16. Let G be the group of symmetries of a circle and R be a rotation...
Let D4 be the group of symmetries of the square That is, D4 = {1, R, R2, Rº, T., Ty, T1,3, T2,4} where, in particular, R is a counterclockwise rotation by 90° about the origin and Tx is a reflection about the x-axis (the group and its elements were defined in class). (a) Show that D4 is generated by {R, Tx}, that is, D4 = (R, Tx). (b) Construct the Cayley graph Cay(D4, {R, Tx}).
Let o and be the following symmetries of the regular pentagon: 240° rotation reflection 5 6 (a) (6 points) Express o and T as permutations of the vertices (i.e., in two-row notation). (b) (9 points) Compute the following permutations. Express answers in two-row notation. i. OOT ii. τοσ iii. TOOOT (c) (5 points) Determine the symmetry represented by TOOOT. Give the axis of symmetry if a reflection, or the degrees of clock-wise rotation if a rotation.
Let Ds be the group of symmetries of the square. (a) Show that Ds can be generated by the rotation through 90° and any one of the four reflections. (b) Show that Dg can be generated by two reflections. (c) Is it true that any choice of a pair of (distinct) reflections is a generating set of Dg?
Let o and 7 be the following symmetries of the regular pentagon: 2. 2 240° rotation o reflection т 5 6 6 (a) (6 points) Express o and 7 as permutations of the vertices (i.e., in two-row notation). (b) (9 points) Compute the following permutations. Express answers in two-row notation. i. OOT ii. Too iii. TOOOT (c) (5 points) Determine the symmetry represented by TOOOT. Give the axis of symmetry if a reflection, or the degrees of clock-wise rotation if...
21. List as many subgroups as you can of the group of symmetries of a circle.
Problem 3 () (2 marka) Prove that the group R and the circle group St are not isomsorphic to each other. Hind เตบ๐s fad element of order 2 m S., Hou about RV (a)(2marks) Let n 2 be an integer, give an escample (including explanatlon) of a group G and a subgroup FH with IG: H-nsuch that H is not normal in G. (iii) (S marks) Let G-16:l : a,b,c ER, a 7.0, eyh 아 You are given that G...
Consider the rectangle shown, and let A be the eight points listed. The symmetry group, G, of this rectangle has four elements: the identity j a flip over a horizontal axis through its centre v a flip over a vertical axis through its centre r a rotation about its centre by 180 degrees. We regard G as a subgroup of SA Page 2 (a) Find G(a) and G(b), the orbits of a and b. (b) Find Ga and Gb, the...
Let G be a group of order 16, such that each element can be written (uniquely) in the form rasb, where a є {0, ,7} and b є {0,1). The elements r and s satisfy the relations: r"=1; s2 = 1; sr=r38. (The final relation means that an s can be moved past an r if we raise the r to the third power.) Let H = {1,s). Let . : G × G/H → G/H be the usual action...
Example: Let D6 be the group of symmetries of the regular hexagon (see Exercise 6.2.15). 7. Determine the orders of the elements of De, and count the elements of each order. Decide which ones are a. conjugate (make a table summarizing your results, as in the text) What are the normal subgroups of D6? b. order of element geometric description #(conjugates) identity 180° rotations preserving edges 180° rotations preserving faces 1 1 2 2 3 3 +120° rotations 8 +90°...
ANSWER 2 & 3 please. Show work for my understanding and upvote. THANK YOU!! 2. Given a regular n-gon, let r be a rotation of it by 2π/n radians. This time, assume that we are not allowed to flip over the n-gon. These n actions form a group denotecd (a) Draw a Cayley diagram for Cn for n-4, n-5, and n-6 (b) For n 4, 5, 6, find all minimal generating sets of C.· [Note: There are minimal generating sets...