A homomorphism is a function from one AlgebraicGroup into another which preserves the group structure. Formally, if *G* and *H* are groups with operations * and #, respectively, and *f* is a function from *G* to *H*, then *f* is a homomorphism if

*a***b*=*c*implies*f*(*a*) #*f*(*b*) =*f*(*c*)

A homomorphism maps the identity element of *G* onto the identity element of *H*. If *h* is the image of *g* under a homomorphism, then the inverse of *h* is the image of the inverse of *g*.

The definition above is not the usual one, but it is equivalent to it. The usual definition is that *f*(*a***b*) = *f*(*a*) # *f*(*b*) for all *a* and *b* in *G*.

If a homomorphism is one-to-one and onto, then its inverse is also a homomorphism, and it is called an *isomorphism*. Two groups are considered to be the same if there is an isomorphism between them.

If *H* is a group with identity element *e*, and *f* is a homomorphism from *G* to *H*, then the set of all elements *g* in *G* such that *f*(*g*) = *e* is called the *kernel* of the homomorphism. The kernel of a homomorphism is always a NormalSubgroup? of *G*.

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