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.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter4: More On Groups
Section4.4: Cosets Of A Subgroup
Problem 3E: In Exercises 3 and 4, let G be the octic group D4=e,,2,3,,,, in Example 12 of section 4.1, with its...
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Please do the following questions with handwritten working out
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.
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