21. List as many subgroups as you can of the group of symmetries of a circle.
21. List as many subgroups as you can of the group of symmetries of a circle.
16. Let G be the group of symmetries of a circle and R be a rotation of the circle of V2 degrees. What is IRI?
Consider the additive group ℤ(20). (a) How many subgroups does ℤ(20) have? List all the subgroups. For each of them, give at least one generator. (b) Describe the subgroup < 2 > ∩ < 5 > (give all the elements, order of the group, and a generator). (c) Describe the subgroup <2, 5> (give all the elements, order of the group, and a generator).
4. (a) (3 points) List all the subgroups of the symmetric group S3. (b) (4 points) List all the normal subgroups of Sz. (c) (3 points) Show that the quotient of S3 by any nontrivial normal subgroup is a cyclic group.
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?
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°...
Order and Cyclic Subgroups: Problem 5 Previous Problem Problem List Next Problem (1 point) Let x be an element of order 91 in a group G (not necessarily cyclic, finite, or Abelian). How many distinct subgroups of G are contained in (x)?
19. Let G be a group with no nontrivial proper subgroups. (a) Show that G must be cyclic. (b) What can you say about the order of G?
TaHs has been predicted to have Cav symmetry. Determine the symmetries of the ligand group orbitals constructed from the 1s orbitals on each of the H atoms in this molecule. Describe using only words (not an MO diagram) which Hs group orbitals can interact with the s, p, and d valence atomic orbitals on the central Ta atom based solely on symmetry 2.
TaHs has been predicted to have Cav symmetry. Determine the symmetries of the ligand group orbitals constructed...
(4) This exercise outlines a proof that [21 KI 1//IIKİ whenever H and K are subgroups of a group G. (Note that HK-{hk | he H and k E K). The set HK is not always a subgroup of G.) Let -{hK | h є H). Define an action . . H x Ο Ο by the rule hị . ћК hihi. (You may assume that this is an action.) (a) Prove that OH(X). (b) Prove that HK-Hn K. (Here...
please illustrate each stepbof the solution so I can
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4) Given the diagram 2 7 3 6 4 (a) Up to symmetries, in how many ways can we paint the numbered vertices, if two colors (red and blue) are available? (b) How many ways, if additionally we want that at most two numbered vertices are painted red?
4) Given the diagram 2 7 3 6 4 (a) Up to symmetries, in how many ways can we paint...