Some properties of E(G,W,FTG) and an application in the theory of splittings of groups
Let us consider W a G-set and M a Z₂G-module, where G is a group. In this paper we investigate some properties of the cohomological the theory of splittings of groups. Namely, we give a proof of the invariant E(G,W,M), defined in [5] and present related results with independence of E(G,W,M) with res...
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irk-123456789-1885622023-03-07T01:26:48Z Some properties of E(G,W,FTG) and an application in the theory of splittings of groups Fanti, E.L.C. Silva, L.S. Let us consider W a G-set and M a Z₂G-module, where G is a group. In this paper we investigate some properties of the cohomological the theory of splittings of groups. Namely, we give a proof of the invariant E(G,W,M), defined in [5] and present related results with independence of E(G,W,M) with respect to the set of G-orbit representatives in W and properties of the invariant E(G,W,FTG) establishing a relation with the end of pairs of groups ê(G, T), defined by Kropphller and Holler in [15]. The main results give necessary conditions for G to split over a subgroup T, in the cases where M = Z₂(G/T ) or M = FTG. 2020 Article Some properties of E(G,W,FTG) and an application in the theory of splittings of groups / E.L.C. Fanti, L.S. Silva // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 2. — С. 179–193. — Бібліогр.: 19 назв. — англ. 1726-3255 DOI:10.12958/adm1246 2010 MSC: 20E06, 20J06, 57M07 http://dspace.nbuv.gov.ua/handle/123456789/188562 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Let us consider W a G-set and M a Z₂G-module, where G is a group. In this paper we investigate some properties of the cohomological the theory of splittings of groups. Namely, we give a proof of the invariant E(G,W,M), defined in [5] and present related results with independence of E(G,W,M) with respect to the set of G-orbit representatives in W and properties of the invariant E(G,W,FTG) establishing a relation with the end of pairs of groups ê(G, T), defined by Kropphller and Holler in [15]. The main results give necessary conditions for G to split over a subgroup T, in the cases where M = Z₂(G/T ) or M = FTG. |
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Article |
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Fanti, E.L.C. Silva, L.S. |
| spellingShingle |
Fanti, E.L.C. Silva, L.S. Some properties of E(G,W,FTG) and an application in the theory of splittings of groups Algebra and Discrete Mathematics |
| author_facet |
Fanti, E.L.C. Silva, L.S. |
| author_sort |
Fanti, E.L.C. |
| title |
Some properties of E(G,W,FTG) and an application in the theory of splittings of groups |
| title_short |
Some properties of E(G,W,FTG) and an application in the theory of splittings of groups |
| title_full |
Some properties of E(G,W,FTG) and an application in the theory of splittings of groups |
| title_fullStr |
Some properties of E(G,W,FTG) and an application in the theory of splittings of groups |
| title_full_unstemmed |
Some properties of E(G,W,FTG) and an application in the theory of splittings of groups |
| title_sort |
some properties of e(g,w,ftg) and an application in the theory of splittings of groups |
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Інститут прикладної математики і механіки НАН України |
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2020 |
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http://dspace.nbuv.gov.ua/handle/123456789/188562 |
| citation_txt |
Some properties of E(G,W,FTG) and an application in the theory of splittings of groups / E.L.C. Fanti, L.S. Silva // Algebra and Discrete Mathematics. — 2020. — Vol. 30, № 2. — С. 179–193. — Бібліогр.: 19 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT fantielc somepropertiesofegwftgandanapplicationinthetheoryofsplittingsofgroups AT silvals somepropertiesofegwftgandanapplicationinthetheoryofsplittingsofgroups |
| first_indexed |
2025-07-16T10:40:11Z |
| last_indexed |
2025-07-16T10:40:11Z |
| _version_ |
1837799741603184640 |
| fulltext |
© Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 30 (2020). Number 2, pp. 179–193
DOI:10.12958/adm1246
Some properties of E(G,W,FTG)
and an application in the theory of splittings
of groups∗
E. L. C. Fanti and L. S. Silva
Communicated by L. A. Kurdachenko
Abstract. Let us consider W a G-set and M a Z2G-module,
where G is a group. In this paper we investigate some properties
of the cohomological the theory of splittings of groups. Namely, we
give a proof of the invariant E(G,W,M), defined in [5] and present
related results with independence of E(G,W,M) with respect to the
set of G-orbit representatives in W and properties of the invariant
E(G,W,FTG) establishing a relation with the end of pairs of groups
ẽ(G, T ), defined by Kropphller and Holler in [15]. The main results
give necessary conditions for G to split over a subgroup T , in the
cases where M = Z2(G/T ) or M = FTG.
