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1 (7 pages) MAY 011 Time : Three hours Maximum : 100 marks PART A (10 3 = 30 marks) Answer any TEN questions. 1. If H is a subgroup of G and N is a normal subgroup of G, show that subgroup of H. H N is a normal G GßÓ S»zvÀ H EmS», N ºø EmS» GÛÀ H N, H À ºø EmS» GÚ {¹.. Define automorphism of a group. Give an example. S»zvß uß J ø ø Áøµ Özx GkzxUPõmk u P. a b 3. Let G be the set of all matrices 0 d where ad 0 under matrix multiplication. Let 1 b N = prove that N is a normal subgroup 0 1 of G.

2 a b G = ad 0 AoPÎß ö UPø» 0 d 1 b ö õözx G J S» ØÖ N = GÛÀ N 0 1 J ºø EmS» GÚ {¹ Find the orbit and cycles of PõsP ß JÊUS ØÖ _ÇÀPøÍU 5. Determine the conjugacy class of (1, ) in S 3. S 3 À EÒÍ (1, ) ß Cøn ÁS ø U PõsP. 6. Define a division ring. Give an example. ÁSzuÀ ÁøÍ Áøµ Ö. GkzxUPõmk JßÖ u P. 7. Find all ideals of the ring (, + ) (, + ) 6 6, 6 z. 6 6, 6 z GßÓ ÁøÍ zvß GÀ»õ ^º [PøͲ Psk i.

3 8. Prove that (1, 0, 0), (1, 1, 0), (0, 0, ) are linearly (3) independent in R where R is the set of reals. (3) R Gß x ö ö spîß Pn R À (1, 0, 0), (1, 1, 0) ØÖ (0, 0, ) Gß x À \õº ØÓøÁ GÚ {ÖÄP. 9. Define inner product space. Give an example. EÒ ö UPÀ öáî ß Áøµ øó u P. GkzxUPõmk u P. 10. If V is finite dimensional over f and if T A(V ) is singular, then there exist an S 0 in A (V ) such that ST = TS = 0. V Gß x iäö Ai õn Eøh x ØÖ T A(V ) ØÖ J ø õúx GÛÀ ST = TS = 0 GßÖ Aø ² õö A (V ) À S IU Põn i² GßÖ {¹. 11. If T, S A( V ) and if S is regular, prove that T and 1 ST S have the same minimal polynomial. T, S A( V ) ØÖ S JÊ[Søh x GÛÀ T ØÖ 1 ST S Gß x J µ ]Ö À¾Ö U PõøÁ Eøh Ú GÚ {ÖÄP. 3

4 1. Let T A( R ) find the matrix of T defined by T( x, y) = (x + 3y, 4x y) with respect to the basis (1, 0) and (0, 1). T A( R ) ØÖ T( x, y) = (x + 3y, 4x y). (1, 0) ØÖ (0, 1) Eøh Ai õnzøu ö õözx T ß Aoø U PõsP. PART B (5 6 = 30 marks) Answer any FIVE questions. 13. If G is a finite group and N is a normal subgroup G O( G) of G, prove that O =. N O( N) G GßÓ iäö S»zvß J ºø EmS» N G GÛÀ O = N O( G) GßÖ {ÖÄP. O( N) 14. Show that Kernel of a group homomorphism is a normal subgroup. J S» ö\ ö»õ ø ß EmP J ºø EmS»ö Ú {ÖÄP. 15. Let φ be a homomorphism of G onto G with Kernel K. Let N be a normal subgroup of G G G N = { x G / φ ( x) N }. Prove that. N N 4 [P.T.O.]

5 φ : G G S» J ø ØÖ K Auß EmP. N Gß x G ß ºø EmS». ¾ N = { x G / φ ( x) N } GÛÀ G G GßÖ {ÖÄP. N N 16. Prove that N (a) is a subgroup of G. N (a) G ß EmS» GÚ {¹. 17. Let R be a commutative ring with unit element and M an ideal of R. If M R is a field prove that M is a maximal ideal of R. R J A»S Eøh õøö ÁøÍ. M Auß R ^º BS. J PÍ GÛÀ M J «ö M ^º GßÖ {¹. 18. Let R be a Euclidean ring. Suppose that for a, b, c R, a / bc but ( a, b) = 1. Then prove that a / c. R Gß x ³U¼i ß ÁøÍ. a, b, c R À a / bc ØÖ ( a, b) = 1 GÛÀ a / c GÚ {¹. 5

6 19. Prove that any two finite dimensional vector spaces over F of the same dimension are isomorphic. iäö Ai õn öpõsh öáuhº öáî À \ õú Ai õn öpõshøá GÛÀ AøÁ J ø Eøh Ú GÚ {¹. PART C (4 10 = 40 marks) Answer any FOUR questions. 0. State and prove Cayley s theorem. öp ¼ ß uøózøu TÔ {ÖÄP. 1. Derive class equation. ÁS a \ ß õmøh Á ÂUP.. Show that for a prime p,, +, ) is a field. ( p p p p J Põ Gs GÛÀ, +, ) J PÍö Ú {ÖÄP. ( p p p 3. State and prove unique factorization theorem on Euclidean ring. ³U¼i ß ÁøÍ zvà J µ Põµo uøózøu TÔ {ÖÄP. 6

7 4. If V is a finite-dimensional inner product space, prove that V has an orthonormal basis. V Gß x iäö Ai õn Eøh J EÒö UPÀ öáî V US J ö Ô ö\[szx A»S i õn EÒÍx GÚ {ÖÄP. 5. If V is n-dimensional over F and if T A(V ) has all its characteristic roots in F prove that T satisfies a polynomial of degree n over F. V Gß x F À iäö Ai õn Eøh x T A(V ) ß GÀ»õ ]Ó»[PÐ F À EÒÍÚ GÛÀ T BÚx F À n i Eøh À¾Ö U PõøÁø {øóä ö\ ² GßÖ {¹. 7