Abstract Algebra Answers

State, with a reason, the given statement is true or false .
{ α1, α2, ...., αn } is a set only if all the αis
follow a given rule.

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.

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

State the following statement are True? 1. If G = <x> is of order 25 , then x^(alpha) generates G , where alpha is a factor of 25

Find the sum of the 5th roots of unity?

Which of the following statement are True?1. If ( R, *_1 ,*_2) is a ring ,then psi : R ×R --> psi (r_1 , r_2) = r_1 *_2 r_1 is a binary operation?