Group isomorphism

In abstract algebra, given two groups (G, *)(H, @)group isomorphism from (G, *)(H, @) isbijective group homomorphism from GH. Spelled out, this means thatgroup isomorphism isbijective function f : G -> H such thatall uvGholds that

f(u * v) = f(u) @ f(v).

If there exists an isomorphism betweengroups GH, thengroupscalled isomorphic. Fromstandpointgroup theory, isomorphic groups havesame propertiesneed not be distinguished.

Tablecontents

1 Examples 2 Consequences 3 Automorphisms

Examples

The groupall real numbersaddition, (R,+),isomorphic togroupall positive real numbersmultiplication (R⁺,×) viaisomorphism

f(x) = exp(x)

(see exponential function).

The group Zintegers (with addition) issubgroupR, andfactor group R/Zisomorphic togroup S¹complex numbersabsolute value 1 (with multiplication); an isomorphismgiven by

f(x + Z) = exp(2πxi)

for every xR.

The Klein four-groupisomorphic todirect producttwo copiesZ/2Z (see modular arithmetic).

Consequences

Fromdefinition,follows that f will mapidentity elementG toidentity elementH,

f(e_G) = e_H

thatwill map inversesinverses,

f(u^-1) = f(u)^-1

for all uG, and thatinverse map f^-1 : H -> Galsogroup isomorphism.

The relation "being isomorphic" satisfies allaxiomsan equivalence relation. If fan isomorphism between GH, then everything thattrue about G can be translated via f intotrue statement about H,vice versa.

Automorphisms

An isomorphism fromgroup GGcalled an automorphismG. The compositiontwo automorphismagain an automorphism,with this operationsetall automorphisms ofgroup G, denoted by Aut(G), forms itselfgroup,automorphism groupG.