Abstract Algebra Answers

Prove that every non-trivial sub group of cyclic group has finite index. Hence prove that (Q,+) is not cyclic.

Find the multiplicative inverse of 197 modulo 3000. Highlight each step clearly.

Let G be a finite group.let S={g€G|g^5=e}, where e is identity element of G.show that |S| is odd.

Prove that an ideal M neq R in a commutative ring R with identity is maximal if and only if for every r in R-M, there exists x in R such that 1_R - rx in M.

the set of all functions of {1 2 3...,10} to itself forms a group w.r.t the composition of function.true or false

Let S be the set of all polynomial with real coefficient ,if f, g€S, define f~g if f¹=g¹ where f¹ is the derivative of f, show that ~ is an equivalence relation. Describe the equivalence classes of S

let i=<x,2> and=<x,3> be ideal in z[x] .prove that IJ=<x,6>

If an ideal is contained in union of two ideals then show that it is wholly contained in one of them