NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND
GROUP HOMOLOGY
ROMAN MIKHAILOV AND INDER BIR S. PASSI

arXiv:1611.01313v1 [] 4 Nov 2016

Dedicated to the memory of Chander Kanta Gupta and Narain Gupta
Abstract. This paper is about application of various homological methods to classical
problems in the theory of group rings. It is shown that the third homology of groups
plays a key role in Narain Gupta’s three normal subgroup problem. For a free group
F and its normal subgroups R, S, T, and the corresponding ideals in the integral group
ring Z[F ], r = (R − 1)Z[F ], s = (S − 1)Z[F ], t = (T − 1)Z[F ], a complete description
of the normal subgroup F ∩ (1 + rst) is given, provided R ⊆ T and the third and the
fourth homology groups of R/R ∩ S are torsion groups.

1. Introduction
It is well-known that the second (co)homology of groups plays an important role in the
theory of groups; in particular, in the theory of central extensions. The third cohomology
of a group classifies k-invariants for crossed modules or homotopy 2-types ([10], [11], [19]).
However, it is not easy to find an explicit application of the third (co)homology in grouptheoretical questions which are formulated without the language of homological algebra.
In this paper, we show that the third homology of groups plays a key role in the solution
of a problem in free group rings.
Let F be a free group and Z[F ] its integral group ring. For every two-sided ideal a in
Z[F ], we have a normal subgroup D(F, a) := F ∩ (1 + a) of F . The identification of such
normal subgroups in free groups is a recurring problem in the theory of group rings (see
[9], [14], [18]). As demonstrated in our works ([15], [16], [17]), homology of groups and
derived functors of non-additive functors can provide a useful tool for investigating such
subgroups. In the present article we use this homological approach to address Narain
Gupta’s problem ([9], Problem 6.3, p. 119) in free group rings which, in general, has been
rather intractable so far.
Given a normal subgroup R of F , let r denote the two-sided ideal of Z[F ] generated
by the augmentation ideal ∆(R) of the integral group ring Z[R], i.e., r = ∆(R)Z[F ].
Clearly D(F, r) = R. For two normal subgroups R, S of F , it is known that D(F, rs) =
[R ∩ S, R ∩ S], the derived subgroup of R ∩ S ([4]; [9], Theorem 1.6, p. 3). If R, S, T
are three normal subgroups of F , a currently open problem formulated by Narain Gupta
(loc. cit.) asks for the identification of the normal subgroup D(F, rst). The answer to
this general problem is known for the following special cases:
1

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

D(F, rfs) =

2

D(F, rfr) = γ3 (R) ([7]; [9], p. 116),

(1.1)

D(F, rrs) = [R′ ∩ S, R′ ∩ S]γ3 (R ∩ S) [12],

(1.2)

p
[R′ ∩ S ′ , R ∩ S][R ∩ S ′ , R ∩ S ′ ][R′ ∩ S, R′ ∩ S] ([7], [16]);
D(F, frf) =

p

[R′ , F ] [21],

(1.3)
(1.4)

√
where, for groups H ⊆ G, H ′ and H denote respectively the derived subgroup of H
and the isolator in G of the subgroup H, and {γi (G)}i≥1 is the lower central series of G.
Given a triple of subgroups R, S, T of a free group F , set
p
I(R, S, T ) := [(R ∩ S)′ ∩ (S ∩ T )′ , R ∩ T ](R ∩ (S ∩ T )′ )′ ((R ∩ S)′ ∩ T )′ .
Observe that, for all the above-mentioned known identifications of D(F, rst), we have
D(F, rst) = I(R, S, T ).

(1.5)

The object of the present work is to investigate the case when R ⊆ T (or equivalently,
in view of the canonical anti-automorphism of Z[F ], when T ⊆ R). It is easy to see
that a complete answer for this case, together with the known results (1.2, 1.3, 1.4), will
provide identification of D(F, rst) whenever one of the three normal subgroups R, S, T
is contained in either of the other two. Our main result is the following
Theorem 1.1. If R, S, T are normal subgroups of a free group F , such that R ⊆ T , and
the integral homology groups H3 (R/R ∩ S), H4 (R/R ∩ S) are torsion groups, then
p
D(F, rst) = I(R, S, T ) = (R ∩ (S ∩ T )′ )′ [(R ∩ S)′ , R].
Our proof of the above theorem involves a mix of homological and combinatorial
arguments, which we develop in Section 2, and it is completed in Section 3. A striking feature to note here is the role played by the third homology in the identification of
normal subgroups determined by ideals in free group rings. In Section 4 we bring out
further the role of integral homology and give an example with D(F, rsf)/I(R, S, F )
non-zero, thus showing that (1.5) does not hold in general.
In Section 5, we prove (Theorem 5.2), using combinatoral arguments, that if the normal
subgroup R is contained in both S and T , and
a ∈ D(F, srt + trs),
then
a2 ∈ D(F, rrs + srr + trr + rrt)

and therefore, by [21], Theorem 4, a4 ∈ [R, R, ST ]. Thus, in particular, we have a
combinatorial proof of one of Ralph Stöhr’s results, which is implicit in his homological
approach [21] to Gupta’s problem, namely that if a ∈ D(F, frf ), then a2 ∈ D(rrf + frr).

