What do you mean by interactivity in educational communication? Explain the different levels of interactivity with suitable examples.

Select a topic on which you would like to write a Self-Learning Material (SLM). Briefly write an SLM by clearly indicating the introductory (first) part, main (middle) part, and concluding (end) part.

 Discuss the pedagogical potential of audio and video programmes in the teaching-learning process.

 What do you mean by learning design? Explain it with a suitable example.

 Construct a field with 125 elements.  

Check whether or not  is a field.  

Find all the units of .  

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 .
ii) If R is an integral domain and I is an ideal of R, then .
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.

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  is a ring isomorphic to R.  
PART-C (MM: 20 Marks)
(Based on Block 4.)

Is , for any two ideals I and J of a ring R? Give reasons for your answer. 

For an ideal I of a commutative ring R, define
. Show that
i)  is an ideal of R.
ii) .
iii)  in some cases. 

 For an ideal I of a commutative ring R, define
. Show that
i)  is an ideal of R.
ii) .
iii)  in some cases. 

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.
iv) If  is a ring homomorphism, then it is a group homomorphism from (R, +) to (S, +).
v) If R is a ring, then any ring homomorphism from  into R is surjective.

 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
i) 
ii) 

iii)   
The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’.

PART-B (MM: 30 Marks)
(Based on Block 3.)

 Give the smallest  for which A_n is non-abelian. Justify your answer. 

List two distinct cosets of  in D₁₀, where r is a reflection in D₁₀. 

Let  be a fixed odd permutation in S₁₀. Show that every odd permutation in S₁₀ is a product of  and some permutation in A₁₀. 

Using Cayley’s theorem, find the permutation group to which a cyclic group of order 12 is isomorphic.

Prove that a cyclic group with only one generator can have at most 2 elements.

Let G be an infinite group such that for any non-trivial subgroup H of . Then prove that
i)  or H is infinite;
ii) If , then o(g) is infinite.