BMTC-134 English Medium Solved Assignment 2026

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BMTC-134 English Medium Solved Assignment 2026 Available

Part-A (MM: 50 Marks) (To be done after studying Blocks 1 and 2.)

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 xy = yx then x = c or y = e where e is the identity of G with respect to ..

iii) There exists a unique non-abelian group of prime order.

iv) If where A is a group, then o((a, b)) = o(a) * o(b) (a, b) \in AA

v) If H and K are normal subgroups of a group G, then hk=kh vee heH . k \in K

2. a) Prove that every non-trivial subgroup of a cyclic group has finite index. Hence prove that (Q. +) is not cyclic. (7)

b) Let G be an infinite group such that for any non-trivial subgroup H of G. |G / H| < ∞ Then prove that

1) H<= G Rightarrow H = \{e\} or H is infinite;

ii) If g \in G g ne e. then o(g) is infinite.

c) Prove that a cyclic group with only one generator can have at most 2 elements. (3)

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

b) Lett be a fixed odd permutation in S_{10} Show that every odd permutation in S_{10} is a product of t and some permutation in A (2)

c) List two distinct cosets of < I > in D_{10} where r is a reflection in D_{10}

(2) d) Give the smallest n \in N for which A * 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

1) H’ (H cap K^ prime ) triangleleft H^ prime (H cap K)

ii) K’ (H^ prime cap K) triangleleft K^ prime (H cap K)

im) H’ (H cap K) H^ prime (H cap K^ prime ) = K^ prime (H cap K) K^ prime (H^ prime cap K) = (H cap K) (H^ prime cap K)(H cap K^ prime )

(15)

The situation can be represented by the subgroup diagram below, which explains the name “butterfly”.

K

H’ (H cap K)

H

H cap K

H’ (H cap K^ prime )

HOK

K’ (H cap K)

K'(HOK)

L=(HOK)(HOK)

HOK’

Part-B

5. Which of the following statements are true, and which are false? Give reasons for your answers

1) For any ring R and a, beR, (a + b) ^ 2 = a ^ 2 + 2ab + b ^ 2

in) 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 f:RS is a ring homomorphism, then it is a group homomorphism from (R,+) to (S, +).

If R is a ring, then any ring homomorphism from RR into R is surjective. v)

6. a) For an ideal 1 of a commutative ring R, define sqrt(I) =\ x in mathbb R |x^ * in I for some ne N). Show that

1) VI is an ideal of R.

1 subset sqrt(t) .

iii) 1 = sqrt(1) in some cases.

b) R/I * R/J = (RR)/(IJ) for any two ideals I and I of a ring R? Give reasons for your answer.