Group isomorphism

A group isomorphism is a mathematical object from algebra that is particularly considered in group theory . Analogous to other definitions of isomorphisms , the group isomorphism is defined as a bijective group homomorphism . A group isomorphism that maps a group to itself is a group automorphism .

Group isomorphisms are used, for example, in isomorphism sentences .

definition

Be and two groups . A group homomorphism is called a group isomorphism if it has an inverse mapping , that is, if there is a group homomorphism with and . Equivalent to this is the requirement that the group homomorphism is bijective . ${\ displaystyle \ left (G, \ ast \ right)}$${\ displaystyle \ left (H, \ star \ right)}$ ${\ displaystyle f \ colon G \ to H}$${\ displaystyle f}$${\ displaystyle g \ colon H \ to G}$${\ displaystyle f \ circ g = \ operatorname {id} _ {H}}$${\ displaystyle g \ circ f = \ operatorname {id} _ {G}}$${\ displaystyle f}$

If the group isomorphism maps from to , i.e. the domain of definition and the set of images are the same, the group isomorphism is also called group automorphism. ${\ displaystyle \ left (G, \ ast \ right)}$${\ displaystyle \ left (G, \ ast \ right)}$

properties

${\ displaystyle \ operatorname {Ker} \ left (f \ right) = \ left \ {e_ {G} \ right \}}$
• His image is the whole group, i.e. H.:
${\ displaystyle \ operatorname {im} \ left (f \ right) = H}$
• For each group isomorphism there is a clearly defined inverse function .${\ displaystyle f \ colon G \ to H}$ ${\ displaystyle f ^ {- 1} \ colon H \ to G}$

Isomorphism of groups

Groups between which such a group isomorphism exists are called isomorphic to one another: they differ only in the designation of their elements and are the same for almost all purposes.

It can easily be shown that the isomorphism of groups forms an equivalence relation .

Examples

• For each group G is the identical mapping , a group automorphism.${\ displaystyle \ operatorname {id} \ colon G \ to G, x \ mapsto x}$
• The exponential function is a group isomorphism.${\ displaystyle \ exp \ colon \ left (\ mathbb {R}, + \ right) \ to \ left (\ mathbb {R} _ {> 0}, \ cdot \ right), x \ mapsto e ^ {x} }$
• The conjugation describes a group automorphism.