Introduction
The theory of splittings of groups is closely related to the theory of
ends of groups and pairs of groups. The first result relating splittings of
groups to the theory of ends of groups was given by Stallings in [19], where
it is proved that if G is finitely generated, then G splits over some finite
subgroup if only if e(G) > 2. The number of ends e(G) of a group G was
introduced by Freudenthal - Hopf ([12], [13]) for finitely generated groups
∗The authors’ research was supported by FAPESP (grant 12/24454-8, 16/24707-4)
and CAPES.
2010 MSC: 20E06, 20J06, 57M07.
Key words and phrases: cohomology of groups, cohomological invariants, split-
tings and derivation of groups.
https://doi.org/10.12958/adm1246
180 Some properties of E(G,W,FTG)
and it was motivated by theory of ends of topological spaces. Later, Specker
([18]) extended the definition to cover arbitrary groups. In the case where G
is infinite, one has a cohomological formula e(G) = 1+dimZ2
H1(G;Z2G).
A natural extension of e(G) is the invariant end e(G, T ) for pairs of group,
T a subgroup of G, which was introduced independently by Houghton
([14]) and Scott ([16]).
Invariant ends for pairs of groups in a more general set were studied
by Kropholler and Roller, for example in [15]. The authors defined the
invariant end e(G,S,M) for S a subgroup of G and M, a Z2G-module
and presented an interesting study for ẽ(G,S) := e(G,S,P(S)), where
P(S) is the power set of all subsets of S. In an attempt to obtain a
cohomological formula for e(G, T ), Andrade and Fanti ([2]) defined an
invariant end, E(G,S,M), where S is a family of subgroups of G and M
is a Z2G-module. Afterwards, in [5] the authors adapted the definition of
E(G,S,M) by using the cohomology theory of Dicks and Dunwoody to
pairs (G,W ), where W is a G-set, and defining the invariant E(G,W,M),
they obtained general properties and results about splittings and duality
of groups, particularly by considering E(G,W,Z2).
In this work, we present properties of E(G,W,M) and by using deriva-
tion we prove that the definition of this invariant is independent of the
set of G-orbit representatives in W . We consider the case where M is the
Z2G-module FTG and we show that, under certain conditions for T , there
is a relation between E(G,W,FTG) and the invariant ẽ(G, T ). Finally,
we present necessary conditions to G splits over a, non necessary finite,
subgroup T where M = Z2(G/T ) or M = FTG.
1. Preliminaries
Let us recall definitions and results important for this work. We will
consider R = Z or Z2.
Definition 1. A group G is defined by generators X = {xk} and relations
R = {rj = 1}, if G ≃ F/H, where F is a free group generated by X and
H is the smallest normal subgroup of F generated by {rj}. In this case,
〈X;R〉 is a presentation of G.
Definition 2. Consider the groups G1 and G2 with presentations Gk =
〈Xk;Rk〉 , k = 1, 2, where Xk is a set of generators and Rk is a set of
relations for Gk.
1) If T1 ⊂ G1 and T2 ⊂ G2 are subgroups and θ : T1 → T2 is an
isomorphism from T1 into T2, then the free product G1 ∗T G2 of G1 and
E. L. C. Fanti, L. S. Silva 181
G2 with the amalgamated subgroup T = T1 = T2 is defined by
G1 ∗T G2 := 〈X1, X2;R1,R2, t = θ(t), ∀t ∈ T1〉 .
2) Let G1 be a group with presentation G1 = 〈X;R1〉. If T and T ′ are
subgroups of G1 and θ : T → T ′ is an isomorphism, then the HNN -group
G1∗T,θ over a base group G1, with respect to θ and stable letter p, is given
by
G1∗T,θ :=
〈
X1, p;R1, p
−1tp = θ(t), ∀t ∈ T
〉
.
Remark 1. These classes of groups arise naturally when we calculate the
fundamental group of certain spaces.
Definition 3. A group G splits over a subgroup T if G = G1 ∗T G2 with
G1 6= T 6= G2 or G = G1∗T,θ.
Example 1. 1) G = G1 ∗G2 = G1 ∗{1}G2, in particular, Z∗Z = Z∗{1}Z.
2) Z = {1}∗{1}, id =
〈
{1}, p, psp−1 = s, ∀s ∈ {1}
〉
= 〈p〉.
3) The fundamental group G of the Torus is Z⊕ Z which is the HNN
group, Z ⊕ Z = 〈a〉 ⊕ 〈b〉 = “Z∗Z, id”. We can see this by considering
H = 〈a〉 ≃ Z, b the stable letter, T = T ′ = H and σ = id : T → T .