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

3

We conclude with a few observations on the corresponding four normal subgroup
problem including the identification
D(F, rsfr) = D(F, rfsr) = [R ∩ S ′ , R ∩ S ′ , R]γ4 (R),

provided R ⊆ S, which is a generalization of a result of Chander Kanta Gupta [8].
.
2. Homological and Combinatorial Preliminaries
Theorem 2.1. If F is a free group, and R, S its normal subgroups with S ⊆ R, then
there is a natural isomorphism
fs ∩ rf
H2 (F/S, f/r) ∼
.
=
fsf + rs
Proof. Consider the Gruenberg free resolution ([5], p. 34)
· · · → sf/s2 f → s/s2 → f/sf → Z[F/S] → Z → 0

of Z viewed as a trivial left F/S-module. On tensoring this resolution with the right
F/S-module f/r, we have the complex
· · · → f/r ⊗F/S sf/s2 f → f/r ⊗F/S s/s2 → f/r ⊗F/S f/sf → f/r.

For a free group F , and ideals b ⊂ a, d ⊂ c, we have ([20], Lemma 4.9)
ac
.
(a/b) ⊗F (c/d) ∼
=
bc + ad
Thus the above complex reduces to the following complex
fs
f2
fsf
→ 2
→
→ f/r.
··· → 2
fs f + rsf
fs + rs
fsf + rf
Hence
fs ∩ rf
H2 (F/S, f/r) ∼
.
=
fsf + rs


Remark 2.1. Note that the natural composition
H2 (R/S) ⊗ f/r /
/

H2 (R/S, f/r)
/

H2 (F/S, f/r)



∆2 (R)∩∆(S)Z[R]
∆(R)∆(S)+∆(S)∆(R)

⊗ f/r
/

fs
fsf +rs

is induced by the map
(α, β) 7→ βα, α ∈ ∆2 (R) ∩ ∆(S)Z[R], β ∈ f.
Lemma 2.1. (see [23], Theorem 1.1) If R, S, T are normal subgroups of a free group F
and R ⊆ T , then

∆(R)∆(S)∆(T ) ∩ ∆(R)∆(R ∩ S) = ∆(R)∆(R ∩ S)∆(R) + ∆(R)∆(R ∩ (S ∩ T )′ ). (2.1)

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

4

Proof. We can assume that F = ST = T S. Let
u ∈ ∆(R)∆(S)∆(T ) ∩ ∆(R)∆(R ∩ S).

Since the ideal ∆(R)Z[F ] is a free ringht Z[F ]-module,
X
u=
(y − 1)uy ,

(2.2)

y∈Y

where Y is a free basis of R and

uy ∈ ∆(S)∆(T )Z[F ] ∩ ∆(R ∩ S)Z[F ] = ∆(S)∆(T ) ∩ Z[F ]∆(R ∩ S).

(2.3)

Let Z be a transversal for T in F = ST . Since ∆(R)∆(R ∩ S) ⊆ Z[T ], projecting the
above equation (2.2) under the map θ : Z[F ] → Z[T ] induced by
f = ts 7→ t

for f ∈ F. t ∈ T, s in a right transversal for S ∩ T in S, it follows that
uy ∈ Z[T ].

(2.4)

uy ∈ ∆(S ∩ T )∆(T ) ∩ Z[T ]∆(R ∩ S).

(2.5)

u ∈ ∆(R)∆(R ∩ S)∆(T ) + ∆(R)∆(R ∩ (S ∩ T )′ ).

(2.6)

u ∈ ∆(R)∆(R ∩ S)∆(R) + ∆(R)∆(R ∩ (S ∩ T )′ ).

(2.7)

Similarly, using the projection Z[F ] → Z[T ] induced on using left transversal for S ∩ T
in S, the inclusion (2.3) shows further that
Therefore, uy ∈ ∆(R ∩ S)∆(T ) + ∆(R ∩ (S ∩ T )′ ) [Lemma 2.1 [23], by setting K = S ∩ T ,
H = R]. Consequently
Because u ∈ ∆(R), projecting (2.6) under Z[T ] → Z[R], it follows that

We have thus proved that the left hand side of (2.1) is contained in the right hand side;
the reverse inclusion being obvious, the proof of the Lemma is complete.

A similar analysis, as above, yields the following intersection lemma.
Lemma 2.2. Let R ⊆ S. Then

∆(R)∆(F )∆(S)∆(R) ∩ ∆3 (R) = ∆4 (R) + ∆(R)∆(R ∩ S ′ )∆(R). ✷

For a free group F and its normal subgroup R, and any left Z[F/R]-module M, there
are isomorphisms
(2.8)
Hi (F/R, R/R′ ⊗ M) ∼
= Hi+2 (F/R, M), i ≥ 1.
This is a well-known fact which follows easily from the Magnus embedding R/R′ ֒→ f/fr
of the relation module R/R′ .

