Find all the possible abelian groups, up to isomorphism, of order 900.
To find all possible abelian groups of order 900 up to isomorphism, we can use the fundamental theorem of finitely generated abelian groups. This theorem states that any finitely generated abelian group is isomorphic to a direct product of cyclic groups.
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Check whether or not the ring
i) is finite;
ii) has zero divisors;
iii) has nilpotent elements.
Give two distinct rings whose quotient field is {a ib a, b }. + Justify your answer.
Write as a product of O(3) and an element of B3 (
).
Check whether or not is a Euclidean domain
Construct the free group on the set Further, check if it is a free abelian group or not.
Which of the following statements are true? Give reasons for your answers. Marks will only be given for valid justification of your answers.
(i) If G is a finite abelian group and p is a prime factor of o(G), then the number of Sylow p-subgroups of G is a prime.
ii) The minimum polynomial of 1/3 5 over Q is x1/3 .
iii)
iv) If G is a finite group and m o(G), then G has a subgroup of order m.
v) If and
are two 15th roots of unity, then
vi) There exists an extension field of of order 25.
vii) Every group of order 18 has a normal subgroup of order 2.
viii) If I and J are ideals of a ring R, then IJ =
ix) If f : R S → is a ring homomorphism and I is an ideal of R, then f (I) is an ideal of S.
x) Every prime ideal of an integral domain is a maximal ideal.
Let G be a group of automorphisms of a field K. Is the fixed field KG a subfield of K ? Why, or why not?