Glossarygroup theory

Please refergroup theory forgeneral description oftopic. See also listgroup theory topics.

A group isset togetheran associative operation such that every element has an inverse.

Throughoutarticle, we use edenoteidentity element ofgroup.

Tablecontents

1 Basic definitions 2 Typesgroups 3 Miscellaneous

Basic definitions

Order ofgroup. Order ofgroup (G,*) iscardinality (i.e.size)G. A groupfinite ordercalledfinite group.

Orderan element ofgroup. Suppose x∈Gthere existspositive integer m such that x^m = e, thensmallest possible mcalledorderx. The order offinite groupdivisible byorderevery element.

Subgroup. A subset H ofgroup (G,*) which remainsgroup whenoperation *restrictedHcalledsubgroupG.

Givenset SG. We denote by <S>smallest subgroupG containing S.

Normal subgroup. H isnormal subgroup G ifall gGhH, g * h * g⁻¹ also belongsH.

Both subgroupsnormal subgroupsgiven group formcomplete lattice under inclusionsubsets; this propertysome related resultsdescribed bylattice theorem.

Group homomorphism. Thesefunctions f : (G,*) → (H,×) that havespecial property that

f(a * b) = f(a) × f(b)

for any elements abG.

Kernel ofgroup homomorphism. It ispreimage ofidentity incodomain ofgroup homomorphism. Every normal subgroup iskernel ofgroup isomorphismvice versa.

Group isomorphism. Group homomorphisms that have inverse functions. The inversean isomorphism,turns out, must also behomomorphism.

Isomorphic groups. Two groupsisomorphic if there existsgroup isomorphism mapping from one toother. Isomorphic groups can be thoughtas essentiallysame, onlydifferent labels onindividual elements. One offundamental problemsgroup theory isclassificationgroups up to isomorphism.

Factor group, or quotient group. Givengroup G andnormal subgroup NG,quotient group isset G/Nleft cosets {aN : a∈G} together withoperation aN*bN=abN. The relationship between normal subgroups, homomorphisms,factor groupssummed up infundamental theorem on homomorphisms.

Direct product, direct sum,semidirect productgroups. Thesewayscombining groupsconstruct new groups; please refer tocorresponding linksexplanation.

Typesgroups

Abelian group. A group (G,*)abelian if *commutative, i.e. g*h=h*gall g,h∈G.

Finitely generated group. If there existsfinite set S such that <S> = G, then Gsaidbe finitely generated. If S can be takenhave just one element, G iscyclic groupfinite order, an infinite cyclic group, or possiblygroup {e}just one element.

p-group. If pprime, thenp-groupjustgrouporder p^msome m.

p-subgroup. A subgroup whichalso p-group.

The studyp-subgroups iscentral object ofSylow theorems.

Simple group. Simple groupsthose groups{e}itself asonly normal subgroups. The namemisleading as its structure could be extremely complex. An example ismonster group,grouporder more than one million. Every finite groupbuilt up from simple groups throughusegroup extensions, sostudyclassificationfinite simple groupscentral tostudyfinite groupsgeneral. Asresultextensive effort ovesecond half of20th century,finite simple groups have all been classified.

The structureany finite abelian grouprelatively simple; every finite abelian group isdirect sumcyclic p-groups. This can be extended tocomplete classificationall finitely generated abelian groups, thatall abelian groups thatgenerated byfinite set.

The situationmuch more complicated fornon-abelian groups.

Free group. Given any set A, one can definemultiplicationwords as follows: (abb)*(bca)=abbbca. The free group generated by A issmallest group containing this semigroup.

Every group (G,*)basicallyfactor group offree group generated by G. Please referpresentation ofgroupmore explanation. One can then ask algorithmic questions about these presentations, such as:

• Do these two presentations specify isomorphic groups?; or
• Does this presentation specifytrivial group?

The general casethis isword problem,severalthese questionsin fact unsolvable by any general algorithm.

General linear group. Denoted by GL(n, F), isgroupn-by-n invertible matrices, whereelements ofmatricestaken fromfield F such asreal numbers orcomplex numbers.

Group representation. (notbe confused withpresentation ofgroup). A group representation ishomomorphism fromgroup togeneral linear group. One basically tries"represent"given abstract group asconcrete groupinvertible matrices whichmuch easierstudy.

Miscellaneous

Composition series

Normal series