I'm currently doing some work on GT from a book, and I'm stuck on the following questions. Can anyone please help?
1. "Let G be a group with a binary operation written multiplicatively, and H be a subgroup of G.
a) Shouw that the relation ~ defined by: x ~ y if there are , (element) H such that x = y is an equivalence relation.
b) For x (element of) G, show that [x], the equivalence class of x is equal to the coset xH"
2. "Let G be a group with a binary operation written multiplicatively. Suppose H is a non-empty subset of G such that for all x,y (element of) H, we have (element of) H. Prove that H is a subgroup of G."
These are the last 2 questions in the chapter and I honestly have no clue how to do them
It's from a series of notes from a friend of mine at university already. I'm trying to learn the subject on my own, but can't get my head around it at all, and being able to get this done will be good because I am taking the subject over summer (southern hemisphere).
I dont even have a glue where to start.
Well, you know what an equivalence relation is to begin with? It is a relation ~ which fulfills
1) Reflexivity: for any element a of the set A we're dealing with, a ~ a
2) Symmetry: for any two elements a,b in A, a ~ b iff b ~ a (this means, if a ~ b then b~ a and if b ~ a then a ~ b)
3) transitivity: if a,b,c in A and if a ~ b and b ~ c, then also a ~ c.
Check these to begin with.