Answer to Question #87219 in Abstract Algebra for rohan

Question

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 .

Answer 1

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