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In the following determine whether the systems described are groups. If they are not, point out...

In the following determine whether the systems described are groups. If they are not, point out which of the group axioms fail to hold.

(a) G = set of all integers, a· b = a - b.

(b) G = set of all positive integers, a · b = ab, the usual product of integers.

(c) G = a0 , a 1 , ... , a6 where ai · a i = ai + i if i + j < 7, ai · a i = ai + i _ 7 if i + j ;:::: 7 (for instance, a5 • a 4 = a 5 +4 _ 7 = a2 since 5 + 4 = 9 > 7).

(d) G = set of all rational numbers with odd denominators, a· b _ a + b, the usual addition of rational numbers.

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