symmetrigruppen

Symmetrigruppen, or symmetry groups, are mathematical structures in group theory that formalize the concept of symmetry. For a given object or system, a symmetry is a transformation that preserves its essential properties. The set of all such transformations, with composition as the operation, forms a group: it contains the identity, every symmetry has an inverse, and the composition of two symmetries is again a symmetry. This framework applies to concrete objects as well as to abstract structures.

Among finite symmetry groups are the cyclic groups C_n, consisting of rotations by multiples of 360/n, and the dihedral groups D_n, which include those rotations together with reflections. The symmetric group S_n contains all permutations of n elements, while the alternating group A_n consists of the even permutations. Continuous symmetries give rise to Lie groups, such as the special orthogonal group SO(n) and the special unitary group SU(n).

Group actions describe how a symmetry group acts on a set in a way that preserves structure. Orbits are the sets of elements reachable by the group's actions, and stabilizers are subgroups fixing a given element. The orbit-stabilizer theorem links their sizes. Cayley’s theorem states that every finite group is isomorphic to a subgroup of a symmetric group.

Symmetrigruppen appear across mathematics and the sciences. They underpin geometry, combinatorics, and topology, and they provide the language for invariants and conservation laws in physics. In chemistry and crystallography, point groups and space groups classify molecular and crystal symmetries. The study of symmetry groups continues to inform both theoretical insights and practical computations.