity in the group; andii. ( (-b) + (-a) ) + (a + b) = e.