What is meant by representation of a group?
What is meant by representation of a group?
The term representation of a group is also used in a more general sense to mean any “description” of a group as a group of transformations of some mathematical object. More formally, a “representation” means a homomorphism from the group to the automorphism group of an object.
What is the action of a group?
A group action is a representation of the elements of a group as symmetries of a set. Many groups have a natural group action coming from their construction; e.g. the dihedral group D 4 D_4 D4 acts on the vertices of a square because the group is given as a set of symmetries of the square.
What is a representation in representation theory?
Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures.
What is a faithful group action?
A group action is called faithful if there are no group elements (except the identity element) such that for all . Equivalently, the map induces an injection of into the symmetric group . So. can be identified with a permutation subgroup. Most actions that arise naturally are faithful.
Which is an equivalent representation?
Two representations of an algebra are sometimes called equivalent, or isomorphic, if their kernels coincide; two representations of a topological group are called equivalent if the induced representations of some group algebra of this group are isomorphic.
What is the orbit group theory?
In celestial mechanics, the fixed path a planet traces as it moves around the sun is called an orbit. When a group acts on a set (this process is called a group action), it permutes the elements of . Any particular element moves around in a fixed path which is called its orbit.
What is stabilizer of a group?
From Encyclopedia of Mathematics. of an element a in a set M. The subgroup Ga of a group of transformations G, operating on a set M, (cf. Group action) consisting of the transformations that leave the element a fixed: Ga={g∈G:ag=a}.
How do you prove something is a group action?
Theorem 2 Let G be a finite group, and let H be a subgroup of G such that [G : H] = p, where p is the smallest prime dividing |G|. Then H is a normal subgroup of G. define an action of G on X by g · aH = gaH, for g ∈ G and aH ∈ X. That is, we let G act on the left cosets of H in G by left multiplication.
What is a transitive group action?
A group action is transitive if it possesses only a single group orbit, i.e., for every pair of elements and , there is a group element such that . In this case, is isomorphic to the left cosets of the isotropy group, . The space.
What are examples of representation?
Representation is the act of speaking on someone’s behalf, or depicting or portraying something. When a lawyer acts on behalf of a client, this is an example of representation. When you make a drawing of your mother that is meant to look like her, this is an example of a representation of your mother.
What is the difference between group action and group representation?
We can view a group G as a category with a single object in which every morphism is invertible. A (left) group action is then nothing but a (covariant) functor from G to the category of sets, and a group representation is a functor from G to the category of vector spaces.
What are the applications of group representations?
Finite groups — Group representations are a very important tool in the study of finite groups. They also arise in the applications of finite group theory to crystallography and to geometry.
What is a representation of a group called?
A representation of a group G on a vector space V over a field K is a group homomorphism from G to GL ( V ), the general linear group on V. That is, a representation is a map Here V is called the representation space and the dimension of V is called the dimension of the representation.
What is an example of group action?
The standard example of a group action is when GGG equals the symmetric group SnS_nSn (((or a subgroup of Sn)S_n)Sn) and X={1,2,…,n}X = \\{1,2,\\ldots,n\\}X={1,2,…,n}.