State whether each of the following is a group

State whether each of the following is a group.
(a) All the real numbers with the rule of combination for forming the product of two elements being addition.
(b) All positive integers with the rule of combination being multiplication.
(c) All real numbers except zero, with the rule of combination being multiplication.

Expert Solution

(a) Since the sum of two real numbers is a real number, closure is satisfied. The identity element is 0. The inverse is the negative of the real number. (0 is its own inverse.) Addition is associative. Hence this is a group.

(b) The identity element is 1. The inverse of 2 is 1 / 2, which is not an element in the proposed group, so this is not a group.

(c) Multiplication is associative. The identity element is 1. The inverse is the reciprocal of the number, and every member has an inverse. The product of two real nonzero numbers is a real nonzero number, so closure is satisfied.

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