Note: Answer all questions.
9. (a) Every subgroup of a cyclic group is cyclic more over if |< a>|=n then the order of any subgroup of < a > is a divisor of n and for each positive divisor k of n, the group < a > has exactly one subgroup of order K namely <an/k>.
(b) Prove that the Alternating group An has order n!/2 if n>1.
--- Content provided by FirstRanker.com ---
10. (a) Prove that the group of rotations of a cube is isomorphic to S4.
(b) Let G be a group and let G/Z(G) is cyclic then G is abelian.
1. State and prove Lagrange theorem.
--- Content provided by FirstRanker.com ---
2. (a) Let G be a group and e be an identity element. If a,b ? H and a-1b ? H, then prove that H is a subgroup of G.
3. (a) Let R be a commutative ring with unity. Prove that A is an ideal of R if and only if R/A is an integral domain.
(b) If D is an integral domain then D[x] is an integral domain.
--- Content provided by FirstRanker.com ---
4. (a) If R is a ring with unity and the characteristic of R is 0 then R contains a subring isomorphic to Q. If the characteristic of R is n>0 then R contains a subring isomorphic to Zn.
(b) Let R[x] denotes the subset of all polynomials with real coefficients and let A denote the set of all polynomials with constant term 0 then prove that A is an ideal of R.
5. (a) Let f: R ? R' be a ring homomorphism. Show that f-1(0) is an ideal of R.
--- Content provided by FirstRanker.com ---
(b) Let S = {(_a b) / a,b ? R}. Show that f: R[x] ? S defined by f(a+bx) = (_a b) is a ring isomorphism.
--- Content provided by FirstRanker.com ---
This download link is referred from the post: OU B-Sc Last 10 Years 2010-2020 Question Papers || Osmania University