Group Theory(Nov.2006)

1.a)If phi:G ->G' is a homomorphism of the group G into the group G' then with usual notations prove that

i)phi(1)=1

ii)phi(xn)=[phi(x)]n, x belongs to G, n belongs to Z

iii)phi preserves all group relations.

b)If G is of finite order n then prove that the group of automorphisms of G (i.e.AutG) is isomorphic to the group of those positive integers <n, which are co-prime to n, with binary operation as multiplication modulo n and order of Aut(G)=phi(n)(phi is Euler's phi function).

Hence or otherwise show that

Aut(Zn,+n)is isomorphic to (Zncross,cross).

c)Construct an abelian group G of order 27 in which x3 = 1 for all x in G.

2.a)i)Define a cycle of length l(i.e.l-cycle).

Prove that number of l-cycles in Sn is {(n)(n-1)...(n-l+1)}/l

ii) If sigma is a cycle of length l, then prove that tsigmat-1 is also a cycle of length l for any t belongs to Sn.

b)i) Define symmetric polynomial in n variables x1,x2,...xn.

Determine which of the following polynomial is symmetric polynomial

I)f(x, y, z)= x2y+y2z+z2x

II)g(x, y, z)= x2y + xy2 + y2z + yz2 + z2x + zx2.

ii)Find the group of symmetries of the polynomial f(x1,x2,x3)=product of(xi -xj), i<j.

c)Determine the elements of order n in ccross, the group of non-zero complex numbers under multiplication.

3.a)Define the cyclic structure of element sigma in Sn.

Prove that two elements of Sn are conjugate to each other if and only if they have the same cyclic structure.

b)Determine the conjugacy classes in S3 and A3 and hence write down their class equation.

c)i)Show that in any group G, ab and ba are conjugate to each other. What can you say about the orders of ab and ba ?

ii)Rule out as many of the following as possible as class equations for a group of order 10.

1+1+1+2+5, 1+2+2+5, 1+2+3+4, 1+1+2+2+2+2.

4.a)i)Show that a simple abelian group is cyclic of prime order.

ii)Show that for K=Z,Q,R or C, SL(n,K) is a normal subgroup of GL(n,K).

b)State Cauchy's theorem for finite abelian group. Use this to prove that the converse of Lagrange's theorem for finite abelian group holds.

c)Show that a group of order 4 is either cyclic of isomorphic to V4.

5.a)If N,H are subgroups of a group G such that N is normal in G then prove that HN/N is isomorphic to H/(H intersection with N)

b)If N and C are respectively the normalizer and centralizer of a subgroup H of a group G, then show that C is normal in N and that N/C is isomorphic to a subgroups Aut(H).

c)If Z is the multiplicative group of complex numbers of absolute value 1 and Ri the additive group of reals then show that R/Z is isomorphic to T.

What happens if we replace R by Q?

6.a)Define the direct product of the subgroups H and k if and only if i)every x in G can be uniquely expressed as x = hk, h in H, k in K ii) hk = 1 if h in H, k in K.

b)Give an example of a group G and normal subgroups N1,N2,..,Nk such that G = N1N2...Nk and Ni intersection with Nj = {e} for all i is not equal to j and yet G is not direct product of Ni...Nk.

c)Prove that for n = kl, (k, l)=1

(z/nz) = (kz/nz) + (lz/nz)(direct).

7.a)Define p-subgroup of a group G (p is a prime).

Prove that for any prime p, every p-subgroup H of a finite group G is contained in a sylow p-subgroup of G.

b)Let p>q be primes and p is not congruent with 1 (modulo q). By using sylow's theorems show that every group of order pq is cyclic.

c)Determine all sylow subgroups of A4 , alternating group of degree 4.

8.a)Prove that a group G is soluble if and only if the derived series

G=G0containing G(1) containing G(2) containing ...of G ends at E in a finite number of steps.

b) If H and K are normal subgroups of G then show that G/(H intersection with K) is soluble if and only if G/H and G/K are both soluble.

c)i) Define a central series in a group G. Find the central series for a group Q8, group of quaternian.

ii) Is the group An(n>=4)nilpotent? why?

Book Name :

I.B.S.Passi and I.S.Luther

Please solve these questions and send answer.

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