one more try with more info

Suppose a simple group G of order 660 is a subgroup of the symmetric group S11 ( here S is with 11 as a subscript ) and x = {1,2,3,4,5,6,7,8,9,10,11}( a permutation like 1 goes to 2 and so on, I think ) in G. If P equals the span of x ( or ) determine the permutations which generate NGP ( here G is written as the subscript o ...continues

Ideals

Let R be a ring, and suppose that I and J are ideals of R. Let I + J= {x+y: xEI, yEJ} . Prove that I + J is an ideal of R.

Factor Group/ Torsion Group

Heres my problem. Consider the group (reals under addition) and its normal subgroups Z (integers) and Q (rationals0. (These are normal because R is abelian, of course.) (i) Find an element of Q/Z of order 350. (ii) Show that Q/Z is the torsion subgroup of R/Z. This problem is quite straightforward if you use the defini ...continues

permutation groups

Here's my problem: Let (i1, i2, . . . , ik) be a k-cycle (k less or equal to n) element of Sn and let sigma be an element of Sn. (i) Find a precise expression for sigma * (i1, i2, . . . , ik)* sigma-inverse. Hint: experiment a little, perhaps, then take a guess and prove it. (ii) Describe precisely the set {sigma * (1, ...continues

Kernel and Homomorphism

Here's my problem: If A and B are subsets of a group G, define AB = {ab | a 2 A, b 2 B}. Now suppose phi: G -> G0 is a homomorphism of groups and N = Ker(phi) is its kernel. (i) If H is a subgroup of G, show that HN = NH. (Warning: this is an equation of sets; proceed accordingly; do not assume that G is abelian.) (i ...continues

Normalizer of of a sylow p-subgroup of G

Let Sp be a sylow P-subgroup G. Prove that the normalizer of the normalizer of Sp satisfied N(N(Sp))=N(Sp).

Prove that a group of order 120 is not simple.

Prove that a group of order 120 is not simple.

Homomorphism of cyclic group

Let G be a group and h:G-->h(G) a homomorphism.Prove or Disprove If G is cyclic,then h(G)is cyclic.

Automorphism of Zp+Zp ,p prime

let G=Zp+Zp .how many automorphism does G have. Please explain clearly the counting principle.

Groups and Generators

Let G =(x,y|x²ⁿ = e, xⁿ= y², yˉ¹ *x*y= x^ˉ¹). Show that Z(G) = {xⁿ , e} . Assume |G| = 4n, show that G/Z(G) ≈ Dn