© Hindawi Publishing Corp.
GROUP THEORY BASED AT ANY POINT
M. A. ALBAR and S. A. HUQ
(Received 22 September 1997 and in revised form 20 July 1998)
Abstract.If(G,·)is a group with identitye, we callG, the group based ate. In this paper, we aim to release the present day group theory which is based ate, by replacingeby an arbitrary element of the group.
2000 Mathematics Subject Classification. 20A05, 20F12.
1. Introduction. If (G,·)is a group with the identitye, we callG a group based ate. We can change this base toany other elementa∈G, by defining a new sandwich multiplication,a∗, called themultiplication basedata, as
p∗aq=pa−1q; p,q∈G (1.1)
making(G,a)∗ , a group now based ata. It was previously known [1,2] that all these structures at different based points are isomorphic.
Thus the group theoretic concepts, presently available at(G,·)based ateare rather special cases of the more general concepts based at an arbitrary pointa∈G, called now theconceptsbased ata. The idea here is parallel tothat in differential geome-try, where one chooses a point on a smooth manifold and then studies differential geometric concepts like that of, say, a “tangent bundle” at that point [3].
If a concept based ataholds for alla∈G, we call this aneverywhere conceptinG. In this paper, we briefly underline these ideas. Our study arose while looking at concepts for pregroups mentioned in our previous work [2]. We use the notation of [2], by replacing ˆa, by∗afor simplification.
Throughout, we denote(G,·)for the group based ate, andadenotes an arbitrary fixed element ofG.
2. Based subgroups
Definition2.1. A nonempty subsetHof a groupGis called ana-based subgroup of Gif
p,q∈H impliespq−1a∈H. (2.1)
We use the notationH≤aG.
It is quite clear that for two distinct a,b∈G, ana-based subgroup need not be b-based, as can be seen from the following example.
Example 2.2. The positive rationals, with respect to multiplication, denoted by (Q+,·)is an identity based subgroup of(R+,·)of all positive reals, but not anr-based subgroup of an irrationalr∈R+.
Our next proposition determines when a nonempty subsetHof a group(G,·)based ateis ana-based subgroup fora∈G and is an adapted version of a result due to Certaine [1].
Theorem2.3. A nonempty subsetHof a groupGis ana-based subgroup ofGif H=UaoraU for the subgroupU=Ha−1(ora−1H). Conversely, ifH=Ua(oraU) for any subgroupUofG, thenHis ana-based subgroup ofG.
Proof. Suppose thatHis ana-based subgroup. Then forp,q∈H;pq−1a∈H, o r in other words,
p,q∈H impliespq−1∈Ha−1. (2.2)
This is precisely the condition that
U=Ha−1 is a subgroup ofG. (2.3)
For if u1,u2 ∈ U, then u1 =ha−1 and u2= ha−1; h, h ∈ H. Then u1u−21=
ha−1ah−1
=hh−1∈Ha−1by condition (2.2). SoU=Ha−1is a subgroup ofGand
Ua=H.
Conversely, suppose thatH=Uafor a subgroupUofG, then forp,q∈H, p=ua, g=ua, thus
pq−1a=uaa−1u−1a=uu−1∈Ua=H. (2.4)
ThusHis ana-based subgroup.
The proof when H is a left a-coset is similar. Thus the a-based subgroups are nothing buta-cosets (left and right).
The next result is rather simple.
Proposition2.4. For two elementsa,band a subgroupHofG, Hisa-based implies Hisb-based, if and only ifa−1b∈H, that is,bH=aH (orab−1∈H,i.e.,Ha=Hb). That is, ifa,b∈GandH≤aG, thenH≤aGif and only ifa−1b∈H.
Thus for two elementsaandb belonging to a subgroupH,H is a-based always impliesHis b-based. We notice that any subgroupH is an everywhere subgroup if and only ifH=G, itself, and the group consisting of a single element is only that element based subgroup that is anowheresubgroup.