Let R, S be normal subgroups of F . For the group G := R/R ∩ S ∼
= RS/S, one can
consider two different relation modules and a natural map between them:
R∩S
→ S/S ′ .
′
(R ∩ S)

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

5

This map can be naturally extended to a map between the corresponding relation
sequences ([6], p. 7):
R∩S
(R∩S)′

/

∆(R)
∆(R∩S)∆(S)

/

//

∆(R)
∆(R∩S)Z[R]
≃




S/S ′ /
/



∆(RS)
∆(S)∆(RS)

//

∆(RS)
∆(S)Z[RS]

∆(R)
∆(RS)
The Z[G]-modules ∆(R∩S)∆(S)
and ∆(S)∆(RS)
are free, hence for any Z[G]-module G, there
are natural isomorphisms


R∩S
⊗M ∼
Hi R/R ∩ S,
= Hi (R/R ∩ S, S/S ′ ⊗ M) ∼
= Hi+2 (R/R ∩ S, M), i ≥ 1.
′
(R ∩ S)
(2.9)

We need some well-known facts about certain quadratic endofunctors on the category
of abelian groups, namely:
⊗2 tensor square,

SP2 symmetric square,
Λ2 exterior square,
˜ 2 antisymmetric square,
⊗

Γ2 divided square.

For a survey of the properties of these functors and their derived functors, see ([1], [2]).
Recall that, for an abelian group A, by definition,
SP2 (A) = ⊗2 (A)/ha ⊗ b − b ⊗ a, a, b ∈ Ai,
Λ2 (A) = ⊗2 (A)/ha ⊗ a, a ∈ Ai,

˜ 2 (A) = ⊗2 (A)/ha ⊗ b + b ⊗ a, a, b ∈ Ai.
⊗

The divided square functor Γ2 is also known as the Whitehead quadratic functor. Given an
abelian group A, the abelian group Γ2 (A) is generated by symbols γ(x), x ∈ A, satisfying
the following relations for all x, y, z ∈ A:
γ(0) = 0;
γ(x) = γ(−x);
γ(x + y + z) − γ(x + y) − γ(x + z) − γ(y + z) + γ(x) + γ(y) + γ(z) = 0.
The exterior and the antisymmetric squares are connected as follows. For every abelian
group A, we have a short exact sequence
2

˜ (A) → Λ2 (A) → 0.
0 → A ⊗ Z/2 → ⊗

(2.10)

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

6

Similarly, connecting the symmetric and divided square functor, we have the following
short exact sequence:
0 → SP2 (A) → Γ2 (A) → A ⊗ Z/2 → 0.

(2.11)

Let E be a free abelian group, I its subgroup and E/I = A. Then there is a natural
exact sequence
0 → L1 SP2 (A) → Λ2 (E)/Λ2 (I) → E ⊗ E/I → SP2 (A) → 0

(2.12)

where L1 SP2 is the first derived functor of SP2 in the sense of Dold-Puppe, and L1 SP2 (A)
is equal to the quotient of Tor(A, A) by the subgroup generated by the diagonal elements.
We refer the reader to [16] and [17] for the proof and applications of above kind of
sequences in the theory of groups rings.
Another ingredient that we need is the following analog of the results from [13] on
Koszul sequences.
Lemma 2.3. For a free abelian group E, I ⊆ E, E/I = A, the homology of the naturally
defined complex
˜ 2 (E)
SP2 (I) ֌ I ⊗ E → ⊗
satisfies the following:
˜ 2 (A),
H0 = ⊗
0 → Tor(A, Z/2) → H1 → L1 Λ2 (A) → 0.

(2.13)

Proof. The description of H0 follows from the natural commutative diagram
SP2 (E)

SP2 (E)




I ⊗E /

2

⊗ (E)
/


//

2

⊗ (A)




I ⊗E
/

˜ 2 (E)
⊗

//

˜ 2 (A)
⊗

since the image of SP2 (E) in ⊗2 (A) is the same as the image of SP2 (A) in ⊗2 (A).
Recall that the first homology of the Koszul sequence
Γ2 (I) → I ⊗ E → Λ2 (E)
is naturally isomorphic to the derived functor L1 Λ2 (A) (see [13]). Observe that the kernel
of the map I ⊗ Z/2 → E ⊗ Z/2 is naturally isomorphic to Tor(A, Z/2). The short exact

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

7

sequence (2.13) follows from the following commutative diagram:
E ⊗ Z/2

A ⊗ Z/2

//






2

SP (I) /

I ⊗E
/





2

˜ (E)
⊗
/

//




Γ2 (I) /
/

I ⊗E
/

Λ2 (E)

2

˜ (A)
⊗


//

Λ2 (A)



I ⊗ Z/2.


