Recent content by PennState666

how might I show that Aut(Z) has order 2?

Homework Statement prove Aut(Z) has order 2. Homework Equations none The Attempt at a Solution The generators for Z = <-1, 1>. if f is a mapping in Aut(Z), f(x)= x or f(x) = -x

the generator for the group Z(sub 2) x Z(sub 3) = <[1],[1]> and since the operation is addition, the generator for a group in the form Zn x Zm = <[1], [1]>. so Zn x Zm is cyclic and abelian.

yes of course I just wanted to be refreshed and reassured... Z2 x Z3 = {([0],[0]), ([0],[1]), ([0],[2]), ([1],[0]), ([1],[1]), ([1],[2])}... I cannot grasp how this could be generated by a single element though?

What would the elements of Z2 x Z3 look like again?

mhmm i believe so, but i don't know where to start since I am dealing with a cross product...

Isomorphisms isn't my thing, I do not know many facts about Z(sub mn). I can say the contents of Z(sub mn) are {[0],[1], [2],... [mn]}. generated by a single element? I am not sure...Abelian? yes i believe so...

Homework Statement Allow m,n to be two relatively prime integers. You must prove that Z(sub mn) ≈ Z(sub m) x Z(sub n) Homework Equations if two groups form an isomorphism they must be onto, 1-1, and preserve the operation. The Attempt at a...

Homework Statement Let P be a prime integer, prove that Aut(Z sub P) ≈ Z sub p-1 Homework Equations none The Attempt at a Solution groups must preserve the operation, be 1-1, and be onto and they can be called an isomorphism. Z sub p-1 has one less element in it so and all the...

that (Z x W, I) is reflexive anti-symmetric, and transitive

that is it a partially ordered set

Homework Statement (Z, Q) and (W, S) are two partially ordered sets. There is a relation I on Z x W (Z cross W) that is defined... for all (a, b), (c, d) in Z x W, set (a, b) I (c, d) if and only if aQc and bSd. How does one prove that (Z x W, I) is a partially ordered set? Homework...

yes the law is correct. What was given to me in the problem is misleading regarding the law of associativity which is why i am so stumped. as for j, I am not sure what putting one of my variables in for j will do, but i will mess with it and see where it gets me. Thanks!

Homework Statement a function 'd' is a closed binary operation on a set called 'T'. There is an identity element named j. for all elements a, b, and c in the set 'T', we have d(a, d(b,c)) = d((a,c), b) can anyone help me show that d is commutative and associative? Homework...

we don't know enough about a to be able to say that?