1. Which of the following statements are true? Give reasons for your answers.
i) If a group G is isomorphic to one of its proper subgroups, then G = Z.
ii) If x and y are elements of a non-abelian group (G, *) such that x * y = y * x, then x=e or y=e, where e is the identity of G with respect to *.
iii) There exists a unique non-abelian group of prime order.
v) If H and K are normal subgroups of a group G, then
2. a) Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that (Q+1) is not cyclic.
b) Let G be an infinite group such that for any non-trivial subgroup H of
Then prove that
c) Prove that a cyclic group with only one generator can have at most 2 elements.
3. a) Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.
be a fixed odd permutation in S10. Show that every odd permutation in S10 is a product of
and some permutation in A10.
c) List two distinct cosets of <r> in D10, where r is a reflection in D10.
d) Give the smallest
for which An is non-abelian. Justify your answer.
4. Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma:
Let H and K be subgroups of a group G and H’ and K’ be normal subgroups of H and K, respectively. Then
The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’.
5. Which of the following statements are true, and which are false? Give reasons for your answers.
i) For any ring R and
ii) Every ring has at least two elements.
iii) If R is a ring with identity and I is an ideal of R, then the identity of R/ I is the same as the identity of R.
is a ring homomorphism, then it is a group homomorphism from
v) If R is a ring, then any ring homomorphism from R x R into R is surjective.
6. a) For an ideal I of a commutative ring R, define
for any two ideals I and J of a ring R? Give reasons for your answer.
7. Let S be a set, R a ring and f be a 1-1 mapping of S onto R. Define + and • on S by:
Show that (S, +, •) is a ring isomorphic to R.
8. Which of the following statements are true, and which are false? Give reasons for your answers.
i) If k is a field, then so is k x k.
ii) If R is an integral domain and I is an ideal of R then, Char (R) = Char (R/I).
iii) In a domain, every prime ideal is a maximal ideal.
iv) If R is a ring with zero divisors, and S is a subring of R then, S has zero divisors.
v) If R is a ring and
is of degree
, then f(x) has exactly n roots in R.
9. a) Find all the units of
b) Check whether or not
is a field.
c) Construct a field with 125 elements.