3. Based abelianness
Definition3.1. A subgroupHof(G,a)∗ is calleda-based abelian, if forp,q∈H,
From the definition it is clear thatHisa-based abelian if
pa−1q=p−1aq−1−1, (3.2)
that is,
p−1aq−1pa−1q=e or ap−1aq−1pa−1q=a. (3.3)
Proposition3.2. For a subgroupHof the group(G,·)based ate, and two distinct elementsa,b∈H, ifHisa-based abelian thenHisb-based abelian also.
Proof. This follows from [2, Theorem 3.2], which asserts(H,a)∗ (H,b)∗ .
Corollary3.3. A subgroupHis abelian if and only if it isa-based, for onea∈H.
Proof. IfHisa-based abelian, then forp,q∈H, pa−1q=qa−1p, soHisa-based abelian.
Conversely, ifHisa-based abelian, thenHise-based abelian also, for the identity elementeofG; that is,His abelian.
Thus an a-based abelian group G, fo r a∈G, is exactly an abelian group in the common group theoretic sense.
Next we examine this concept for ana-based subgroupHof(G,·).
Proposition3.4. Ana-based subgroupHisa-based abelian if and only ifH=Ua (oraU) for an abelian subgroupUofG.
Proof. ByTheorem 2.3,His ana-based subgroup if and only ifHhas the required stated form. Now, forh1,h2∈H,h1=u1aandh2=u2a. Soh1a−1h2=h2a−1h1gives
u1aa−1u2a=u2aa−1u1a, (3.4) that is,u1u2=u2u1. SoUis abelian.
ConverselyUis abelian, givesu1u2a=u2u1afor ana∈Gwhich givesh1a−1h2=
h2a−1h1forh1=u1aandh2=u2a. ThusH=Uaisa-based abelian.
4. Based normality
Definition4.1. A subgroupHofGis calleda-based normal, denoted byH aG, if for anyx∈G,
ax−1ha−1x∈H ∀h∈H. (4.1)
From the definition, it is clear that
(a) a subgroupH isa-based normal, ifa(xH)a−1⊆Hx, that is, the inner auto-morphism byaofG, translates a left coset into a right coset,
(b) a subgroupH is everywhere normal ifHa−1x⊆xa−1H for every pair of ele-mentsa,x∈G, that is,Hab⊆baHfor alla,b∈G.
based at that element. Thus, it becomes more natural toconsider ana-based subgroup which is a-based normal as well. When we wish toindicate this, we write H a G, meaningHis ana-based subgroup for which condition (i) holds.
(ii) If every pair of elements(a,x)of the groupGis inverse commuting (i.e., the inverse of one commutes with the other), then every normal subgroup ofGis every-where normal.
One now asks when is a subgroupa-based normal for ana∈G? The answer is as follows.
Theorem 4.3. IfG is a group,a∈G andH a subgroup ofG thenH aG, if and only if
(i) H G,
(ii) [a−1,G]⊆H, where[a−1,G]⊆Hmeans[a−1,x]∈Hfor everyx∈G.
Proof. Let H aG, then for all x ∈G and allh ∈H, ax−1ha−1x ∈H, that is,
ax−1hxa−1a−1x∈H or (xa−1)−1h(xa−1)[a−1,x]∈H. Choosing h=e, this gives for allx∈G, [a−1,x]∈H. This is (ii).
Conversely, for allx∈G, h∈H, (xa−1)−1h(xa−1)∈H. Next lety∈G, then for allh∈H,
y−1hy=yaa−1−1hyaa−1∈H and thereforeH G. (4.2)
Conversely, letHsatisfy (i) and (ii). Then by (i) for allx∈Gand allh∈H,
xa−1−1hxa−1∈H, (4.3)
and by (ii)
a−1,x∈H; (4.4)
thus(xa−1)−1h(xa−1)[a−1,x]∈H. That is,ax−1ha−1x∈Hand soH aG.