Thus, H∗T, id = 〈a, b; b−1 · a · b = σ(a)〉 = 〈a, b; a · b = b · a〉 = Z ⊕ Z.
Therefore, Z⊕ Z splits over an infinite subgroup.
4) The fundamental group of the Bitorus is (Z∗Z)∗Z (Z∗Z). It follows
from the Seifert-van-Kampen Theorem for appropriate subspaces.
Remark 2. If G = G1 ∗T G2 with G1 6= T 6= G2 or G = G1∗T,θ, then
[G : Gi] = ∞, i = 1, 2 (see [3]).
Proposition 1 ([8]). 1) If G = G1 ∗T G2, one has the following short
exact sequence of RG-modules:
0 → R(G/T )
α
→ R(G/G1)⊕R(G/G2)
ε̄
→ R → 0,
where α is given by α(gT ) = (gG1,−gG2), g ∈ G and ε̄(gG1, 0) =
ε̄(0, gG2) = 1. Here ε̄ is the augmentation map.
2) If G = G1∗T,θ, one has the following short exact sequence of RG-
modules:
0 → R(G/T )
α
→ R(G/G1)
ε̃
→ R → 0,
where α is given by α(gT ) = gG1 − gp−1G1, p is the stable letter of G
and ε̃ is the augmentation map.
182 Some properties of E(G,W,FTG)
Now let us introduce a relationship between the concepts of derivation
groups and of the cohomology group H1(G;M) (see [10]), used to prove the
independence of E(G,W,M) with respect to set of G-orbit representatives
in W .
Definition 4. A derivation of a group G in an RG-module M is a map
d : G → M, such that, d(gh) = d(g) + gd(h), for all g, h ∈ G.
Example 2. For each m ∈ M , the map dm : G → M ; dm(g) := gm−m
is a derivation.
The derivations of the previous example are called principal derivations.
Proposition 2 ([10]). Let G be a group and M an RG-module. Then
H1(G;M) ≃
Der(G,M)
P (G,M)
,
where Der(G,M) is the set of derivations and P (G,M) = {dm, m ∈ M}
is the set of principal derivations of G in M .
Proposition 3 (Shapiro’s Lemma; see [10]). If S is a subgroup of the
group G and M is an RS-module, then H∗(G; CoindGS M) ≃ H∗(S;M).
Example 3. Let G be a group. In P(G), the power set of G, considered
with the operation of symmetric difference we can define an induced G-
action: G× P(G) → P(G); (g,A) 7→ g · A = {g · a; a ∈ A}, which turns
out P(G) a Z2G-module. One has, by Shapiro’s Lemma,
Hn(G;P(G)) ≃ Hn(G; CoindG{1} Z2)
≃ Hn({1};Z2) ≃
{
Z2, se n = 0,
0, se n > 1.
Proposition 4 (Mackey’s Formula; see [10], Proposition III.5.6). Let S
and T be subgroups of a group G and consider E a set of representatives
for the double classes SgT . Then, for any RT -module M , there exists an
RS-isomorphism
ResGS IndGT M ≃
⊕
g∈E
IndSS∩gTg−1 Res
gTg−1
S∩gTg−1 gM.
In particular, if T is a normal subgroup of G and T ⊂ S, then there exists
a Z2T -isomorphism
ResGS IndGT M ≃
⊕
g∈E
IndST gM,
E. L. C. Fanti, L. S. Silva 183
and in the case of T = {1} and L is a set representatives for the classes
Sg, then
ResGS IndG{1}M ≃
⊕
g∈L
IndS{1} gM.
Definition 5 ([11]). Consider G a group, W a G-set and RW the free
R-module generated by W . Let ε : RW → R be the augmentation map,
∆ = ker ε, M an RG-module and P → ∆ a projective RG-resolution of ∆.
The relative cohomology group of the pair (G,W ), with coefficients in M ,
is defined by
Hk(G,W ;M) := Hk−1(HomRG(P,M)), for all k ∈ Z.
For the relative cohomology group of pairs (G,S), where S is a family
of subgroups of G, we define:
Definition 6 ([9]). Let (G,S) be a pair with G a group and S = {Si, i ∈
I} a family of subgroups of infinite index in G, M a Z2G-module, ε′ :
Z2(G/S) → Z2; ε
′(gSi) = 1, the augmentation map, ∆S the kernel of ε′
and F ։ Z2 a Z2G-projective resolution of the trivial Z2G-module Z2.
