3.3. LA ESCUELA Y LA EDUCACIÓN EN VALORES
3.3.2. La educación en valores en los procesos educativos actuales
are not necessarily the same due to the presence of the groups Aut(f). Hence the pattern H∗(FR(G), R)∼=H∗(Gn−1, R), (3.4.3.9) H∗(WH(G), R)∼=H∗(Gn−1oInn(G), R), (3.4.3.10) H∗(PAut(G), R)∼=H∗(Gn−1oAut(G), R) (3.4.3.11) is broken when non-pure symmetric automorphisms are present.
Remark 3.4.10. There is a discrepancy between the three isomorphisms we obtain in (3.4.3.9), (3.4.3.10) and (3.4.3.11) and the results of [28] and [4]. LetG = G1∗. . .∗Gn be a free product with n factors. Calculating the Euler characteristic we find that
χ(FR(G)) =χ(Gn−1), (3.4.3.12)
whereas in [28] it is calculated to be
χ(Gn−1).Y
i
χ(Inn(Gi))−1. (3.4.3.13) Across all of the results there is a factor ofχ(Inn(G)) difference. There is an analogous difference with the results of [4]. We attribute this to a misquoting of Proposition 5.1 of [37] in each of the papers.
Example 3.4.3.2. LetGi =Z and consider the group ΣFR(Fn)∼=FR(Fn)oSn. Using Theorem 3.4.7 we may calculate the homology as
H∗ ΣFR(Fn), R
=H∗(Sn, R)⊕ M
f∈Forests
H∗ Aut(f),Cb∗(f)
(3.4.3.14)
∼=H∗(Sn, R)⊕ M
f∈Forests
t|E(f)|H∗ Aut(f), Rf
, (3.4.3.15)
where t is of degree one and Rf is the one-dimensional ‘determinant’ Aut(f)-module: a factor of −1 is introduced every time two edges off are swapped.
3.5 Categorical interpretation of symmetric automor-phisms
The partial conjugations play an important role in the theory of automorphisms of free products. In this section we aim to show that they can be viewed as generalisations of inner automorphisms in a sense which can be made precise in the language of category theory. In fact we may show that the Whitehead automorphism group WH(G) occurs naturally as automorphisms which can be extended with respect to the free product
functor
∗n−1 :Gps×n→Gps. (3.5.0.16)
In comparison, the inner automorphisms of a group are those automorphisms which extend with respect to the identity functor
IGps :Gps →Gps. (3.5.0.17)
3.5.1 Extendable automorphisms
LetF :C → D be a functor.
Definition 3.5.1. Let C be an object of C and α : F C → F C be an automorphism of F C in D. Then we say that the pair (C, α) is extendable with respect to F if for every morphism f :C → B with source C there exists an automorphism β of F B making the square
F C α //
F f
F C
F f
F B β //F B
(3.5.1.1)
commute. The extendable automorphisms{(C, α)}of F C form a group, which we denote EFC.
Although it is not necessarily the case, in many examples the extendable automor-phisms form a functor EF : C → Gps to the category of groups, where the commuting squares take the form
F Cα∈EFC//
F f
F C
F f
F B(EFf)(α)//F B.
(3.5.1.2)
Example 3.5.1.1. Let F be the identity functor on the category of groups, I :Gps → Gps. Then by a theorem of Schupp [43] the extendable automorphisms consist of the inner automorphisms.
Theorem 3.5.2 (due to Schupp [43]). Let G be a group and α an automorphism of G.
The automorphism α is an inner automorphism of G if and only if α has the property that whenever G is embedded in a group H then α extends to some automorphism of H.
Proof. A subgroupK ≤H is calledmalnormal in H ifhKh−1∩K ={1}for allh∈H\K.
Schupp noted that it was enough to prove that any groupGis embeddable as a malnormal subgroup of a complete groupH, which he then went on to prove. This is enough because any automorphism of H is inner by the completeness of H and it restricts to G only if the conjugating element is inG by the malnormality of the embedding.
3.5. CATEGORICAL INTERPRETATION OF SYMMETRIC AUTOMORPHISMS 83
We record Schupp’s intermediate result as the following lemma, referring back to [43]
for the proof.
Lemma 3.5.3(due to Schupp [43]). Any groupGis embeddable as a malnormal subgroup of a complete group H.
