Answer ALL the questions
1. Define a partially ordered set and give an example.
2. If G is group then prove that the identity element of G is unique.
3. Show that every subgroup of an abelian group is normal.
4. Define order of an element in a group.
5. Express (1,3,5) (5,4,3,2)(5,6,7,8) as a product of disjoint cycles.
6. Define an automorphism of a group with an example.
7. Define an integral domain and give an example.
8. If F is a field, prove that its only ideals are {0} and F itself.
9. Define maximal ideals.
10. Find all the units in 𝑍[𝑖] = {𝑥 + 𝑖𝑦 ∶ 𝑥, 𝑦 ∈ 𝑍}.
Answer any FIVE questions
11. If H and K are subgroups of G, then prove that HK is a subgroup of G if and only if HK=KH.
12. Prove that every subgroup of a cyclic group is cyclic.
13. Prove that a subgroup N of a group G is a normal subgroup of G if and only if the product of two
left cosets of N in G is again a left coset of N in G.
14. If f is a homomorphism of a group G into a group G then prove that kernel of f is a normal
subgroup of G.
15. Prove that every group is isomorphic to a group of permutations.
16. Let Q be the set of all rational numbers and let 𝑄(√2 ) = {𝑎 + 𝑏√2: 𝑎, 𝑏 ∈ 𝑄}. Show that
𝑄(√2 ) is a field under usual addition and multiplication.
17. Prove that every Euclidean ring is a principal ideal domain.
18. Let R be a commutative ring with unity and P an ideal of R .Then prove that P is a prime ideal of R
if and only if R/P is an integral domain.
Answer any two questions
19. a.) If H and K are finite subgroups of a group G then prove that 0(𝐻𝐾) =
b.)If 𝑎 ∈ 𝐺 and 𝑎 = 𝑒, prove that 0(a) divides n.
20. a.) State and prove Lagrange’s theorem.
b.) If H and K are two subgroups of a finite group G and 𝐻 ⊆ 𝐾 , show that
[𝐺: 𝐻] = [𝐺: 𝐾][𝐾: 𝐻].
(12 + 8)
(12 + 8)
21. a.) State and prove Fundamental theorem of homomorphism of a group G.
b.) Show that an arbitrary intersection of ideals of a ring R is an ideal of R.
(12 + 8)
22. a.)Let R be a Euclidean ring. Then prove that every non-zero element of R is either a unit
in R or can be uniquely written as a product of a finite number of prime elements of R.
b.) Let R be a Euclidean ring. Then prove that any two elements a and b in R have a
greatest common divisor d which can be expressed in the form 𝜆𝑎 + 𝜇𝑏 for some 𝜆, 𝜇 ∈ 𝑅.
(12 + 8)