3. Proof of Theorem 1.1
Since
F ∩ (1 + ∆(R)∆(S)) = [R ∩ S, R ∩ S] ⊂ (1 + ∆(R ∩ S)∆(R)),

Lemma 2.1 implies that

D := D(F, ∆(R)∆(S)∆(F )) = F ∩ (1 + ∆(R)∆(R ∩ S)∆(R) + ∆(R)∆(R ∩ (S ∩ T )′ ))
Observe that, for w ∈ D, w − 1 ∈ ∆(R ∩ S)∆(R), and
∆(R)∆(R ∩ S)∆(R) ⊂ ∆(R ∩ S)∆(R).
Therefore
D = F ∩ (1 + ∆(R)∆(R ∩ S)∆(R) + ∆(R)∆(R ∩ (S ∩ T )′ ) ∩ ∆(R ∩ S)∆(R))

(3.1)

We have to prove that the quotient
F ∩ (1 + ∆(R)∆(R ∩ S)∆(R) + ∆(R)∆(R ∩ (S ∩ T )′ ) ∩ ∆(R ∩ S)∆(R))
(R ∩ (S ∩ T )′ )′ [(R ∩ S)′ , R]

(3.2)

is a torsion group. By Theorem 2.1,
∆(R)∆(R ∩ (S ∩ T )′ ) ∩ ∆(R ∩ S)∆(R)
= H2
∆(R)∆(R ∩ (S ∩ T )′ )∆(R) + ∆(R ∩ S)∆(R ∩ (S ∩ T )′ )


∆(R)
R
.
,
R ∩ (S ∩ T )′ ∆(R ∩ S)
(3.3)
Consider the homological Hochschild-Serre spectral sequence for the group extension
1→



R
R
R∩S
→
→
→1
′
′
R ∩ (S ∩ T )
R ∩ (S ∩ T )
R∩S

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

8

∆(R)
with coefficients in ∆(R∩S)
. Three terms of the E 2 -page of the spectral sequence, which
give a contribution to the homology group (3.3), are the following:

E0,2 2

E1,2 1

E2,2 0



;



;

= H0



R
, H2
R∩S



R∩S
∆(R)
,
′
R ∩ (S ∩ T ) ∆(R ∩ S)

= H1



R
, H1
R∩S



∆(R)
R∩S
,
′
R ∩ (S ∩ T ) ∆(R ∩ S)

= H2



∆(R)
R
,
R ∩ S ∆(R ∩ S)



= H3



R
R∩S



.

2
2
By hypothesis, E2,0
is a torsion group. Consider the term E1,1
:




R
R
R∩S
∆(R)
R∩S
2
E1, 1 = H1
= H2
,
⊗
,
.
R ∩ S R ∩ (S ∩ T )′ ∆(R ∩ S)
R ∩ S R ∩ (S ∩ T )′

The exact sequence
0→

S∩T
S
R∩S
→
→
→ 0,
′
′
R ∩ (S ∩ T )
(S ∩ T )
R∩S

yields an exact sequence




R
S
S∩T
R
2
→ E1,1 → H2
.
,
,
H3
R∩S R∩S
R ∩ S (S ∩ T )′

(3.4)

(3.5)

The natural isomorphism
R(S ∩ T )
R
→
S∩T
R∩S
induces an isomorphism (see (2.9))




R
S
∩
T
R
∼
,
.
H4
= H2
R∩S
R ∩ S (S ∩ T )′

By hypothesis, all terms in (3.5) are torsion. Consider the natural map




∆(R)
R∩S
∆(R)
R∩S
=
H
⊗
,
→
f : H2
2
R ∩ (S ∩ T )′ ∆(R ∩ S)
R ∩ (S ∩ T )′
∆(R ∩ S)


R
∆(R)
H2
.
,
R ∩ (S ∩ T )′ ∆(R ∩ S)

The natural image of the map f under the isomorphism (3.3) is generated by elements of
the type (f − 1)([r1 , r2 ] − 1), with f ∈ R, r1 , r2 ∈ R ∩ S (see Remark 2.1). It follows that
the quotient group
∆(R)∆(R ∩ (S ∩ T )′ ) ∩ ∆(R ∩ S)∆(R)
im(f ) + ∆(R)∆(R ∩ (S ∩ T )′ )∆(R) + ∆(R ∩ S)∆(R ∩ (S ∩ T )′ )

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

9

is a torsion group. Since im(f ) ⊆ ∆(R)∆(R ∩ S)∆(R), the quotient

∆(R)∆(R ∩ S)∆(R) + ∆(R)∆(R ∩ (S ∩ T )′ ) ∩ ∆(R ∩ S)∆(R)
∆(R)∆(R ∩ S)∆(R) + ∆(R ∩ S)∆(R ∩ (S ∩ T )′)

is a torsion group. Therefore, the quotient (3.2) is torsion if and only if the quotient
F ∩ (1 + ∆(R)∆(R ∩ S)∆(R) + ∆(R ∩ S)∆(R ∩ (S ∩ T )′ ))
(R ∩ (S ∩ T )′ )′ [(R ∩ S)′ , R]

