Question

1. Which of the following statements are true? Justify your answers. (This means that if you
think a statement is false, give a short proof or an example that shows it is false. If it is
true, give a short proof for saying so.) (20)
i) f(n) = n −1 "nÎN, where f is the Euler-phi function.
ii) If 1 G and 2 G are groups, and 1 2 f :G ®G is a group homomorphism, then
o(G ) o(G ) 1 2 = .
iii) If G is an abelian group, then G is cyclic.
iv) If G is a group and HDG , then | G: H | = 2 .
v) Every element of n S has order at most n .
vi) If R is a ring and I is an ideal of R , then xr = rx " xÎI and rÎR .
vii) If S (n 3) n sÎ ³ is a product of an even number of disjoint cycles, then sign (s) =1.
viii) If a ring has a unit, then it has only one unit.
ix) The characteristic of a finite field is zero.
x) The set of discontinuous functions from [0, 1] to R form a ring with respect to pointwise
addition and multiplication.