Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Question 6 Recall the symmetries of the square, consisting of: a d b (a) Let H be the set of those symmetries which send the line ac to itself. That is, H = {σ € D4 : σ({a,c}} = {a,c}}. Show that H is a subgroup of D4. You can do this by determining which of the above symmetries are in H and then checking that they form a subgroup, but that isn't the only approach. (b) Find the centre Z(D4). (See the tutorial problems) . • id- the identity transformation. Pi-rotation by 90° clockwise. ⚫ P2 - rotation by 180°. • ⚫ P3 - rotation by 270° clockwise. ⚫ μ₁ - vertical reflection (switches the top and the bottom). • μ2-horizontal reflection (switches left and right). • S₁-diagonal reflection switching a and c. 82-diagonal reflection switching band d. This group of symmetries is known as D4. Tutorial exercises Question 1 Let G be a group. Define Z(G) = {xЄG: gx=xg for all g€ G}. Prove that Z(G) is a subgroup of G (this is called the centre of G). Question 2 Show that if G is a group with no proper nontrivial subgroups, then G is cyclic. Question 3 Recall that Sy is the set of all permutations of {1,2,3), which is the same (iso- morphic!) as the symmetries of an equilateral triangle. (i) Find all the subgroups of $3. (ii) For each of the proper, nontrivial subgroups H, find a geometric feature of the triangle such that H is the set of precisely those symmetries which preserve that feature (map it to itself). Question 4 Recall that a subset H of a group G is a subgroup precisely if it satisfies the following three properties: (i) H is closed under the group operation; (ii) H contains G's identity element; and (iii) For every aЄH, a¹€ H. For every possible combination, find a subset of (Z,+) which satisfies two of these properties but not the third.

Question 6 Recall the symmetries of the square, consisting of: a d b (a) Let H be the set of those symmetries which send the line ac to itself. That is, H = {σ € D4 : σ({a,c}} = {a,c}}. Show that H is a subgroup of D4. You can do this by determining which of the above symmetries are in H and then checking that they form a subgroup, but that isn't the only approach. (b) Find the centre Z(D4). (See the tutorial problems) . • id- the identity transformation. Pi-rotation by 90° clockwise. ⚫ P2 - rotation by 180°. • ⚫ P3 - rotation by 270° clockwise. ⚫ μ₁ - vertical reflection (switches the top and the bottom). • μ2-horizontal reflection (switches left and right). • S₁-diagonal reflection switching a and c. 82-diagonal reflection switching band d. This group of symmetries is known as D4. Tutorial exercises Question 1 Let G be a group. Define Z(G) = {xЄG: gx=xg for all g€ G}. Prove that Z(G) is a subgroup of G (this is called the centre of G). Question 2 Show that if G is a group with no proper nontrivial subgroups, then G is cyclic. Question 3 Recall that Sy is the set of all permutations of {1,2,3), which is the same (iso- morphic!) as the symmetries of an equilateral triangle. (i) Find all the subgroups of $3. (ii) For each of the proper, nontrivial subgroups H, find a geometric feature of the triangle such that H is the set of precisely those symmetries which preserve that feature (map it to itself). Question 4 Recall that a subset H of a group G is a subgroup precisely if it satisfies the following three properties: (i) H is closed under the group operation; (ii) H contains G's identity element; and (iii) For every aЄH, a¹€ H. For every possible combination, find a subset of (Z,+) which satisfies two of these properties but not the third.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question

Please do the following questions with handwritten working out 

 
Question 6 Recall the symmetries of the square, consisting of: a d b (a) Let H be the set of those symmetries which send the line ac to itself. That is, H = {σ € D4 : σ({a,c}} = {a,c}}. Show that H is a subgroup of D4. You can do this by determining which of the above symmetries are in H and then checking that they form a subgroup, but that isn't the only approach. (b) Find the centre Z(D4). (See the tutorial problems) . • id- the identity transformation. Pi-rotation by 90° clockwise. ⚫ P2 - rotation by 180°. • ⚫ P3 - rotation by 270° clockwise. ⚫ μ₁ - vertical reflection (switches the top and the bottom). • μ2-horizontal reflection (switches left and right). • S₁-diagonal reflection switching a and c. 82-diagonal reflection switching band d. This group of symmetries is known as D4. Tutorial exercises Question 1 Let G be a group. Define Z(G) = {xЄG: gx=xg for all g€ G}. Prove that Z(G) is a subgroup of G (this is called the centre of G). Question 2 Show that if G is a group with no proper nontrivial subgroups, then G is cyclic. Question 3 Recall that Sy is the set of all permutations of {1,2,3), which is the same (iso- morphic!) as the symmetries of an equilateral triangle. (i) Find all the subgroups of $3. (ii) For each of the proper, nontrivial subgroups H, find a geometric feature of the triangle such that H is the set of precisely those symmetries which preserve that feature (map it to itself). Question 4 Recall that a subset H of a group G is a subgroup precisely if it satisfies the following three properties: (i) H is closed under the group operation; (ii) H contains G's identity element; and (iii) For every aЄH, a¹€ H. For every possible combination, find a subset of (Z,+) which satisfies two of these properties but not the third.
Transcribed Image Text:Question 6 Recall the symmetries of the square, consisting of: a d b (a) Let H be the set of those symmetries which send the line ac to itself. That is, H = {σ € D4 : σ({a,c}} = {a,c}}. Show that H is a subgroup of D4. You can do this by determining which of the above symmetries are in H and then checking that they form a subgroup, but that isn't the only approach. (b) Find the centre Z(D4). (See the tutorial problems) . • id- the identity transformation. Pi-rotation by 90° clockwise. ⚫ P2 - rotation by 180°. • ⚫ P3 - rotation by 270° clockwise. ⚫ μ₁ - vertical reflection (switches the top and the bottom). • μ2-horizontal reflection (switches left and right). • S₁-diagonal reflection switching a and c. 82-diagonal reflection switching band d. This group of symmetries is known as D4. Tutorial exercises Question 1 Let G be a group. Define Z(G) = {xЄG: gx=xg for all g€ G}. Prove that Z(G) is a subgroup of G (this is called the centre of G). Question 2 Show that if G is a group with no proper nontrivial subgroups, then G is cyclic. Question 3 Recall that Sy is the set of all permutations of {1,2,3), which is the same (iso- morphic!) as the symmetries of an equilateral triangle. (i) Find all the subgroups of $3. (ii) For each of the proper, nontrivial subgroups H, find a geometric feature of the triangle such that H is the set of precisely those symmetries which preserve that feature (map it to itself). Question 4 Recall that a subset H of a group G is a subgroup precisely if it satisfies the following three properties: (i) H is closed under the group operation; (ii) H contains G's identity element; and (iii) For every aЄH, a¹€ H. For every possible combination, find a subset of (Z,+) which satisfies two of these properties but not the third.
Expert Solution
steps

Step by step

Solved in 1 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS