Is the set {0,-1,1} a group with respect to usual addition of numbers?
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Q: Hi, could you please break this down for me?
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- The set {0, 3, 6, 9, 12, 15, 18, 21} is a group under the binary operation +24. Determine all its distinct subgroups, justifying your answer.Let Zeven represent the set of even integers, {... -6, -4, -2, 0, 2, 4, 6, ...}. Determine if the system (Zeven, +): (i) is closed; (ii) is associative; (iii) has an identity element; (iv) has inverses; and (v) is commutative. Is (Zeven, +) a semi-group? Explain Is (Zeven, +) a group? ExplainCreate 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.
- The character table of every point group contains more than one representation all of whose characters are 1. True False(5.2) Challenge Modern Applied Algebra. Show work.Determine if the following sets with operations are groups: (a) R with the operation given by r * y = -1 - y. (b) The subset of R® given by H = {(r,y, 2) € R* |x+ y+z = 0}, with the operation (x, y, 2) + (a, b, c) = (r +a, y +b, z + c).

