The theory uses Galois groups to describe all possible extension fields of a given field by means of a correspondence with subgroups of the Galois group. Groups are used pervasively in mathematics, and they often express symmetry properties of other sets or objects. A homomorphism from an abstract group to a general linear group . Group representations have numerous theoretical and applied uses, since matrix groups have well-known properties and are easy to compute with. The terms are used interchangably.
Homeomorphic spaces have essentially the same topological properties.
Isomorphic objects are essentially the same with respect to the preserved structure. The kernel is always a sub-object of appropriate type. E. g. the kernel of a group homomorphism is a subgroup. A discrete subgroup of the additive group of complex numbers. L-function A complex function which can usually be represented as a Dirichlet series or Euler product and which expresses arithmetic properties of some mathematical construct such as an elliptic curve or modular function .