Abstract Algebra Answers

If G = < x > is of order 25, then x^a generates G , where α is a factor of 25.
State the given statement is true or false, give reasons for your answer.

If G is a group such that o(G) = 2m , where m∈N , then G has a subgroup of order m.
State the given statement is true or false, give reasons for your answer.

If σ is an even permutation, then σ² = I .
State the given statement is true or false, give reasons for your answer.

Show that d: Q [x] /{0} --> N Union {0} : d(f) =2 ^(deg f) is a Euclidean valuation on Q [x] .

Consider the ring Homomorphism phi : Z[x] --> (Z/3):phi (summation (n, i =0 ) of a _i x^i ) bar a_0 .Show that Ker phi = (x,3) . What does the Fundamental Theorem of Homomorphism say in this case?

Check whether f : ( 4Z, +) --> ( Z_4, +) : (4m) = bar m is a group Homomorphism or not? If it is, what does the Fundamental Theorem of Homomorphism gives us in this case? If f is not a Homomorphism , obtain the range of f.

How many Sylow 5 - subgroups , Sylow 3 - subgroups and Sylow 2- subgroups can a group of order 200 have ? Give reason for your answer.

Let G be a group , H ∆= G and beta <=(G /H ) . Let A = { x belongs to G | Hx belongs to beta } . Show that (1) . A<=G (2).H ∆= A (3). Beta = (A\H ) .

For x belongs to G , define H _x = { g^(-1) x g | g belongs to G } . Under what conditions on x will H_x <= G ? Further, if H _x <= G , will H_x ∆= G ? Give reason for your answer.

Define a relation R on Z by R={(n,n+3k)| k€Z}.
Check whether R is an equivalence relation or not. If it is, find all the distinct equivalence classes. If R is not an equivalence relation, define an equivalence relation on Z