The relative cohomology group of (G,S), with coefficients in M , is
defined for all n by
Hn(G,S;M) := Hn−1(G; HomZ2
(∆S ,M))
= Hn−1(HomZ2G(F,HomZ2
(∆S ,M)).
The next result provides a relation between the relative cohomology
groups of the pair (G,W ) and the relative cohomology groups of the pair
(G,S).
Theorem 1 ([7]). Let G be a group, M an RG-module and W a G-set.
Consider E = {wi, i ∈ I} a set of orbit representatives in W and let
S = {Si := Gwi
, i ∈ I} be the family of stabilizer subgroups of wi in E.
Then,
Hk(G,W ;M) ≃ Hk(G,S;M).
Proposition 5. Let (G,W ) be a pair where G is a group and W is a G-
set. Consider E a set of G-orbit representatives in W , Gw the G-stabilizer
of w, for each w ∈ E and M an RG-module. Then, we have the long exact
sequence:
0 → H0(G;M) → H0(W ;M)
δ
→ H1(G,W ;M)
J
→
J
→ H1(G;M)
resG
W→ → H1(W ;M) → · · · ,
184 Some properties of E(G,W,FTG)
where
resGW : Hn(G;M)→Hn(W ;M)
not.
=
∏
w∈E
Hn(Gw;M)
resGW ([f ]) :=
(
resGGw
[f ]
)
w∈E
,
and resGGw
: Hn(G;M) → Hn(Gw;M) is the restriction map induced by
the inclusion map Gw →֒ G, w ∈ E.
Proof. The above sequence is the exact sequence of the pair (G,S), where
S is a family of subgroups of G, given in [9], using the notation of Dicks
and Dunwoody.
2. Properties of E(G,W,M)
Let us consider G a group, W a G-set, E a set of G-orbit representatives
in W and M a Z2G-module. Suppose that [G : Gw] = ∞, for all w ∈ E.
Let us recall the definition of E(G,W,M) and some general properties.
The main goal of this section is to show that E(G,W,M) is independent
of E, using derivation of groups.
Definition 7 ([5]). Define E(G,W,M) := 1 + dimker resGW , where,
resGW : H1(G;M) → H1(W ;M) =
∏
w∈E
H1(Gw;M);
resGW ([f ]) =
(
resGGw
[f ]
)
w∈E
and
resGGw
: H1(G;M) → H1(Gw;M)
is the restriction map induced by the inclusion map Gw →֒ G, w ∈ E.
Remark 3. 1) In the previous definition, the requirement [G : Gw] = ∞
implies that G is always an infinite group.
2) The definition of E(G,W,M) and other invariants for pairs of groups
were, in general, motivated by the definition of e(G) and e(G, T ), where T
is a subgroup of G. For more details see [13], [16] and [17]. Since e(G) = 0
if G is finite and e(G, T ) = 0 if [G : T ] < ∞, it is natural to require
[G : Gw] = ∞, ∀w ∈ E in the definition of E(G,W,M) presented.
3) We can prove that E(G,W,M) is a cohomological algebraic in-
variant in the sense that if C is the category whose objects are the pairs
((G,W ),M) in the above conditions, and ((G,W ),M) is isomorphic to
((G′,W ′),M ′) in C then E(G,W,M) = E(G′,W ′,M ′). In the category
E. L. C. Fanti, L. S. Silva 185
whose objects are the pairs ((G,S),M)) it is proved in [1] that E(G,S,M)
is a cohomological algebraic invariant.
The next proposition gives us a property of E(G,W,M) that will
be useful to prove its independence with respect to the set of G-orbit
representatives in W . It is an adaptation of the result presented in [6], for
pairs (G,S), where S is a family of subgroups of G.
Proposition 6. E(G,W,M) = 1 + dim
⋂
w∈E ker resGGw
.
Lemma 1. Let us consider W a G-set, w and w′ elements of W and Gw
and Gw′ the respective isotropic subgroups. If the G-orbits G(w) = G(w′)
and w′ = g0w with g0 ∈ G, then Gw′ = g0Gwg
−1
0 .
Proof. Suppose that G(w′) = G(w) and w′ = g0w. Then
g ∈ Gw′ ⇐⇒ gw′ = w′ ⇐⇒ gg0w = g0w ⇐⇒ g−1
0 gg0w = w
⇐⇒ g−1
0 gg0 ∈ Gw ⇐⇒ g ∈ g0Gwg
−1
0 .
