Group action on a set. Coursework of the symmetry group of regular polyhedra Group actions on the set

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A group G acts (on the left) on a set X if an element gx X is defined for any elements g and х X, moreover, g2 (g1х) \u003d (g2 g1) х and ex \u003d х for all х X, g1, g2 G. The set

Gx \u003d (gx | g G)

is called the orbit of the element x. The orbits of any two elements of X either coincide or do not intersect, so that the set X is split into disjoint orbits. If there is one orbit - the entire set X, then C is said to act transitively on X. In other words, the group G acts transitively on the set X if for any two elements x, x "from X there is an element g from G such that gx \u003d x ".

The stabilizer of an element x from X is a subgroup

StG (x) \u003d (g G | gx \u003d x).

The set of fixed points of an element g from G is the set

Fix (g) \u003d (x X | gx \u003d x).

The cardinality of the orbit is equal to the index of the stabilizer in the group G.

Let K be a fixed cube in three-dimensional Euclidean space, G be the group of all motions of this space that preserve the orientation and transfer K to K. The group G has an identical motion, rotations by 120 ° and 240 ° around four axes passing through opposite vertices cube, 180 ° rotation about axes passing through the midpoints of opposite edges, and 90 °, 180 °, and 270 ° rotation about axes passing through the centers of opposite edges. So, we have found 24 elements in the group G. Let us show that there are no other elements in G. The group G acts transitively on the set K0 of vertices of the cube K, since any two vertices from K can be "connected by a chain of neighbors", and adjacent ones can be translated into each other by a suitable rotation. The stabilizer of the x vertex must also keep in place the vertex x "farthest from it. Therefore, it consists of identical motion and rotations about the xx axis" by 120 ° and 240 °. Therefore, | G | \u003d | К ° | * || \u003d 8 * 3 \u003d 24 and, therefore, all the above rotations constitute a group G.

The group G is called the rotation group of the cube. Let us prove that Rotations from G permute the four longest diagonals of the cube. A homomorphism arises: q: G\u003e. The kernel of this homomorphism is (e), since only the identical motion leaves each diagonal of the cube in place. Therefore, G is isomorphic to a subgroup of the group. Comparing the orders of these groups, we see that G.

Symmetry groups

One of the most commonly used examples of groups and, in particular, permutation groups, are groups that "measure" the symmetry of geometric figures, both flat and spatial.

Symmetry group of a tetrahedron.

The tetrahedron (Fig. 1) has 4 axes of symmetry l1, l2, l3, l4 of the third order, passing through its vertices 1, 2, 3, 4 and the centers of opposite faces. Around each axis, besides the identical one, two more rotations are possible. The following permutations correspond to them:

around the l1 axis

around the l2 axis

around the l3 axis

around the l4 axis

In addition, there are 3 axes of symmetry of the 2nd order, passing through the midpoints A, B, C, D, E, F of crossing edges. Therefore, there are 3 more (according to the number of pairs of crossing edges) nonidentical transformations, which correspond to the permutations:

around the AB axis,

around the CD axis,

around the EF axis.

So, together with the identity transformation, we get 12 permutations. Under the indicated transformations, the tetrahedron self-aligns, turning in space; in this case, its points do not change their position relative to each other. The set of 12 permutations written out is closed with respect to multiplication, since the sequential execution of rotations of the tetrahedron will again be a rotation. Thus, we get a group, which is called the group of rotations of the tetrahedron.

Under other transformations of space, which are self-alignments of the tetrahedron, the internal points of the tetrahedron move relative to each other. Namely: the tetrahedron has 6 planes of symmetry, each of which passes through one of its edges and the middle of the opposite edge. Symmetries with respect to these planes correspond to the following transpositions on the set of vertices of the tetrahedron:

Already on the basis of these data, it can be argued that the group of all possible symmetries of the tetrahedron consists of 24 transformations. Indeed, each symmetry, self-aligning the tetrahedron as a whole, must somehow rearrange its vertices, edges, and faces. In particular, in this case, symmetries can be characterized by permutations of the vertices of the tetrahedron. Since a tetrahedron has 4 vertices, its symmetry group cannot consist of more than 24 transformations. In other words, it either coincides with the symmetric group S4 or is a subgroup of it. The symmetries of the tetrahedron with respect to the planes described above determine all possible transpositions on the set of its vertices. Since these transpositions generate the symmetric group S4, we obtain what is required. Thus, any permutation of the vertices of a tetrahedron is determined by some of its symmetry. However, the same cannot be said about an arbitrary permutation of the edges of the tetrahedron. If we agree to denote each edge of the tetrahedron by the same letter as its midpoint, then, say, the permutations on the set of edges

correspond, respectively, to two rotations around the l1 axis, and to a rotation around the AB axis. Having written out the permutations on the set (A, B. C, D, E, F) for all symmetry transformations, we obtain a certain subgroup of the symmetric group S6, consisting of 24 permutations. The group of permutations of the vertices of the tetrahedron and the group of permutations of its edges - different groups permutations because they act on different sets. But behind them one and the same group is "visible" - the group of space transformations that leave the tetrahedron in place.

