consisting of symmetries of given mathematical objects

Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions.

groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above

Symmetry groups are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below).[51]

a rich structure [16] including regular families and sporadic or exceptional phenomena

Like many other mathematical concepts, symmetry groups have a rich structure including regular families and sporadic or exceptional phenomena [16].

defining properties

A symmetry group obeys the defining properties of any group.

structure

Groups are a structure that encode the mathematical idea of symmetry.

The symmetry operations

The symmetry operations of a molecule (or other object) form a group.

symmetry transformations

In mathematical terms, the symmetry transformations make up a group.

something about symmetry groups

(you may have learned something about symmetry groups at one point)?

a group (in the algebraic sense) formed by the set of symmetry operations of a given object

A symmetry group is a group (in the algebraic sense) formed by the set of symmetry operations of a given object.

they encode the symmetry appearing within any particular setting

Groups are important mathematical objects that arise in many contexts since they encode the symmetry appearing within any particular setting.

fundamental kinship

Groups share a fundamental kinship with the notion of symmetry.

an abstraction used to describe the symmetries of an object

In mathematics, a symmetry group is an abstraction used to describe the symmetries of an object.

a group of symmetry-preserving operations, i.e., rotations, reflections, and inversions

A symmetry group is a group of symmetry-preserving operations, i.e., rotations, reflections, and inversions.

the framework

The symmetry groups provide the framework for all the possible ways these fundamental particles and forces can interact, enabling physicists to write down equations, called Lagrangians, predicting these possible interactions.

a group whose elements are transformations on some objects

A symmetry group is a group whose elements are transformations on some objects, and whose operation is the composition operation \(\circ\).

a group consisting of all permutations of given finite set, with composition of permutations as the binary operation

A symmetric group is a group consisting of all permutations of given finite set, with composition of permutations as the binary operation.

all symmetries of objects

In mathematics, a symmetry group describes all symmetries of objects.

In mathematics of objects

In mathematics, a symmetry group describes all symmetries of objects.

In math-speak to the collection of ways one can slide, reflect or rotate an object so that an object's final appearance is the same as an object's starting one

In math-speak, “symmetry groups” refers to the collection of ways one can slide, reflect or rotate an object so that an object's final appearance is the same as an object's starting one.

a group of spatial transformations ; that leaves an object unchanged

A mathematical symmetry group is a group of spatial transformations that leaves an object unchanged;