D.K.M COLLEGE FOR WOMEN (AUTONOMOUS), VELLORE DEPARTMENT OF MATHEMATICS

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1 D.K.M COLLEGE FOR WOMEN (AUTONOMOUS), VELLORE DEPARTMENT OF MATHEMATICS SUB: ALGEBRA SUB CODE: 5CPMAA SECTION- A UNIT-. Defne conjugate of a n G and prove that conjugacy s an equvalence relaton on G. Defne normalzer of a n G and prove that N(a) s a subgroup of G. G) 3. If G s a fnte group then prove that C a= N( a)) 4. Prove that the number of elements conjugate to a n G s the ndex of the normalzer of a n G. 5. State and prove class equaton. G) 6. Prove that G)= N( a)) 7. If a Z f and only f N(a)=G. If G s fnte, a Z ff N(a))= G). 8. If G)=p n, p s a prme number then Z(G) {e}. 9. If G)=p, where p s a prme number then G s abelan.. State and prove Cauchy s theorem for fnte group.. If p s a prme number and p/g) then G has an element of order p.. If p m G), p m+ G) then G has a subgroup of order p m. 3. Prove that n(k)= +p+ +p k-. 4. If A and B are fnte subgroup of G then AxB)= 5. Defne nternal and external drect product A). B) A xbx ) 6. Suppose that G s the nternal drect product of N,N, N n then for j, N N j={e} and f a N, b N j then ab=b 7. Let G be a group and suppose that G s the nternal drect product of N,N, N n. Let T= N XN X XN n then G and T are somorphc. 8. Obtan class equaton of S State and prove thrd part of sylow s theorem.. Prove that the number of p- sylow subgroup n G for a gven prme s of the form +kp. State and prove frst part of sylow s theorem.. State and prove second part of sylow s theorem. 3. State and prove thrd part of sylow s theorem.

2 4. State and prove fundamental theorem on fntely generate R-module. 5. If P s a prme number and p α G) then G has a subgroup of order p α UNIT-II SEC-A. Defne module and gve one example.. Defne solvable and gve one example. 3. G s solvable ff G (k) ={e} for some nteger K. 4. Prove that homomorphc mage of a solvable group s solvable. 5. Prove that subgroup of a solvable group s solvable. 6. If G s solvable group and f G s a homomorphc mage of G then G s solvable. 7. Prove that any fnte abelan group s the drect product of cyclc groups. 8. Prove that S n s not solvable for n Let G= S n, n 5 then G (k), K=,.. Contans every 3-cycle of S n. UNIT-III SEC-A. If W V s nvarant under T then T nduces a lnear transformaton T on W V defned by (V+W) T =VT+W. f satsfes q(x) F[x] then so does T. If P(x) s the mnmal polynomal for T over F and f P(x) s that for T then P x) P ( ). ( x. If V s n-dmensonal over F and f T A(V) has all ts characterstc roots n F then T satsfes a polynomal of degree n over F. 3. If V V V... Vk, where each subspace V s of dmenson n and s nvarant under T, an element A(V) then a bass ofv can be found so that the matrx of T n ths s of the A form A A k where each A s an n xn matrx of the lner transformaton nduced by T on V. 4. If T A(V) s nlpotent then α +α T+.α mt m where α F s nvertble f α. n k 5. If u V s such that ut = where <k<n, then u=u T k for some u V. 6. If M of dmenson m s cyclc wth respect to T then the dmenson of MT K s m-k for all k<m. 7. Suppose that V V V where V and V are subspaces of V nvarant under T. Let T and T be the lnear transformatons nduced by T on V and V respectvely. If the mnmal polynomal of T over F s P (x) whle that of T s P (x) then the mnmal polynomal for T over F s the LCM of P (x) and P (X)

3 . Let T A(V) has all ts characterstcs roots n F then the s a bass of V n whch the matrx of T s trangular.. If A F n has all ts characterstcs roots n F then there s a matrx C F n such that CAC - s trangular. 3. If T A(V) s nlpotent of ndex of nlpotence n then a bass of V can be found such that M the matrx of T n the bass has the form n n... nr dmv. n M n M n r where n n... n r and 4. Prove that there exsts a subspace W of V s nvarant under T such that V V W. 5. Prove that the two nlpotent lner transformaton are smlar ff and only f they have the same nvarant. UNIT-IV SEC-A. If V V V... Vk where each V s nvarant under T and f P (x) s the mnmal polynomal over F of T, the lner transformaton nduced by T on V, then the mnmal polynomal of T over F s the L.C.M. of p (x),p (x),...p k(x).. State and prove Jordan theorem 3. Prove that for each =,,...k, V and V V V... Vk the mnmal polynomal of T s q (x) 4. If all the dstnct characterstc roots... k of T le n F then V V V... Vk where V l v V \ v T and where T has only one characterstc root on V. 5. Let T A(V) has all ts characterstc roots...n n F then a bass of V can be found n J whch the matrx T s of the form where B B... B r J J k where each are basc Jordon blocks belongng to B J B B r and 6. Suppose that T n A(V) has mnmal polynomal over F the polynomal P x r r x... rx x Then there s a bass of V over F, m T... r Sec-B

4 . State and Prove Jordon theorem?. If Px qx e then mt C q x e C q t 3. If T A(V) has P x q x... q x tk then T R C q x e C q x e r k x e r R m R R k where 4. Let V and W be two vector spaces over F and suppose that s a somorphsm of V onto W suppose S A F(V), T A F(W) such thatvs V T elements dvsors. Then S and T have the same Unt-V SEC-A. For A,B F n and F, ) tr A tra ) tr(a+b)=tra+trb )tr(ab)=tr(ba).. If T A(V) then trt s the sum of the characterstc roots of T. 3. If F s a feld of characterstc and f T A(V) s such that trt for all then T s nlpotent. 4. If F s of characterstc o and f S,T A(V) are such that ST-TS commutes wth S then ST- TS s nlpotent. 5. For all A,B F n, ) (A ) =A ) (A+B) =A +B ) (AB) =B A 6. If T A(V) s such that (UT,V)= & v V then T= 7. If T A(v) them v V, there exsts an element w v dependent on v and T such that (ut,v)=(u,w) for all u V 8. If T A(v) then T * A(v) then ) T * * T ) S T * S * T * S * S * v) ST * T * S * 9. T A(v) s untary ff TT *. If S A(v) and f USS*= then US=. If T s Hermtan vt k = for K, then VT=. ). If N s normal Lnear transformaton and f UN= for u V then UN*=. 3. If N s normal and f UN k = then UN=o 4. If & u are dstnct characterstc roots of N then VN=V, WN=uw & (u,w)= 5. If N s normal & AN=NA then AN*=N*A Sec-B

5 . The Lnear transformaton T on V s untary f and only f t takes an orthonormal bass of V nto an orthonormal bass of V.. If {v,v,...v n} s an orthonormal bass of V and f T A(V) n ths bass (α j) then m(t*) s j j 3. If T A(v) s Hermtal then all ts characterstc roots are real. 4. If N s normal then UNU - (UNU*) s dagonal where V s a untary matrx. 5. The normal transformaton N s () Hermtan ff ts characterstc roots are real () Untary ff ts characterstc roots are all of absolute value. 6. State and Prove Sylvester s law