isomorphic groups

Two groups (X1,*1) and (X2,*2) are said to be isomorphicPlanetmathPlanetmathPlanetmath if there is a group isomorphism ψ:X1X2.

Next we name a few necessary conditions for two groups X1,X2 to be isomorphic (with isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath ψ as above).

  1. 1.

    If two groups are isomorphic, then they have the same cardinality. Indeed, an isomorphism is in particular a bijection of sets.

  2. 2.

    If the group X1 has an element g of order n, then the group X2 must have an element of the same order. If there is an isomorphism ψ then ψ(g)X2 and (ψ(g))n=ψ(gn)=ψ(e1)=e2 where ei is the identity elementsMathworldPlanetmath of Xi. Moreover, if (ψ(g))m=e2 then ψ(gm)=e2 and by the injectivity of ψ we must have gm=e1 so n divides m. Therefore the order of ψ(g) is n.

  3. 3.

    If one group is cyclic, the other one must be cyclic too. Suppose X1 is cyclic generated by an element g. Then it is easy to see that X2 is generated by the element ψ(g). Also if X1 is finitely generatedMathworldPlanetmathPlanetmathPlanetmath, then X2 is finitely generated as well.

  4. 4.

    If one group is abelianMathworldPlanetmath, the other one must be abelian as well. Indeed, suppose X2 is abelian. Then


    and using the injectivity of ψ we conclude  g*1h=h*1g.

Note.Isomorphic groupsMathworldPlanetmath are sometimes said to be abstractly identical, because their “abstract” are completely similar — one may think that their elements are the same but have only different names.

Title isomorphic groups
Canonical name IsomorphicGroups
Date of creation 2013-03-22 14:01:58
Last modified on 2013-03-22 14:01:58
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 10
Author alozano (2414)
Entry type Definition
Classification msc 20A05
Defines isomorphic
Defines abstractly identical