Proposition 7. If W is a G-set, w and w′ are elements of the same G-
orbit and w′ = g0w with g0 ∈ G, then by considering the maps restriction
below:
resGGw
: H1(G;M) → H1(Gw;M)
resGG
w′
: H1(G;M) → H1(Gw′ ;M)
one has ker resGGw
= ker resGG
w′
.
Proof. By Proposition 2, we have
H1(G;M) ≃
Der(G,M)
P (G,M)
and H1(Gw;M) ≃
Der(Gw,M)
P (Gw,M)
.
We will indicate an element d+ P (G,M) of H1(G,M) by [d]. Then,
we have resGGw
: H1(G;M) → H1(Gw;M), [d] 7→ [d|Gw
]. Thus,
[d] ∈ ker resGGw
⇐⇒ resGGw
([d]) = 0 ⇐⇒ [d|Gw
]=0 ⇐⇒ d|Gw
∈P (Gw,M).
Therefore, there exists m ∈ M such that d|Gw
= dm : Gw → M .
Similarly, [d] ∈ ker resGG
w′
if there exists m′ ∈ M such that d|G
w′
=
dm′ : Gw′ → M . Let us show that ker resGGw
⊂ ker resGG
w′
. Suppose [d] ∈
ker resGGw
and let m ∈ M such that d|Gw
= dm.
By the previous lemma, since w′ = g0w, we have Gw′ = g0Gwg
−1
0 . Let
m′ := g0m− d(g0). We can see that [d] ∈ ker resGG
w′
. Indeed if g′ ∈ Gw′ ,
186 Some properties of E(G,W,FTG)
then g′ = g0gg
−1
0 , for some g ∈ Gw. Thus, since in a derivation d(g) =
−gd(g−1), one has d|G
w′
(g′) = dm′(g′). Hence, ker resGGw
⊂ ker resGG
w′
.
Analogously, we have ker resGG
w′
⊂ ker resGGw
. Therefore, ker resGGw
=
ker resGG
w′
.
Corollary 1. The definition of E(G,W,M) is independent of G-orbit
representatives set in W .
Proof. Let W be a G-set and let E, E′ be sets of G-orbit representatives
in W . Note that if w′ = g0w, with w ∈ E and w′ ∈ E
′, then
[G : Gw] = ∞ ⇐⇒ [G : Gw′ ] = ∞,
because [G : Gw] = |G(w)| = |G(w′)| = [G : Gw′ ]. Hence, the result
follows by the previous proposition and Proposition 6.
We note that the previous result was proved in [7] using different
arguments.
Proposition 8 ([5]). If the Z2-vector spaces: H0(G;M), H0(W ;M) =∏
w∈E H0(Gw;M)and H1(G,W ;M) have finite dimensions, then
E(G,W,M) = 1+ dimH0(G;M)− dimH0(W ;M) + dimH1(G,W ;M).
Proposition 9. Let G be an infinity group and W a G-set. If the G-
action in W is free and E is a set of G-orbit representatives in W , then
E(G,W,M) = 1 + dimH1(G;M).
Proof. Since the action is free, we have Gw = {g ∈ G; gw = w} = {1}
for all w ∈ E, so, H1(Gw;M) = H1({1};M) = 0 for all w ∈ E and
ker resGW = H1(G;M). Therefore, E(G,W,M) = 1 + dimH1(G;M).
The above result was presented in [5] for the particular invariant
E(G,W,Z2) but, as we have shown, it is true for any Z2G-module M .
The next proposition is similar to that introduced in [2], for pairs of
groups (G,S), where S is a subgroup of the group G with [G : S] = ∞
and it is adapted here for pairs (G,W ).
Proposition 10. Let G be a group and let W be a G-set with [G : Gw] =
∞, ∀w ∈ W . Consider E a set of G-orbit representatives in W and let
N and M be Z2G-modules. If there exists a Z2G-homomorphism φ :
N → M such that the induced map φ∗ : H1(G;N) → H1(G;M) is a
monomorphism, then E(G,W,N) 6 E(G,W,M).
E. L. C. Fanti, L. S. Silva 187
3. E(G,W,FTG) and splittings of groups
In this section, we initially introduce some notations, briefly recalling
the definition of ẽ(G, T ), where T is a subgroup of G (for more details
see [15]), and we present some properties of the invariant E(G,W,FTG)
establishing a relation with the end of the pair ẽ(G, T ). Then, we prove
the two main results of this work, which give applications in the theory of
splittings of groups.
The following subsets are Z2G-submodules of P(G)
FG := {F ⊂ G; F is finite} ⊂ P(G),
FTG := {A ∈ P(G); A ⊂ F.T, for some F ∈ FG},
QG := {A ⊂ G; A+ g ·A ∈ FG, ∀g ∈ G},
where G, “+” denotes the operation of symmetric difference in P(G) and
g ·A := {ga, a ∈ A}.