Symmetry group of a cube. Symmetries of a cube, like those of a tetrahedron, are divided into two types - self-alignment, in which the points of the cube do not change their position relative to each other, and transformations that leave the cube as a whole in place, but move its points relative to each other. Transformations of the first type will be called rotations. All rotations form a group called the cube rotation group.

There are exactly 24 rotations of the cube around different axes of symmetry.

Indeed, when the cube is rotated, any of the 6 cube faces can take the place of the bottom face (Fig. 2). For each of the 6 possibilities - when it is indicated which face is located at the bottom - there are 4 different positions of the cube, corresponding to its rotations around an axis passing through the centers of the upper and lower faces, at angles 0, p / 2, p, Zp / 2. Thus, we get 6Х4 \u003d 24 cube rotations. Let us indicate them explicitly.

The cube has a center of symmetry (the point of intersection of its diagonals), 3 axes of symmetry of the fourth order, 4 axes of symmetry of the third order, and 6 axes of symmetry of the second order. It is enough to consider rotations around the axes of symmetry.

a) The axes of symmetry of the fourth order are the axes passing through the centers of the opposite faces. Around each of these axes there are three non-identical rotations, namely rotations through the angles p / 2, p, 3p / 2. These rotations correspond to 9 permutations of the cube vertices, in which the vertices of the opposite faces are permuted cyclically and consistently. For example, permutations

correspond to rotations around the axis

b) The axes of symmetry of the third order are the diagonals of the cube. Around each of the four diagonals,,, there are two non-identical rotations at the angles 2p / 3, 4p / 3. For example, rotations around the diagonal define such permutations of the cube vertices:

We get 8 such rotations in total.

c) The axes of symmetry of the second order will be straight lines connecting the midpoints of the opposite edges of the cube. There are six pairs of opposite edges (for example,,), each pair defines one axis of symmetry, that is, we get 6 axes of symmetry of the second order. There is one non-identical rotation around each of these axes. Only 6 spins. Together with the identity transformation, we get 9 + 8 + 6 + 1 \u003d 24 different rotations. All cube rotations are indicated. Rotations of a cube define permutations on the sets of its vertices, edges, faces, and diagonals. Consider how the rotation group of a cube acts on the set of its diagonals. Different rotations of the cube permute the diagonals of the cube in different ways, that is, they correspond to different permutations on the set of diagonals. Therefore, the rotation group of the cube defines a group of permutations on the set of diagonals, consisting of 24 permutations. Since the cube has only 4 diagonals, the group of all such permutations coincides with the symmetric group on the set of diagonals. So, any permutation of the diagonals of the cube corresponds to some of its rotation, and different permutations correspond to different rotations.

Let us now describe the entire symmetry group of the cube. The cube has three planes of symmetry passing through its center. Symmetries about these planes, combined with all the rotations of the cube, give us 24 more transformations, which are self-alignments of the cube. Therefore, the full symmetry group of the cube consists of 48 transformations.

Symmetry group of the octahedron. Octahedrodine of five regular polyhedra. It can be obtained by connecting the centers of the faces of the cube and considering the body bounded by the planes, which are determined by connecting lines for adjacent faces (Fig. 3). Therefore, any symmetry of the cube is at the same time the symmetry of the octahedron and vice versa. Thus, the symmetry group of the octahedron is the same as the symmetry group of the cube, and consists of 48 transformations.

The symmetry group of a regular polytope consists of 2l transformations, where l is the number of its flat angles. This statement holds for all regular polytopes, it can be proved in general viewwithout finding all the symmetries of the polyhedra.

Let G be a group, X some set, and f: G × X → X

- display. We denote f (g, x) by gx. We say that an action of G on X (or G acts on X) is given if (gh) x \u003d g (hx) and ex \u003d x for all g, h G, x X. In this case, the set X is called a G-set.

Comment. More precisely, so specific action called left. Under the right action, a mapping f: X × G → X is considered, the notation f (x, g) \u003d xg is introduced, and the following conditions are required: x (gh) \u003d (xg) h and xe \u003d x. It is clear that everything said below about the left action is also true (with appropriate changes) for the right one. Moreover, note that the formula xg \u003d g − 1 x establishes a one-to-one correspondence between the left and right actions of G on X (that is, roughly speaking, the left and right actions of groups are “one and the same”). Right action will naturally arise in chapter 10.

A subset Y X is called a G-subset if GY Y (i.e., gy Y for all g G, y Y).

