# Difference between revisions of "Group homomorphism"

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==Isomorphism== | ==Isomorphism== | ||

− | We say that two groups are ''isomorphic'' if there is a [[bijective mapping|bijective]] | + | We say that two groups are ''isomorphic'' if there is a [[bijective mapping|bijective]] homomorphism of one onto the other : the mapping is called an '''[[group isomorphism|isomorphism]]'''. Isomorphic groups have identical structure and are often thought of as just being relabelings of one another. |

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+ | ==References== | ||

+ | * {{cite book | author=A.G. Howson | title=A handbook of terms used in algebra and analysis | publisher=[[Cambridge University Press]] | year=1972 | isbn=0-521-09695-2 | pages=34-37 }} |

## Revision as of 16:11, 22 December 2008

In group theory a **group homomorphism** is a map from one group to another group that preserves the group operations.

Formally, therefore, a map is a homomorphism if

although the first two are in fact consequences of the third.

The **kernel of a homomorphism** is the set of all elements of the domain that map to the identity element of the codomain. This subset is a normal subgroup, and every normal subgroup is the kernel of some homomorphism.

An embedding or monomorphism is an injective homomorphism (or, equivalently, one whose kernel consists only of the identity element).

## Isomorphism

We say that two groups are *isomorphic* if there is a bijective homomorphism of one onto the other : the mapping is called an **isomorphism**. Isomorphic groups have identical structure and are often thought of as just being relabelings of one another.

## References

- A.G. Howson (1972).
*A handbook of terms used in algebra and analysis*. Cambridge University Press, 34-37. ISBN 0-521-09695-2.