Let O be the group of rotational symmetries of the cube (see figure of cube below left); we know it has order 24 (follows from the orbit-stabilizer theorem, see Chapter 1). (1). O contains rotations of order 2, 3 and 4. Give a brief description of these and determine how many there are in each conjugacy class. You will need to introduce some notation for this discussion, and a geometric discussion is sufficient: you do not need to prove the conjuga- cies. [Hint: there are 5 conjugacy classes of elements, one of which is the identity {I}.] (2). Suppose in general a finite group G acts on two sets X and Y. Let Z = X x Y, and consider the so-called diagonal action of G on Z given by g. (x, y) = (g.x, g.y). Write down Burnside's formula for the number of orbits of a G-action. Find Z8 in terms of X8 and Y%, explaining your answer carefully, and deduce an expression for the number of orbits for the G action on Z in terms of the actions on X and Y. (3). Consider now the action of 0 on V, the set of vertices of the cube, and on E, the set of its edges. Let Z = V × E be the set of all 96 edge-vertex pairs of the cube. (i) Use (a) and (b) to find the number of orbits of the action of 0 on Z. (ii) Find a single representative of each of the orbits, and explain briefly why they are all in distinct orbits.
Let O be the group of rotational symmetries of the cube (see figure of cube below left); we know it has order 24 (follows from the orbit-stabilizer theorem, see Chapter 1). (1). O contains rotations of order 2, 3 and 4. Give a brief description of these and determine how many there are in each conjugacy class. You will need to introduce some notation for this discussion, and a geometric discussion is sufficient: you do not need to prove the conjuga- cies. [Hint: there are 5 conjugacy classes of elements, one of which is the identity {I}.] (2). Suppose in general a finite group G acts on two sets X and Y. Let Z = X x Y, and consider the so-called diagonal action of G on Z given by g. (x, y) = (g.x, g.y). Write down Burnside's formula for the number of orbits of a G-action. Find Z8 in terms of X8 and Y%, explaining your answer carefully, and deduce an expression for the number of orbits for the G action on Z in terms of the actions on X and Y. (3). Consider now the action of 0 on V, the set of vertices of the cube, and on E, the set of its edges. Let Z = V × E be the set of all 96 edge-vertex pairs of the cube. (i) Use (a) and (b) to find the number of orbits of the action of 0 on Z. (ii) Find a single representative of each of the orbits, and explain briefly why they are all in distinct orbits.
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
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![Let O be the group of rotational symmetries of the cube (see figure of cube below left); we know it
has order 24 (follows from the orbit-stabilizer theorem, see Chapter 1).
(1). O contains rotations of order 2, 3 and 4. Give a brief description of these and determine how
many there are in each conjugacy class. You will need to introduce some notation for this
discussion, and a geometric discussion is sufficient: you do not need to prove the conjuga-
cies.
[Hint: there are 5 conjugacy classes of elements, one of which is the identity {I}.]
(2). Suppose in general a finite group G acts on two sets X and Y. Let Z = X x Y, and consider
the so-called diagonal action of G on Z given by g. (x, y) = (g.x, g.y). Write down Burnside's
formula for the number of orbits of a G-action. Find Z8 in terms of X8 and Y%, explaining
your answer carefully, and deduce an expression for the number of orbits for the G action on
Z in terms of the actions on X and Y.
(3). Consider now the action of 0 on V, the set of vertices of the cube, and on E, the set of its
edges. Let Z = V × E be the set of all 96 edge-vertex pairs of the cube.
(i) Use (a) and (b) to find the number of orbits of the action of 0 on Z.
(ii) Find a single representative of each of the orbits, and explain briefly why they are all in
distinct orbits.
(4). Let T < O be the subgroup consisting of rotations of the tetrahedron, where the tetrahedron
is inscribed in the cube as in the diagram below right. Determine the number of orbits of
the action of the subgroup T on same set Z with 96 elements as before. Finally, give a brief
explanation for the relationship between the O-orbits and the T-orbits.
Z
y](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aa27a28-efdb-4d0f-aad3-266fe3536dfa%2Fb4285d88-4652-418c-aca0-8a054adde8bc%2F620nzu9_processed.png&w=3840&q=75)
Transcribed Image Text:Let O be the group of rotational symmetries of the cube (see figure of cube below left); we know it
has order 24 (follows from the orbit-stabilizer theorem, see Chapter 1).
(1). O contains rotations of order 2, 3 and 4. Give a brief description of these and determine how
many there are in each conjugacy class. You will need to introduce some notation for this
discussion, and a geometric discussion is sufficient: you do not need to prove the conjuga-
cies.
[Hint: there are 5 conjugacy classes of elements, one of which is the identity {I}.]
(2). Suppose in general a finite group G acts on two sets X and Y. Let Z = X x Y, and consider
the so-called diagonal action of G on Z given by g. (x, y) = (g.x, g.y). Write down Burnside's
formula for the number of orbits of a G-action. Find Z8 in terms of X8 and Y%, explaining
your answer carefully, and deduce an expression for the number of orbits for the G action on
Z in terms of the actions on X and Y.
(3). Consider now the action of 0 on V, the set of vertices of the cube, and on E, the set of its
edges. Let Z = V × E be the set of all 96 edge-vertex pairs of the cube.
(i) Use (a) and (b) to find the number of orbits of the action of 0 on Z.
(ii) Find a single representative of each of the orbits, and explain briefly why they are all in
distinct orbits.
(4). Let T < O be the subgroup consisting of rotations of the tetrahedron, where the tetrahedron
is inscribed in the cube as in the diagram below right. Determine the number of orbits of
the action of the subgroup T on same set Z with 96 elements as before. Finally, give a brief
explanation for the relationship between the O-orbits and the T-orbits.
Z
y
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