Distinct elements and relations

[journey85]

Homework Statement

The dihedral group D_2n has elements e, x, x², ..., x^n-1, y, xy, x²y, ..., x^n-1y and relations xⁿ=e, y²=e (where e is identity) and yx=x^n-1y

(a) Show that D_2n={ elements listed above} i.e. show that these elements are distinct

(b) Show that xy=yx^n-1

(c) Is there an integer m between 1, ..., n-1 such that yx^m=x^my?


The problem I have is simply getting started as it's been so long since I've had any type of linear algebra/modern algebra courses.. If anyone could help me get started on this problem I'd greatly appreciate it.. How does one prove distinction?

 

[lippyka]

Homework Equations None applicableThe Attempt at a Solution (a) Show that D2n={ elements listed above} i.e. show that these elements are distinct: We can prove that the elements in D2n are distinct by showing the order of each element. Since the elements of D2n are composed of x, y, and e where e is the identity element, we know that any element with an exponent of 0 is equal to e. Thus, the order of any element in D2n is equal to n. We can also show that any element with an exponent of n-1 will be equal to y. Therefore, each element in D2n is distinct and the group contains all elements listed above.