(3.6)

is torsion. Since F ∩ (1 + ∆(R)∆(R ∩ S)∆(R) + ∆(R ∩ S)∆(R ∩ (S ∩ T )′ )) ⊆ (R ∩ S)′ ,
and ∆(R ∩ S)∆(R ∩ (S ∩ T )′ ) ⊂ ∆2 (R ∩ S),
F ∩ (1 + ∆(R)∆(R ∩ S)∆(R) + ∆(R ∩ S)∆(R ∩ (S ∩ T )′ )) =

F ∩ (1 + ∆(R)∆(R ∩ S)∆(R) ∩ ∆2 (R ∩ S)Z[R] + ∆(R ∩ S)∆(R ∩ (S ∩ T )′)).

Recall that


R
∆(R)∆(R ∩ S)∆(R) ∩ ∆2 (R ∩ S)Z[R]
H4
=
R∩S
∆2 (R ∩ S)∆(R) + ∆(R)∆2 (R ∩ S)
is finite by hypothesis. Therefore, the quotient (3.6) is torsion if and only if the quotient


F ∩ (1 + ∆2 (R ∩ S)∆(R) + ∆(R)∆2 (R ∩ S) + ∆(R ∩ S)∆(R ∩ (S ∩ T )′))
(R ∩ (S ∩ T )′ )′ [(R ∩ S)′ , R]

is torsion.

We next observe that the following commutative diagram with exact rows follows from
(2.12) (with E = (R ∩ S)/(R ∩ S)′ , I = (R ∩ (S ∩ T )′ )/(R ∩ S)′ ∩ (S ∩ T )′ , and the fact that
the derived functor L1 SP2 in (2.12) vanishes since the quotient (R ∩ S)/(R ∩ (S ∩ T )′ ⊆
(S ∩ T )/(S ∩ T )′ is torsion-free):
Λ2





R∩S
(R∩S)′

R∩(S∩T )′
(R∩S)′ ∩(S∩T )′

Λ2


/

(R∩S)′
(R∩(S∩T )′ )′ [(R∩S)′ ,R∩S]

/

/

/

R∩S
R∩(S∩T )′

⊗

R∩S
(R∩S)′

∆2 (R∩S)
∆3 (R∩S)+∆(R∩S)∆(R∩(S∩T )′ )

2

//

SP

//

SP2





R∩S
R∩(S∩T )′

R∩S
R∩(S∩T )′





.

On passing to the homology H∗ (R/R ∩ S, −), we obtain the exact sequence


R∩S
R∩S
⊗
→
H1 R/R ∩ S,
R ∩ (S ∩ T )′ (R ∩ S)′


2
H1 R/R ∩ S, SP

R∩S
R ∩ (S ∩ T )′



→

(R ∩ S)′
→
(R ∩ (S ∩ T )′ )′ [(R ∩ S)′ , R]

∆2 (R ∩ S)
(3.7)
∆2 (R ∩ S)∆(R) + ∆(R)∆2 (R ∩ S) + ∆(R ∩ S)∆(R ∩ (S ∩ T )′ )

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

10

That is, we get an epimorphism



R∩S
2
H1 R/R ∩ S, SP
→
R ∩ (S ∩ T )′
F ∩ (1 + ∆2 (R ∩ S)∆(R) + ∆(R)∆2 (R ∩ S) + ∆(R ∩ S)∆(R ∩ (S ∩ T )′ ))
. (3.8)
(R ∩ (S ∩ T )′ )′ [(R ∩ S)′ , R]

The next commutative diagram follows from Lemma 2.3:
SP2



R∩S
R∩(S∩T )′




/

S∩T
(S∩T )′

/

⊗

2

SP2



SP

S∩T
(S∩T )′

/



S∩T
(S∩T )′



/

S∩T
(S∩T )′

S∩T
(S∩T )′

//

˜2
⊗



S∩T
(S∩T )′



˜2
⊗



⊗




S∩T
(S∩T )′
R∩S
)
SP2 ( R∩(S∩T
)′

/

˜2
⊗







K /

/







R∩S
R∩(S∩T )′

S∩T
(S∩T )′

/

⊗

S∩T
(R∩S)(S∩T )′

//

S∩T
(R∩S)(S∩T )′



,

where K lives in the short exact sequence




S∩T
S∩T
2
→ 0;
, Z/2 → K → L1 Λ
0 → Tor
(S ∩ T )′ (R ∩ S)
(S ∩ T )′ (R ∩ S)
in particular, K is a torsion group. Recall that the homology in dimension ≥ 1 of a group
with coefficients in the symmetric (or exterior, or antisymmetric) square of its relation
module is a 2-torsion group. Hence,






