Question

A) Let R be the rectangle in R² (i.e., in the xy-plane) with vertices at the points (2,1),(−2,1),(−2,−1),(2,−1). By the group of the symmetries of R we mean the set G of all linear isometries f:R²→R² satisfying f(R)=R where the group operation is the function composition. We know that a linear isometry g:R²→R² is either a rotation about the origin or a reflection over a line passing through the origin. (For instance, if h:R²→R² denotes the reflection over the line y=0, then h(x,y)=(x,−y) and h(R)=R. For another instance, if h:R²→R² denotes π/2 degree rotation about the origin in the counter dockwise direction then h(x,y)=(−y,x) and h(R)=R). Draw the multiplication table of the group G.

b) Let G be a group of order 80 and g be an element of G of order 20. Show that there is no element x of G satisfying x³⁰=g.

Answer 1

No element x in the group G of order 80 satisfying x^{30}=g can exist because the order of element x would have to be a multiple of 30, which is not a divisor of 80, the order of group G.

Step-by-step explanation:

To answer part (b), let's first understand the context of the question. In abstract algebra, group theory studies algebraic structures known as groups, which are sets equipped with an operation that combines any two elements to form a third element, satisfying certain axioms. Specifically, in the scenario of a group G with an element g of order 20, the group G has 80 elements in total. According to Lagrange's theorem, the order of any element in a finite group must divide the order of the group. Since 20 is a divisor of 80, it's valid for G to have an element of order 20.

However, if there were an element x in G such that x30 = g, then the order of x would have to be a multiple of 30, because 30 would be the smallest positive integer k for which xk = e, where e is the identity element. But the maximum order that any element x could have in this group is 80, as that's the order of the group itself. Since no multiple of 30 is a divisor of 80, no element x that satisfies x30 = g could exist within this group. Thus, we can conclude there is no such x in G.