Remark 4. If T is a subgroup of the group G, then:
1) FTG = P(G) when [G : T ] < ∞.
2) If S is a subgroup of G, with T ⊂ S and [S : T ] < ∞, then
FSG = FTG.
3) FTG ≃ IndGT P(T ).
4) If T is a normal subgroup of G, then gP(T ) ≃ P(T ).
5) If T is a finite subgroup of G, then FTG = FG ≃ Z2G ≃ IndG{1} Z2.
Definition 8 ([15]). Let T be a subgroup of the group G. Then
ẽ(G, T ) := dimH0(G;P(G)/FTG) = dim(P(G)/FTG)G.
Lemma 2 ([15]). If [G : T ] = ∞, then ẽ(G, T ) = 1 + dimH1(G;FTG).
Proposition 11. If [G : T ] < ∞, then E(G,W,FTG) = 1.
Proof. Using known results and the hypothesis of the proposition, we
obtain
H1(G;FTG)
Remark 4.3
≃ H1(G; IndGT P(T ))
[G:T ]<∞
≃ H1(G; CoindGT P(T ))
Shapiro
≃ H1(T ;P(T ))
Example 3
= 0.
Thus, ker resGW,FTG = 0.
Therefore, E(G,W,FTG) = 1 + dimker resGW,FTG = 1.
Proposition 12. Let G be a group and let W be a G-set. If the G-action in
W is transitive and Gw, w ∈ W is finitely generated and normal subgroup
188 Some properties of E(G,W,FTG)
of G, with [G : Gw] = ∞, then
E(G,W,FGw
G) = 1 + dimH1(G;FGw
G) = ẽ(G,Gw).
Proof. From Remark 4.3, Proposition 4 and Remark 4.4 one has
H1(Gw;FGw
G)
Remark 4.3
≃ H1(Gw; Ind
G
Gw
P(Gw))
Prop. 4
≃ H1(Gw;
⊕
g∈G/Gw
gP(Gw)) ≃
⊕
g∈G/Gw
H1(Gw; gP(Gw))
Remark 4.4
≃
⊕
g∈G/Gw
H1(Gw;P(Gw))
Example 3
≃ 0.
Hence, resGW : H1(G;FGw
G) → H1(Gw;FGw
G) is the null map and
ker resGW = H1(G;FGw
G). Therefore,
E(G,W,FGw
G) = 1 + dimH1(G;FGw
G)
Lemma 2
= ẽ(G,Gw).
Proposition 13. Let (G,W ) be a pair where G is a group and W is
a G-set. Consider E a set of G-orbit representatives in W such that
[G : Gw] = ∞, ∀w ∈ E and let T be a subgroup of G with [G : T ] = ∞.
Then, E(G,W,Z2(G/T )) 6 E(G,W,FTG).
Proof. Consider the short exact sequence of Z2T -modules:
0 → Z2 ≃ {∅, T} → P(T ) → Q =
P(T )
{∅, T}
→ 0.
Since Z2G is a free Z2T -module, we obtain the following exact sequence:
0 → Z2G⊗Z2T Z2
φ
→ Z2G⊗Z2T P(T )
π
→ Z2G⊗Z2T Q → 0.
This sequence induces the long exact sequence in cohomology groups:
0 → H0(G; IndGT Z2) → H0(G; IndGT P(T )) → H0(G; IndGT Q)
→ H1(G; IndGT Z2)
φ∗
→ H1(G; IndGT P(T )) → H1(G; IndGT Q) → · · ·
By hypothesis [G : T ] = ∞, then
H0(G; IndGT Z2) = H0(G; IndGT P(T )) = H0(G; IndGT Q) = 0.
So, φ∗ : H1(G; IndGT Z2) → H1(G; IndGT P(T )) is a monomorphism,
hence by Proposition 10, E(G,W, IndGT Z2) 6 E(G,W, IndGT P(T )).
Since Z2(G/T ) ≃ IndGT Z2, and FTG ≃ IndGT P(T ), it follows that
E(G,W,Z2(G/T )) 6 E(G,W,FTG).
E. L. C. Fanti, L. S. Silva 189
In the following main theorems, we will give necessary conditions for G
to split over a, not necessarily finite, subgroup T , considering M = Z2(G/T )
or M = FTG.