S∩T
S∩T
2
2
H1 R/R ∩ S, SP
= H1 R(S ∩ T )/(S ∩ T ), SP
(S ∩ T )′
(S ∩ T )′
is a 2-torsion group. Thus we have exact sequences
H3



R
,
R∩S

˜2
⊗



S∩T
(R∩S)(S∩T )′





H2



SP2 (S∩T /(S∩T )′ )
R
,
R∩S SP2 ( R∩S ′ )+K
R∩(S∩T )


/

H1



R
,
R∩S

SP

2



R∩S
R∩(S∩T )′


/

(2 − torsion)



H2



R
, S∩T
R∩S (S∩T )′

⊗

S∩T
(R∩S)(S∩T )′



H4



S∩T
R
,
R∩S (R∩S)(S∩T )′



.
(3.9)

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

11



S∩T
S∩T
˜ 2 (R∩S)(S∩T
Observe that, ⊗
are trivial Z[R/R ∩ S]-modules and one
and (R∩S)(S∩T
′
)
)′
can use the Universal Coefficient Theorem to decompose the homology groups in the last
diagram. By hypothesis, the left hand terms in the last diagram are torsion and therefore
the left hand term in (3.8) is a torsion group and so the proof is complete. ✷
4. The subgroup D(F, rsf )
Let us consider the case T = F , and the corresponding subgroup D(F, rsf).
Theorem 4.1. If the cohomological dimension cd(R/R ∩ S) ≤ 3, then


D(F, rsf)
S
2
˜
.
←֓ H3 (R/R ∩ S) ⊗ ⊗
(R ∩ S ′ )′ [(R ∩ S)′ , R]
(R ∩ S)S ′
˜ 2 denotes the antisymmetric tensor square].
[Recall that ⊗

Proof. Let us set

B := ∆2 (R ∩ S)∆(R) + ∆(R)∆2 (R ∩ S) + ∆(R ∩ S)∆(R ∩ S ′ ).

Clearly, B ⊆ ∆(R)∆(S)∆(F ), and we have a natural commutative diagram
F ∩(1+B)
I(R, S, F )

(R∩S)′
I(R, S, F )

/

/

/

∆(R)∆(S)
B




D(F, rsf )
I(R, S, F )

/

/

(R∩S)′
I(R, S, F )

/

∆(R)∆(S)
∆(R)∆(S)∆(F ).

The sequence (3.7) implies that there is an exact sequence


R∩S
R∩S
H1 R/R ∩ S,
→
⊗
R ∩ S ′ (R ∩ S)′


H1 R/R ∩ S, SP

2



R∩S
R ∩ S′



→

F ∩ (1 + B)
→ 0 (4.1)
(R ∩ S ′ )′ [(R ∩ S)′ , R]

Using dimension shifting, we get




R∩S
R∩S
R∩S
= H3 R/R ∩ S,
.
⊗
H1 R/R ∩ S,
R ∩ S ′ (R ∩ S)′
R ∩ S′
The short exact sequence (3.4) implies the exact sequence




S
R∩S
H4 R/R ∩ S,
→ H3 R/R ∩ S,
→ H3 (R/R ∩ S, S/S ′ ) .
(R ∩ S)S ′
R ∩ S′
Again by dimension shifting,
H3 (R/R ∩ S, S/S ′ ) = H3 (RS/S, S/S ′ ) = H5 (R/R ∩ S) = 0.

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

12

Therefore, by the hypothesis on cohomological condition, both sides in the last exact
sequence are zero. Consequently,



R∩S
F ∩ (1 + B)
2
H1 R/R ∩ S, SP
=
′
R∩S
(R ∩ S ′ )′ [(R ∩ S)′ , R]

Invoking the sequences (3.9), we get an isomorphsim






R∩S
S
2
2
˜
= H1 R/R ∩ S, SP
.
H3 R/R ∩ S, ⊗
(R ∩ S)S ′
R ∩ S′
The result now follows from the Universal Coefficient Theorem.



Remark 4.1. Decomposition of the antisymmetric tensor square as 0 → − ⊗ Z/2 →
˜ 2 → Λ2 → 0, leads to the following diagram
⊗
H3 (R/R ∩ S) ⊗

S
(R∩S)S ′

⊗ Z/2



 

S
/
˜ 2 (R∩S)S
H3 (R/R ∩ S) ⊗ ⊗
′
/




S
˜ 2 (R∩S)S
H3 R/R ∩ S, ⊗
′

//

(torsion)



H3 (R/R ∩ S) ⊗ Λ2



S
(R∩S)S ′

We can thus conclude that


2
tf H3 (R/R ∩ S) ⊗ Λ



.

S
(R ∩ S)S ′







= tf H1 R/R ∩ S, SP

Here tf means the torsion-free rank of an abelian group.

2



R∩S
R ∩ S′



.

Example. Let F = F (x1 , x2 , x3 , x4 , x5 ),
R := hx1 , x2 , x3 iF ,

S := h[x1 , x2 ], [x2 , x3 ], [x1 , x3 ], x4 , x5 iF .

