The action of a group on a set. Coursework of the group of symmetries of regular polyhedra Actions of the group on the set

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A group G acts (from the left) on a set X if, for any elements g and x X, an element gx X is defined, and moreover, g2(g1x) = (g2 g1)x and ex = x for all x X, g1, g2 G. The set

Gх = (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 partitioned into non-intersecting orbits. If there is only one orbit - the whole set X, then we say that C acts 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 = x".

The stabilizer of an element x from X is a subgroup

StG(x)=(g G | gx = x).

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

Fix(g) = (x X | gx = x).

The power 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 the group of all motions of this space that preserve orientation and take K to K. In the group G there is an identical motion, rotations of 120° and 240° about four axes passing through opposite vertices cube, 180° rotation about axes passing through the midpoints of opposite edges, and 90°, 180° and 270° rotations about axes passing through the centers of opposite faces. 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 neighbors can be transformed into each other by a suitable rotation. The vertex stabilizer x must also leave in place the vertex x farthest from it. Therefore, it consists of an identical movement and rotations around the xx axis by 120 ° and 240 °. Therefore, |G| = |K°| * || = 8 * 3 = 24 and, therefore, all the above rotations form the group G.

The group G is called the cube rotation group. Let us prove that Rotations from G permute the four longest diagonals of the cube. A homomorphism arises: q: G > . The kernel of this homomorphism is (e), since only the identity move 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 get that G .

Symmetry groups

One of the most 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 the tetrahedron.

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

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 the crossing edges. Therefore, there are 3 more (according to the number of pairs of crossing edges) non-identical transformations, which correspond to permutations:

around the AB axis,

around the CD axis,

around the EF axis.

So, together with the identical transformation, we get 12 permutations. Under these transformations, the tetrahedron is self-aligning, turning in space; 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 successive execution of rotations of the tetrahedron will again be a rotation. Thus, we get a group, which is called the rotation group of the tetrahedron.

Under other transformations of space, which are self-coincidences 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. The 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-combining the tetrahedron as a whole, must somehow rearrange its vertices, edges, and faces. In particular, in this case the symmetries can be characterized by permutations of the vertices of the tetrahedron. Since the 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 written out 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 middle, then, say, permutations on the set of edges

correspond respectively to two rotations around the l1 axis, and 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 the same group is "visible" - a group of space transformations that leave the tetrahedron in place.

Symmetry group of a cube. Cube symmetries, like tetrahedral symmetries, 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, which 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 about different axes of symmetry.

Indeed, when the cube rotates, any of the 6 faces of the cube can take the place of the lower face (Fig. 2). For each of the 6 possibilities - when it is indicated which face is located at the bottom - there are 4 different arrangements of the cube, corresponding to its rotations around an axis passing through the centers of the upper and lower faces, through angles 0, p/2, p, 3p/ 2. Thus, we get 6×4 = 24 rotations of the cube. Let us specify 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 suffices to consider rotations around the axes of symmetry.

a) Axes of symmetry of the fourth order are the axes passing through the centers of opposite faces. Around each of these axes there are three non-identical rotations, namely rotations by angles p/2, p, 3p/2. These rotations correspond to 9 permutations of the cube vertices, in which the vertices of opposite faces are rearranged 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 by angles 2p/3, 4p/3. For example, rotations around the diagonal define the following permutations of the cube's vertices:

In total we get 8 such rotations.

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, i.e. we get 6 axes of symmetry of the second order. Around each of these axes there is one non-identical rotation. Only 6 spins. Together with the identical transformation we get 9+8+6+1=24 different rotations. All cube rotations are indicated. The 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 rearrange the diagonals of the cube in different ways, i.e. they correspond to different permutations on the set of diagonals. Therefore, the cube rotation group defines a permutation group on the set of diagonals, consisting of 24 permutations. Since the cube has only 4 diagonals, the group of all such permutations is the same as 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.

We now describe the entire symmetry group of the cube. The cube has three planes of symmetry passing through its center. The 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 total symmetry group of a cube consists of 48 transformations.

Symmetry group of the octahedron. An octahedron made up of five regular polyhedra. It can be obtained by connecting the centers of the faces of the cube and considering the body bounded by planes, which are determined by the connecting lines for neighboring faces (Fig. 3). Therefore, any symmetry of a cube is simultaneously a symmetry of an 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 polyhedron consists of 2l transformations, where l is the number of its plane angles. This assertion holds for all regular polyhedra and can be proved in general view, without finding all the symmetries of the polyhedra.

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

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

Comment. More precisely, so certain action called left. Under the right action, we consider the mapping f: X × G → X, introduce the notation f(x, g) = xg, and satisfy the conditions: x(gh) = (xg)h and xe = x. It is clear that everything said below about the left action is also true (with appropriate modifications) for the right one. Moreover, note that the formula xg = g−1 x establishes a one-to-one correspondence between the left and right actions of G on X (that is, roughly speaking, left and right actions of groups are “the same thing”). The 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) = (gx | g G) is called an orbit of an element x X. The orbits coincide with the minimal G-subsets of X. The relation “to lie in one orbit” is an equivalence relation on X, so the orbits form a partition sets X.

For a fixed x X, the elements g G such that gx = x form a subgroup of G called the stable

lyzer (or stationary subgroup ) of x and is denoted by St(x).

Orbits and stabilizers are related as follows:

Proposition 7.1 |O(x)| = for any x X.

Example. Let X = G and G acts on X by conjugation, that is, (g, x) 7→gxg−1 . The orbit for such an action is called

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

CG(x)

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

Using this action, we prove

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

7.1. Establish the equivalence of the following two definitions of the action of the group G on the set X:

1) The action of G on X is a mapping G×X → X, (g, x) 7→gx such that (g 1 g2 )x = g1 (g2 x) and ex = 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) = O(y), then St(x) is conjugate to St(y). Is the reverse true?

7.3. Describe the orbits and stabilizers of 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 (that is, (g, x) 7→xg−1 );

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

6) The natural action of the group G = GL(V) of non-degenerate 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 = O(V) of orthogonal linear operators in the Euclidean space V on: a) V , b)

8) G = hσi is a cyclic subgroup of S n , X = (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) = gf(x) for all g G, x X. An action of G on X is said to be transitive if for all x, y X there is g G such that y = gx (that is, 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 an appropriate subgroup H. When are the actions of G on G/H1 and G/H2 are isomorphic?

7.5. Find the automorphism group 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 non-trivial.

7 .8 .* Prove that if |G| = 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| = p3 , then |C(G)| = 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 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 polyhedra

Let O(3) := (A GL(3, R) | At A = E), SO(3) := O(3) ∩

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

Grot (M) = (gSO(3) | gM = M);

the symmetry group M is

Gsym (M) = (g O(3) | gM = M)

(that is, Grot (M) = 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). Here and below, 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 an octahedron. Prove that Grot (M) S4 .

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

Grot (M) A5 .

 

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