Theorem 2. Let (G,W ) be a pair where G is a group that splits over
a subgroup T in G. Suppose that [G : T ] = ∞ and let W be a G-
set. Consider E a set of G-orbit representatives in W and suppose that
H0(T ;Z2(G/T )) ≃ Z2, then if G = G1 ∗T G2 and E = {w1, w2} with
Gwi
= Gi, i = 1, 2, or G = G1∗T,θ and E = {w}, one has
E(G,W,Z2(G/T )) 6 2.
Proof. Suppose that G = G1 ∗T G2. Using the short exact sequence given
in Proposition 1.1, we have ∆ = ker ε ≃ Z2(G/T ) (∗). Thus,
H1(G,W ;Z2(G/T )) ≃ H1(G, {G1, G2};Z2(G/T ))
:= H0(G; HomZ2
(∆,Z2(G/T ))
(∗)
≃ H0(G; HomZ2
(Z2(G/T ),Z2(G/T )))
≃ H0(G; CoindGT (Res
G
T Z2(G/T )))
Shapiro
≃ H0(T ;Z2(G/T ))
Hyp.
≃ Z2.
Furthermore, H0(G;Z2(G/T )) ≃ (IndGT )
G [G:T ]=∞
= 0. Then, by Propo-
sition 5, we have the long exact sequence:
0 → H0(G1;Z2(G/T ))⊕H0(G2;Z2(G/T ))
δ
→ H1(G,W ;Z2(G/T )) ≃ Z2
J
→ H1(G;Z2(G/T )) → · · ·
Therefore, δ is a monomorphism and
H0(W ;Z2(G/T )) = H0(G1;Z2(G/T ))⊕H0(G2;Z2(G/T ))
is a submodule of H1(G,W ;Z2(G/T )) ≃ Z2. Thereby, we have the follow-
ing possibilities: H0(Gi;Z2(G/T )) = 0, i = 1, 2, or H0(Gi;Z2(G/T )) =
0 and H0(Gj ;Z2(G, T )) ≃ Z2, i, j ∈ {1, 2}, i 6= j. In other
words, dimH0(G1;Z2(G/T )) + dimH0(G2;Z2(G/T )) can be 0 or 1,
(dimH0(W ;Z2(G/T )) 6 1). Therefore, by Proposition 8 one has
E(G,W ;Z2(G/T )) = 1− dimH0(W ;Z2(G/T )) + 1 6 2.
If G = G1∗T,θ, the proof is analogous.
Theorem 3. Let (G,W ) be a pair where G is a group and W is a G-set.
Consider E a set of G-orbit representatives in W and T a, not necessarily
finite, normal finitely generated subgroup of G. If G = G1 ∗T G2, E =
190 Some properties of E(G,W,FTG)
{w1, w2} with Gwi
= Gi and [Gi : T ] = ∞, i = 1, 2, or G = G1∗T,θ,
E = {w} with Gw = G1 and [G1 : T ] = ∞, then E(G,W,FTG) = ∞.
Proof. Consider G = G1 ∗T G2. Note that [G : Gi]
Remark 2
= ∞ and
[G : T ] = [G : Gi][Gi : T ] = ∞. Thus, we have
H0(G;FTG)
Remark 4.3
≃ H0(G; IndGT P(T )) ≃ (IndGT P(T ))G
[G:T ]=∞
= 0.
Consider L a transversal for the coset GigT, i = 1, 2. Then,
H0(Gi;FTG)
Remark 4.3
≃ H0(Gi; Ind
G
T P(T ))
Prop. 4
≃ H0(Gi;
⊕
g∈L
IndGi
T gP(T ))
Remark 4.4
≃ H0(Gi;
⊕
g∈L
IndGi
T P(T ))
≃
⊕
g∈L
H0(Gi; Ind
Gi
T P(T )) ≃
⊕
g∈L
(IndGi
T P(T ))Gi
[Gi:T ]=∞
= 0.
Thus, H0(W ;FTG) = H0(Gw1
;FTG) ⊕H0(Gw2
,FTG) = 0 and the
long exact sequence given in Proposition 5 takes the following form:
0 → H1(G,W ;FTG)
J
→ H1(G;FTG)
resG
W→ H1(W ;FTG) → · · ·
Since J is injective and ker resGW = Im J it follows that
dimH1(G,W ;FTG) = dim Im J = dimker resGW . (∗)
On the other hand, from the short exact sequence given in Proposi-
tion 1.1 we have ∆ = ker ε = Imα ≃ Z2(G/T ). So,
H1(G,W ;FTG)
Theorem 1
≃ H1(G, {G1, G2};FTG)
:= H0(G; HomZ2
(∆,FTG)) ≃ H0(G; HomZ2
(Z2(G/T ),FTG))
Shapiro
≃ H0(T ;FTG)
Remark 4.3
≃ H0(T ; IndGT P(T ))
≃ H0(T ;
⊕
g∈G/T
gP(T )) ≃
⊕
g∈G/T
H0(T ; gP(T ))
Remark 4.4
≃
⊕
g∈G/T
H0(T ;P(T )).