The quotient group R/R ∩ S is then a free abelian group of rank three, with the images
S
of x1 , x2 , x3 as generators, the group (R∩S)S
′ is a free abelian group of rank two with the
images of x4 , x5 as generators. Therefore,


S
2
∼
H3 (R/R ∩ S) ⊗ Λ
= Z.
(R ∩ S)S ′
Hence, the quotient
D(F, rsf )
I(R, S, F )
is non-zero.

(4.2)

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

13

At the moment we are not able to indicate explicitly the elements leading to the nontriviality of the quotient (4.2). Here we present candidates for such elements. The recipe
given below shows how the elements from the third homology can be used for constructing
elements belonging to generalized dimension subgroups.
Consider a free group F and elements ri ∈ R, ti ∈ R ∩ S such that
Y
[ri , ti ] = 1.
i

Such elements come from the third homology:
[,]

H3 (R/R ∩ S) = ker{R ∧ (R ∩ S) → R}.

Here the sign ∧ means the non-abelian exterior product in the sense of Brown-Loday [3].
For d, e ∈ S, consider the element
Y
[[ri−1 , d], [ti , e]]
w=
i

Proposition 4.1. w − 1 ∈ ∆(R)∆(S)∆(F ).
Proof. Working modulo ∆(R)∆(S)∆(F ), we get
1 − wi ≡ (1 − [ri−1 , d])(1 − [ti , e]) − (1 − [ti , e])(1 − [ri−1 , d]) ≡

− (1 − [ti , e])(1 − [ri−1 , d]) ≡ −(1 − [ti , e])d−1 ri ((1 − ri−1 )(1 − d) − (1 − d)(1 − ri−1 )) ≡
(1 − [ti , e])(1 − ri )(1 − d) ≡ e−1 t−1
i ((1 − ti )(1 − e) − (1 − e)(1 − ti ))(1 − ri )(1 − d)] ≡
(1 − e−1 )(1 − ti )(1 − ri )(1 − d) ≡

Clearly,

− (1 − e−1 )ri ti (1 − [ri , ti ])(1 − d) ≡ −(1 − e−1 )(1 − [ri , ti ])(1 − d).
1−w ≡

X
i

(1 − wi ).

Therefore,
1 − w ≡ −(1 − e−1 )(

X
i

(1 − [ri , ti ]))(1 − d) ≡ −(1 − e−1 )(1 −

Y
[ri , ti ])(1 − d) = 0.
i



5. A combinatorial proof of a result of Stöhr
As mentioned earlier, the normal subgroup D(F, rfr), which is a special case of Gupta’s
three subgroup problem, has been identified by Stöhr [21]. The following result on free
group rings is implicit in this work based on using homological mehods.
Theorem 5.1. Let F be a free group, and R a normal subgroup of F . If a ∈ D(F, frf ),
then a2 ∈ D(F, rrf + frr).

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

14

In this section we give a combinatiorial proof of the above result, and bring out the
possibility of higher dimensional variations of its statement. Before proceeding further,
let us give a sketch of the main steps from [21] which yield Theorem 5.1.
First identify the tensor square of the relation module R̄ := R/[R, R] as
rr
R̄⊗2 =
frr
and observe that there is a natural Z/2-action on R̄⊗2 , namely the one which permutes
the factors. This action extends to the natural quotient
rr
(R̄⊗2 )F =
.
frr + rrf
One of the main statements in [21] is that the Z/2-action on the subgroup
rr
rr ∩ frf
⊆
H4 (G) =
rrf + frr
frr + rrf
is trivial. The proof in [21], which is homological, uses comparison of different projective
resolutions. Let a ∈ D(F, frf). Then a ∈ R′ , since frf ⊂ fr and D(F, fr) = R′ . The
Z/2-action which permutes the terms in (R̄⊗2 )F sends
a − 1 + frr + rrf 7→ a−1 − 1 + frr + rrf.

Since a ∈ D(F, frf ) and a ∈ R′ , a ∈ D(F, rr ∩ frf). We conclude that
and therefore

a ≡ a−1

mod frr + rrf

a2 ∈ D(F, frr + rrf).
Since the above conclusion is a result purely in group rings, the following questions arise
naturally.
• Does there exist a combinatorial proof of the above fact without the use of homology?
• Is it possible to generalize this result to more complicated ideals and generalized
dimension subgroups?
We answer the first question affirmatively, and offer some remarks on the second question.
Let G = F/R, and choose a transversal {w(g)}g∈G for G in F :

w(g) 7→ g, g ∈ G, w(g) ∈ F, with w(1) = 1.

Then we have a function W : G × G → R defined by

w(g)w(h) = w(gh)W (g, h).

This function satisfies a 2-cocycle condition
W (gh, k)W (g, h)w(k) = W (g, hk)W (h, k) for all g, h, k ∈ G.
−1

Lemma 5.1. For g, h ∈ G, (W (g, h)−1 )w(gh) W (h−1 , g −1 ) ∈ [R, R].

