Group isomorphism

A group iso­mor­phism is a group ho­mo­mor­phism which is bi­jec­tive. We say that two groups are iso­mor­phic if there is an iso­mor­phism be­tween them.

It turns out that iso­mor­phism is a much more use­ful con­cept than true equal­ity of groups, and it cap­tures the idea that “these two ob­jects are the same group”: the iso­mor­phism shows us how to re­la­bel the el­e­ments to see that they are in­deed the same group.

For ex­am­ple, the triv­ial group is in some sense “the only group with one el­e­ment”, but it can be in­stan­ti­ated in many differ­ent ways: as \((\{ a \}, +_a)\), or \((\{ b \}, +_b)\), and so on (where \(+_x\) is the bi­nary func­tion tak­ing \((x, x)\) to \(x\)). They all be­have in ex­actly the same ways for the pur­pose of group the­ory, but they are not liter­ally iden­ti­cal. They are all iso­mor­phic, though: the map \(\{a \} \to \{ b \}\) given by \(a \mapsto b\) is an iso­mor­phism of the re­spec­tive groups.

Two groups are iso­mor­phic if and only if they have the same Cayley table, pos­si­bly with re­ar­range­ment of rows/​columns and with re­la­bel­ling of el­e­ments.

Parents:

  • Group

    The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.

  • Isomorphism

    A mor­phism be­tween two ob­jects which de­scribes how they are “es­sen­tially equiv­a­lent” for the pur­poses of the the­ory un­der con­sid­er­a­tion.