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1. Which of the following statements are true? Give reasons for your answers. i) If f : S->S is 1-1, where S is a set, then f is onto. ii) The signature of the product of k disjoint cycles in S_n is k . iii) R[x] has no maximal ideal. iv) If m| n in Z , then mZ is an ideal of Z/ nZ . v) Every non-trivial subgroup of an infinite group is infinite. vi) The characteristic of the quotient field of any UFD is zero. vii) If H and K are subgroups of a group G , then HK is also a subgroup of G . viii) If the order of a group is even, then it cannot have an element of odd order. ix) If R is an integral domain, then R / I has no zero divisors, where I is an ideal of R .
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