Underaddition, onehomeamorphousandbetweenthemis a lagerofthemfunction.

Wecancheckthatthisis a homomorefizz.

Andbyusingthelawsoflogarithms, thelogof X times y isequaltothelogof X plus a logof y toseeifthisis a niceamorphousum, wehavetotestthefitsofbaijiaction.

Let's firstcheckthatthisfunctionis 1 to 1.

Supposethatlogof X equalslogof y Sincethelogsareequal.

E toeachpowerisequaltothissimplifiesto x equals y Sothisfunctionis 1 to 1.

Next, therangeofthelockfunctionisallrealnumbers.

SothisfunctionisontoSointhisexample, thelogfunctionis a homomorphismandabiejectionThatmakesit a nice a morefizz.

Um, thesetwogroupsareIsomorFIC.

Forournextexample, Thefirstgroupwillbethenonzerocomplexnumbersundermultiplicationwhichwedenoteby a C withthemultiplicationsignhereit's understoodthatyouarenotincludingzerobecausezerodoesnothaveaninverseundermultiplication.

Thesecondgroupwillbethecomplexnumberswithabsolutevalueofoneundermultiplicationwilldenotethisgroupby s one.

Thisis a standardnotationwhentalkingabout n dimensionalspheres.

The S isshortforsphereandtheonetellsusthatdimensioninthiscase s oneisjust a circleonthecomplexplanewith a radiusofone.

Recallthateverycomplexnumbercouldbewritteninpolarformasourtimes e totheeighthAitawhere r isthedistanceof a complexnumbertwo, theOriginandVedaishowfaryouhavetorotatefromthepositive X axistoreachthecomplexnumber.

Withthissetup, wecannowdefine a homo.

Morphisinbetweenthesetwogroups.

Thefunctionis f ofourtimes E totheeyethetaequals e tothe A fatal.

Youcanvisualizethisbytakinganynonzerocomplexnumbers e drawingawayfromtheoriginto Z andmapping Z tothepointwhereitintersects s one.

Butisthis a home?

A morphism?

Let's check.

Letzeequalaretimes e tothealfaand w equal s times e totheeyebeta.

Wewanttocheckthat f of z times w equals f of z times f w tobeginsubstituteinthepolarforms.

Next, multiplythenumbersontheleftusingthedefinitionofthefunction f wecanseewhateachvaluemapstoo.