A subset of a G-set X of the form O (x) \u003d (gx | g G) is called the orbit of an element x X. The orbits coincide with the minimal G-subsets of X. The relation “lie in one orbit” is an equivalence relation on X, therefore the orbits form a partition the set X.

For a fixed x X, elements g G such that gx \u003d x form a subgroup of G, which is called stable

lysator (or stationary subgroup ) of the element x and is denoted by St (x).

The orbits and stabilizers are linked as follows:

Proposition 7.1 | O (x) | \u003d for any x X.

Example. Let X \u003d G and G act on X by conjugation, that is, (g, x) 7 → gxg − 1. The orbit with this action is called

conjugate class , and the stabilizer St (x) -centralizer element x (notation - CG (x)). Obviously, C G (x) \u003d (a G | ax \u003d xa). Moreover, if the group G is finite, then

CG (x)

where, when summing x, runs through the set of representatives of the conjugate classes (i.e., one element from each class is taken).

Using this action, it is proved

Theorem 7.2 (Cauchy's theorem)If the order of a group G is divisible by a prime number p, then G has an element of order p.

7 .1. Establish the equivalence of the following two definitions of the action of a group G on a set X:

1) The action of G on X is a mapping G × X → X, (g, x) 7 → gx such that (g1 g2) x \u003d g1 (g2 x) and ex \u003d x for all g1, g2 G, x X.

2) The action of G on X is a homomorphism G → S (X) (where S (X)

– the group of all bijections of X onto itself).

7 .2. Prove that if O (x) \u003d O (y) then St (x) is conjugate to St (y). Is the opposite true?

7 .3. Describe the orbits and stabilizers for the following actions:

1) Action of G on itself by left shifts (i.e. (g, x) 7 → gx);

2) The action of G on itself by right shifts (i.e. (g, x) 7 → xg−1 );

3) Action of H on G by left (respectively, right) shifts, where H< G;

x X St (x).

4) The action of G by conjugations on the set of its subgroups (that is, (g, H) 7 → gHg−1 );

5) Action of G on the set of right cosets G / H, where H< G (т.е. (g, xH) 7→gxH);

6) Natural action of the group G \u003d GL (V) of nondegenerate linear operators in a linear space V on: a) V, b) V × V, c) the set of all linear subspaces in V;

7) Natural action of the group G \u003d O (V) of orthogonal linear operators in the Euclidean space V on: a) V, b)

8) G \u003d hσi is a cyclic subgroup in Sn, X \u003d (1, 2,..., n).

7 .4. * An isomorphism of actions of a group G on sets X and Y is a bijection f: X → Y such that f (gx) \u003d gf (x) for all g G, x X. An action of G on X is called transitive if for all x, y X there is g G such that y \u003d gx (i.e., X

Is the only orbit of this action). Prove that every transitive action of G on X is isomorphic to an action on G / H for a suitable subgroup H. When are actions of G on G / H1 and G / H2 isomorphic?

7 .5. Find the group of automorphisms of the natural action of the group G on the set G / H.

7 .6. Prove that the orders of the conjugacy classes of a finite group divide its order.

7 .7. * Prove that the center of a finite p-group is nontrivial.

7 .8. * Prove that if | G | \u003d p2, then G is abelian (that is, G is isomorphic to Z (p2) or Z (p) × Z (p)).

7 .9. * Prove that if G is non-Abelian and | G | \u003d p3, then | C (G) | \u003d p.

7 .10. The kernel of the action of G on X is the kernel of the corresponding homomorphism G → S (X).

a) Check that the kernel of the action of G on X is equal to b) Find the kernel of the action of G on G / H, where H< G.

7 .11. * Let H< G, причем = m < ∞. Докажите, что в G существует нормальный делитель N конечного индекса, содержащийся в H, причем делит m! и делится на m.

Symmetry groups of regular polytopes

Put O (3): \u003d (A GL (3, R) | At A \u003d E), SO (3): \u003d O (3) ∩

SL (3, R). Let M R3. Rotation group M is

Grot (M) \u003d (g SO (3) | gM \u003d M);

symmetry group M is

Gsym (M) \u003d (g O (3) | gM \u003d M)

(i.e. Grot (M) \u003d Gsym (M) ∩ SO (3)).

7 .12. Prove that O (3) SO (3) × Z (2).

7 .13. * Find | Grot (M) | and | Gsym (M) | for each of the regular polyhedra (tetrahedron, cube, octahedron, dodecahedron, icosahedron). Hereinafter, it is assumed that M is embedded in R3 so that its center coincides with the origin.

7 .16. * Let M be a cube or octahedron. Prove that Grot (M) S4.

7 .17. * Let M be an icosahedron or dodecahedron. Prove that

Grot (M) A5.

 

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