Corollary4.4. LetGbe a subgroup,a∈GandHa subgroup ofG, thenH aG, if and only if
(i) H G,
(ii) the cosetHais in the center ofG/H.
It follows immediately that a normal subgroupHofGis everywhere normal if and only if the derived groupG=[G,G]⊆H, that is, if and only ifG/His abelian.
As we noticed, the normal closure[a−1,G]of[a−1,G]is ana-based normal sub-group ofG. This is called the a-derived subgroup ofG. In fact this is the smallest a-based normal subgroup ofG.
Notice that the derived groupG Gand[a−1,G]⊆G; in fact [a−1,G]t⊆G for t∈G.
Definition4.5. A groupGis calleda-solvable (of classk), for ana∈G, if[a−1,Gk] ⊆Gk+1=1.
Returning again to based normality, we notice that if inDefinition 4.1, we assume that (i) holds for only a fixedx∈G, we callH aG,relative to the elementx∈G. With this convention we conclude thatifH aG, relative tob∈G, thenH b−1G, relative toa.
Theorem4.6. For anya∈G, (i) H aG,H GimpliesHK aG,
(ii) AaG,B aGimpliesA∩B aG, (this remains valid whenais replaced bya ), (iii) if furthera∈H, thenK aG,H a-subgroup ofGimpliesH∩K aH.
Proof. Since the proofs are routine verification as in common group theory, we exhibit typically the proof of (i).
Tosee (i), noticeHKis a normal subgroup, sinceHandKare, and
a−1,G⊆H⊆HK. (4.5)
Finally, we exhibit a characterization ofa-based subgroups which area-based nor-mal as well. Our result is an adapted version of our previous result [2, Theorem 8.1] for this case, though the proof is completely new.
Theorem4.7. Ana-based subgroup H isa-based normal(H a G) if and only if H=Ua (=aU)whereU G.
Proof. Notice thatHis ana-based subgroup if and only ifH=Ua(oraU) where Uis a subgroup byTheorem 2.3.
Now ifHis ana-based normal subgroup, then for anyx∈G,
ax−1Ha−1x⊆H, (4.6)
that is,ax−1Ux⊆H=Ua, thusax−1Uxa−1⊆Uor in other words
xa−1−1Uxa−1⊆U. (4.7)
ShowingU G, as in the proof ofTheorem 4.3, sinceUa=aU, the result is true for aUalso.
Conversely, ifH=Ua, whereU G, then
ax−1Ha−1x=ax−1Uaa−1x foru∈U =ax−1ux
=au foru=x−1ux∈U ∈aU=Ua=H.
(4.8)
5. Based direct product. Let(Ai)i∈Ibe a family of groups with respect to multipli-cation. Letki∈Ai, then we define the based direct product (ki)i∈IAi, as follows.
For(ai)i∈I,(bi)i∈I, define
(ai)i∈I
∗ ki
i∈I
bii∈I, by
ai
∗ kibi
This makes πAi a direct product based at (ki)i∈I usually denoted by (ki)i∈IAi. Notice that, the inverse[(ai)i∈I]−1=[a−1
i∗ki
]i∈I, wherea−1 i∗ki
denote the inverse of ai with respect tothe multiplication based atki[2].
With this definition of product, usual group theoretic concepts could be extended to hold at an arbitrary point. A simplest theorem of this nature is as follows.
Theorem5.1. Aiai Biimplies (ai)i∈IAi (ai)i∈I (ai)i∈IBi, whereai∈Aifor eachi.
6. Homomorphisms
Definition6.1. LetG and H be twogroups; a∈G, b∈H. We call a mapping φ:G→H, a homomorphism, if forg1,g2∈G,
φg1ag∗ 2
=φ(g1) ∗
bφ(g2), (6.1)
that is,
φg1a−1g2=φ(g1)b−1φ(g2). (6.2) Choosingg1=g2=a, we haveb=φ(a), that is, the homomorphism preserves the base point.
Proposition6.2. Ifφ:G→His a homomorphism, then (i) K a GimpliesKφ b Gφ,
(ii) K b HimpliesK−1φ a
G, can then be formulated.