Since H0(T ;P(T ))
Example 3
≃ Z2 and [G : T ] = ∞, we have
dimH1(G,W ;FTG) = ∞.
E. L. C. Fanti, L. S. Silva 191
Furthermore, it follows from (∗) that
dimker resGW = ∞.
Therefore, E(G,W,FTG) = ∞.
If G = G1∗T,θ, similarly to the previous case, we obtain H0(G;FTG)
= 0 = H0(G1;FTG) and dimH1(G,W ;FTG) = dimker resGW . Besides,
it follows from the short exact sequence given in Proposition 1.2 that
∆ = ker ε = Imα ≃ Z2(G/T ). Thus, as before, we conclude that
dimH1(G,W ;FTG) = H1(G, {G1};FTG) = ∞,
consequently,
dimker resGW = ∞ and E(G,W,FTG) = ∞.
Remark 5. It is important to know on the existence of a subgroup T of
a group G satisfying the conditions of the previous theorem, that is, T
is a finitely generated and normal subgroup of G with [G1 : T ] = ∞ =
[G2 : T ]. We note that Scott and Wall in [17] showed the following fact:
If G = G1 ∗ G2 where G1 and G2 are non-trivial and H is a finitely
generated, normal subgroup of G, then H is trivial or has finite index in G.
Then the authors questioned whether there is an analogous result when
G = G1 ∗T G2 or G = G1∗T,θ. Clearly, if this were true, our result would
not make sense, since in this case, considering H = T with T a finitely
generated and normal subgroup of G, we should have T trivial or [G : T ]
finite. We are interested in the case where T is infinite, so non-trivial, and
[G : T ] = [G : G1][G1 : T ] = ∞. The following example was suggested to
us by G. P. Scott.
Example 4. Consider any group T and G1 and G2 two finitely generated
groups for which T is a normal subgroup. For example, G1 = A× T and
G2 = B × T , where A and B are any finitely generated infinite groups.
A and B can be taken, for example, cyclic groups. Then [G1 : T ] = ∞ =
[G2 : T ] and we can see that T is normal in G := G1 ∗T G2, by using the
Normal Form Theorem, given in [8].
Based on the previous example, we present an application of the
Theorem 3.
Example 5. Given T a finitely generated group, not necessarily finite,
consider G = G1∗TG2 as in the previous example,W = G/G1 ∪̇G/G2 and
a map φ : G×W → W , such that, if w1 = gG1 ∈ W then g′ ·w1 = g′gG1
192 Some properties of E(G,W,FTG)
and if w2 = gG2 ∈ W , then g′ ·w2 = g′gG2. Observe that φ is a G-action
in W . Now, if w1 = 1 ·G1 and w2 = 1 ·G2, one has
G(w1) = {g · 1 ·G1, g ∈ G} = {g ·G1, g ∈ G} = G/G1,
G(w2) = {g · 1 ·G2, g ∈ G} = {g ·G2, g ∈ G} = G/G2.
Furthermore,
Gw1
= {g ∈ G, g · w1 = w1} = {g ∈ G, g ·G1 = G1} = G1.
Similarly, Gw2
= G2. Thus, there exists a G-set W , such that E =
{w1, w2} with Gw1
= G1 and Gw2
= G2. Therefore, by Theorem 3,
E(G,W,FTG) = ∞.
Acknowledgement
We would like to thank to Professor G. P. Scott for the prompt attention
and a suggestion for an example.
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Contact information
Ermínia de Lourdes
Campello Fanti
Department of Mathematics - UNESP - São
Paulo State University, IBILCE, R. Cristovão
Colombo, 2265, CEP 15054-000, São José do
Rio Preto-SP, Brazil
E-Mail(s): erminia.c.fanti@unesp.br
Leticia Sanches
Silva
IFSP - Federal Institute of Technology in São
Paulo, Av. dos Universitários, 145, CEP
17607-220, Tupã-SP, Brazil
E-Mail(s): sanches.leticia@ifsp.edu.br
Received by the editors: 05.09.2018
and in final form 11.07.2020.
mailto:erminia.c.fanti@unesp.br
mailto:sanches.leticia@ifsp.edu.br
E. L. C. Fanti, L. S. Silva
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