Abstract Algebra Answers

*Question: Use the definition of negative numbers to justify that -(-5) equals 5.

Definition of negative numbers: -a is the number when added to a equals 0.

*Throw out and avoid the standard/traditional integer rules that you've learned and use only the knowledge we have gained working with negative numbers using the definition and number lines.
*This is a justification by proof type problem

state what properties you use/defintions/etc...

Let (G,✳) be a group, and let a∈G. Let C(a) = {g∈G: a✳g = g✳a}.

In this problem we will prove that (C(a),✳) is a subgroup of (G,✳).

C(a)⊆G by the definition of C(a).

Prove that if (G,*) is a group, and if the only subgroups of G are G and {e}, then G is cyclic.

If (H,*) is a subgroup of (G,*), and a * b is an element of H, must a is an element of H and b is an element of H? Explain your answer.

if (H,*) is a subgroup of (G,*) and if R is the relation defined on G by aRb iff a * b^-1 is an element of H, then R is an equivalence relation on G, and that aRb iff b is an element of Ha (where Ha={h * a: h is an element of H}), and thus [a] subscript of R=Ha.Let G=S subscript of 4 , and let H=<{(13),(14)}>. Find H, and find all of the equivalence classes of S subscript of 4 under R.

If I=<24,36,42> be an ideal of a ring Z then find a such that I=<a>.

Assume that (G,*) is a group, that g is an element of G and that t is the smallest positive integer such that g^t=e

prove that g^n=e if and only if t |n. use division algorithm and explain why r must = 0

construct cayley table (Z subscript (9) , circle times )and verify its an abelian group.

Assume that a | b times c and that gcf(a,b)=1. Prove that a | c(hint: use the result that gcf(a,b)=1 iff there exist x,y is an element of Z such that a times x+b times y=1)

In the Principal Ideal Domain two non-zero element a,b are coprime if and only if ∃ x,y∈ R such that ax+by=1