Rings : 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 and 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 : Cycles

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 a Sylow p-Subgroup of G

Let Sp be a sylow P-subgroup G. Prove that the normalizer of the normalizer of Sp satisfies 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 automorphisms 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

Group Theory : Cayley Tables, Cyclic Groups and Isomorphisms

• I have the following cayley tables (which is in modulo 9) determine the order of each element . Prove that G is a cyclic group. • Let be the symmetric group of degree 3 together with composition of maps. Is G isomorphic to ? Justify your answer. • Let p be a prime number and G a group of order with identity e ...continues