Are the integers a group under addition? How about multiplication?

There are 4 things we need for a group: associativity, the existence of an identity, inverses in the group and closure. The integers are definitely associative under this operation as addition is associative as a + (b+c) = (a +b) + c. the identity exits as 0 is an integer and for any integer A, A + 0 = A. The inverse exists in the integers as if A is in the integers, - A is too and A + (-A) = 0 = identity, and finally it is also closed as for two integers A and B, A + B is also an integer. Therefore it is a group.For multiplication it is not a group, as the identity for multiplication on the integers is 1, but say we choose an integer A, then the inverse is 1/A as A * 1/A = 1 = identity, but 1/A is in general not an integer, so the integers under multiplication do not form a group.