7. Based commutators. For two elementsp,qofG, we define for a fixed element a∈G, ana-based commutator ofp, andqby
[p,q]a=ap−1aq−1pa−1q. (7.1)
Thus[p,q]efor the identitye, is the usual commutator. From the definition, it is immediate
[p,q]−1
a =q−1ap−1qa−1pa−1. (7.2)
Thus
a[p,q]−1
a a=[q,p]a. (7.3)
Also,
[p,p]a=a. (7.4)
Notice
[p,q]a=pa−1,qa−1ea. (7.5)
Besides these, we record the general commutator identities as follows.
Theorem7.1(see [2]). (i)[p,qa−1r ]
a=[p,r ]aa−1([p,q]a)r. (ii)[pa−1q,r ]a=([p,r ]a)qa−1[q,r ]a, wherexp=p−1
∗ a a
−1xa−1p, wherep−1 ∗ a is the
Proof. As usual, we exhibit the proof of (ii). The left-hand side of (ii)
[pa−1q,r ]
a=apa−1q−1ar−1pa−1qa−1r
=aq−1ap−1ar−1pa−1qa−1r . (7.6)
Alsothe right-hand side
q−1 ∗ a a
−1[p,r ]
aa−1qa−1[q,r ]a=aq−1ap−1ar−1pa−1r a−1qa−1aq−1ar−1qa−1r =aq−1ap−1ar−1pa−1qa−1r . (7.7)
This completes the proof.
Obviously, if[p,q]a=afor anyp,qanda∈G, thenGisa-based abelian therefore abelian byCorollary 3.3ofProposition 3.2.
It is now quite clear how one generalizes various other concepts. For example, in G, twosets S and S area-based conjugate, for ana∈G, if there exists anx∈G, such thatax−1Sa−1x=S. If this is true for everya∈G, we obtain an everywhere conjugate pair of sets.
The study of a -based normalizer, a -based centralizer, and so forth, follows the direction ofa -based normal subgroups (a-based subgroups which area-based normal) and their everywhere counterparts are then a matter of routine. We leave these as exercises.
Remark7.2. (a) LetF be a free group on a setS of symbols. Adjoin a symbolato S togetS∪{a}. The free group onS∪{a}could be considered as a free group onS based ata, denoted byFa and any twosuchFa andFb are isomorphic for different symbolsa,b.
Ifa,b∈S, there is nodifference betweenFa,Fb, o rF. Thus a free group on a setS could be considered freely based at every point ofS, that is everywhere free onS.
(b) Though we have studied(G,a)∗ in terms of the usual(G,·)=(G,∗e)for the iden-titye, with which we are familiar, it was quite possible to study the situation in terms of(G,b)∗ based atb.
For example a version ofTheorem 4.3states that “Ana-based subgroupHaaG∗ a, if and only ifHaaG∗
aand[b∗a−1,G∗a]a⊆Ha.”
We are content to leave our reader to these entertaining exercises again. In a subse-quent paper, we look forward to use our results in differential geometry to study how a principal fiber bundle at any point in a group could be located through that at the identity. Behavior of topological groups under change of base points appears to be an interesting area of study. The group theory where any element of the group could play the role of the identity is of interest to computer science as well.
One notices that our situation generalizes suitably to other algebraic structures.
References
[1] J. Certaine,The ternary operation(abc)=ab−1cof a group, Bull. Amer. Math. Soc.49 (1943), 869–877.MR 5,227f. Zbl 061.02305.
[2] S. A. Huq, Some aspects of pregroups, Arch. Math. (Basel) 47 (1986), no. 6, 522–528.
MR 87m:20191. Zbl 594.20073.
[3] J. A. Lees,Notes on Bundle Theory, Lecture Notes in Series, no. 42, Matematisk Institut, Aarhus Universitet, Aarhus, 1974.MR 50#11235. Zbl 304.55014.