Remark 3.5.4. The inner automorphism group is normal inside the full automorphism group. It might be hoped that the extendable automorphism groupEFCof an objectF C is normal inside the full automorphism group Aut(F C), however this is not necessarily the case. An example will be given by the Whitehead automorphism group WH(Fn) which is not normal inside Aut(Fn). A weaker result holds.
Proposition 3.5.5. Let F :C → D be a functor and C an object of C. Then the subgroup F Aut(C)
≤Aut(F C)normalises the subgroup EFC of extendable automorphisms of F. Proof. Let α be an extendable automorphism of F C and let γ ∈ Aut(C). We need to show that we may complete the square
F CF γαF γ
−1//
F f
F C
F f
F B F B
(3.5.1.3)
for any f ∈Hom(C, B). Redraw the above diagram as F C F γ
−1//
F fFFFFFFF""
F F C α //
F(f γ)
F C
F(f γ)
F γ //F C
||xxxxxxF fxx
F B F B .
(3.5.1.4)
Looking at the central square in this diagram, α may be extended along F(f γ) in order to fill in the lower edge. The same extension serves to extendF γαF γ−1 alongF f. Hence F γαF γ−1 is extendable with respect to F.
3.5.2 A characterisation of W H (G)
The direct product Gps×n of n copies of the category of groups has as objects n-tuples of groups G = (G1, . . . , Gn) and the morphisms are n-tuples of group homomorphisms f = (f1, . . . , fn). The free product functor
∗n−1 :Gps×n→Gps (3.5.2.1)
takes an n-tuple of groups and gives their free product G1∗. . .∗Gn. So an extendable automorphism α of the free productG1 ∗. . .∗Gn with respect to the functor ∗n−1 is an
automorphism α for which each square
G1∗. . .∗Gn α //
f1∗...∗fn
G1∗. . .∗Gn
f1∗...∗fn
H1∗. . .∗Hn //H1∗. . .∗Hn
(3.5.2.2)
may be completed for each n-tuple of group morphisms
(f1, . . . , fn) : (G1, . . . , Gn)→(H1, . . . , Hn). (3.5.2.3) Recall that the Whitehead automorphism group ofG1∗. . .∗Gn is generated by partial conjugations αgij for i 6= j and gj ∈ Gj and by inner factor automorphisms inn(gi) for gi ∈ Gi. Both kinds of generators are extendable; indeed for (f1, . . . , fn), the generator αgij extends to αfij(gj) and the generator inn(gi) extends to inn(fi(gi)). The purpose of this section is to show the converse, that every extendable automorphism is contained in the Whitehead automorphism group.
Theorem 3.5.6. LetG= (G1, . . . , Gn)be ann-tuple of groups andαbe an automorphism of the free product G= G1 ∗. . .∗Gn. Then α is a Whitehead automorphism if and only if it extends with respect to the free product functor ∗n−1. That is for every morphism
f1∗. . .∗fn :G1∗. . .∗Gn→H1∗. . .∗Hn, (3.5.2.4) there exists an automorphism β ofH1∗. . .∗Hn such that the following diagram commutes.
G1∗. . .∗Gn α //
f1∗...∗fn
G1∗. . .∗Gn
f1∗...∗fn
H1∗. . .∗Hn β //H1∗. . .∗Hn.
(3.5.2.5)
Proof. We have already seen that every Whitehead automorphism extends. For the op-posite implication, suppose that α is an automorphism of G which satisfies the extend-ing condition. By Lemma 3.5.3 each Gi is embeddable as a malnormal subgroup of a complete group Hi. Since each Hi is complete, it’s true that each Hi is neither freely decomposable nor isomorphic to Z. And so the automorphism group Aut(H1 ∗. . .∗Hn) is ΣWH(H1∗. . .∗Hn).
We now show that any γ ∈ ΣWH(H1 ∗. . .∗Hn) restricting to G1 ∗. . .∗Gn must restrict to an element of ΣWH(G1∗. . .∗Gn). For g ∈Gi, the action of γ is given by
γ(g) = (g0)hi, (3.5.2.6)
whereg0 =g ∈Gj for some isomorphic factorGj ∼=Giand for some fixedhi ∈H1∗. . .∗Hn.