Abstract Algebra: Cosets

[caeelrod]

Suppose G is a group and H is a subgroup of G

a) prove that the left cosets of H in G are identical with the right cosets of H in G (i.e. For x in G there exists a y in G such that xH = Hy) iff for every g in G we have H = gHg^(-1)

b) Show that if H is a subgroup of index two in G, then every left coset is a right coset.

c) Give an example of a group G and a subgroup H for which the collection of left cosets is distinct from the collection of right cosets.

 

[daon]

1) Showed it.

2) this too

3) ok found one

that all?

seriously, show some work. is there a way ted can prevent people with the same IP for making many usernames? I have feeling people make an account, post a question, rinse & repeat.

 

[DrMike]

@caeelrod : what daon is trying to hint at is : This is not a "free math homework doing forum" but a "free math help forum". You've typed in some homework questions, but you haven't explained to us what you want us to do about them. Where are you stuck? What have you tried?