Problem I: For positive integers m,k let Gm denote the set of invertible elements of the integers mod m and set Hm,k = {gk : g ∈ Gm}.
(a) Show that Hm,k is a subgroup of Gm.
(b) For m=21 and k=2, list all the distinct left cosets for H21,2 in G21.
List all of the distinct right cosets also.
Problem II: Let σ denote the cycle (1 2 3..........n − 1 n) in S(n).
How many left cosets does the subgroup < σ > have in S(n)?
Problem III: Recall that A(n) denotes the subgroup of all even permu-
tations in S(n).
(a) List all the distinct left cosets and all of the distinct right cosets for A(n) in S(n).
(b) Show that gA(n) = A(n)g for all g ∈ S(n).
Hello, Dear friend.
I am expert in matlab, physics, math,...
I have a lot of experiences in many fields.
I can help you well.
Please contact me.
Best regards.
Hello.
We are 2 PhD in Math. We can solve you task for one day. Let speak details about what lemmas and theorems can be used. Which study book do you use?