(5.1)

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

15

Proof. We have
w(h−1 )w(g −1) = w(h−1 g −1 )W (h−1 , g −1 ), g, h ∈ G,

(5.2)

W (g, g −1 )w(g) = W (g −1, g).

(5.4)

w(g −1) = w(g)−1W (g, g −1 ) = W (g −1, g)w(g)−1, g ∈ G,

(5.3)

Substituting in (5.2) we, in turn, have
w(h)−1 W (h, h−1 )w(g)−1W (g, g −1 ) = w(gh)−1W (gh, (gh)−1)W (h−1 , g −1)
−1

w(h)−1 w(g)−1W (h, h−1 )w(g) W (g, g −1) = w(gh)−1W (gh, (gh)−1 )W (h−1 , g −1 )
−1

W (g, h)−1 w(gh)−1W (h, h−1 )w(g) W (g, g −1) = w(gh)−1W (gh, (gh)−1)W (h−1 , g −1)
Finally, we have the following relation:
−1

−1

W (g, h)−w(gh) W (h, h−1 )w(g) W (g, g −1 ) = W (gh, (gh)−1 )W (h−1 , g −1 ).

(5.5)

Equation (5.1) implies that
W (gh, h−1 g −1)W (g, h)w(h

−1 g −1 )

= W (g, g −1 )W (h, h−1 g −1) =
W (g, g −1 )W (h, h−1 )w(g

−1 )

W (h−1 , g −1 ). (5.6)

Substituting the value of W (gh, h−1 g −1 ) from (5.5) to (5.6), we get
−1

[(W (g, h)−1 )w(gh) W (h−1 , g −1)]2 ∈ [R, R].
Since R/[R, R] is free abelian, we conclude that
−1

(W (g, h)−1 )w(gh) W (h−1 , g −1 ) ∈ [R, R].

Now we are ready to prove Theorem 5.1 using only combinatorial tools, and without
homological algebra. In fact, we have the following more general result, from which
Theorem 5.1 follows in case S = T = F .
Theorem 5.2. Let R, S, T be normal subgroups in F , such that R ⊆ S, T . If
a ∈ D(F, srt + trs),
then
a2 ∈ D(F, rrs + srr + trr + rrt).
Proof. Let a ∈ D(F, srt + trs) so that we have an expression
X
X
1−a=
(1 − f )(1 − r)(1 − g) +
(1 − f ′ )(1 − r ′ )(1 − g ′ ),

where the two sums are taken over products with f, g ′ ∈ S, g, f ′ ∈ T, r, r ′ ∈ R. On
opening the brackets, we get
X
X
1−a =
(1−f −r −g +f r +f g +rg −f rg)+ (1−f ′ −r ′ −g ′ +f ′ r ′ +f ′ g ′ +r ′ g ′ −f ′ r ′ g ′).
(5.7)

NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

16

We pick a set of representatives {w(g)}g∈F/R in F for the elements of the quotient group
F/R. Then every element w ∈ F can be written uniquely as w = w(w̄)rw , with w̄ = wR
and rw ∈ R, and so we have
f = w(f¯)rf ,
f ′ = w(f¯′ )rf ′ ,
g = w(ḡ)rg ,

g ′ = w(ḡ ′)rg′ ,

f r = w(f r)rf r,

f ′ r ′ = w(f ′ r ′ )rf ′ r ′

f g = w(f g)W (f, g)rfg rg ,

f ′ g ′ = w(f ′g ′ )W (f ′, g ′ )rfg ′ rg′

rg = w(rg)rg r g ,

r ′ g ′ = w(r ′g ′ )rg′ r ′ ,

f rg = w(f rg)W (f, g)rfg rg r g ,

f ′ r ′ g ′ = w(f ′r ′ g ′ )W (f ′, g ′ )rfg ′ rg′ r ′ .

′

g′

′

g′

Let π : F 7→ R be the projection given by w 7→ rw . Since the element a lies in [R, R],
π(a) = a. The first sum in (5.7), projects under π to the following sum
X
(1 − rf − r − rg + rf r + W (f, g)rfg rg + rg r g − W (f, g)rfg rg r g ) =
X
((1 − rf )(1 − r) − (1 − W (f, g)rfg )rg (1 − r g )) (5.8)
Therefore, modulo r2 t + tr2 we have this sum equivalent to
X
X
−1
(1 − W (f, g))(1 − r g ) ≡
(1 − W (f, g)g )(1 − r).

In the same way we see that, modulo r2 s + sr2 , the second sum in (5.7) is equivalent to
the sum
X
′−1
(1 − W (f ′ , g ′ )g )(1 − r ′ ).
Hence,

1−a≡

X
X
−1
′−1
(1 − W (f, g)g )(1 − r) +
(1 − W (f ′, g ′ )g )(1 − r ′ )

mod r2 s + sr2 + r2 t + tr2 . (5.9)

On applying the involution f 7→ f −1 on F to the equation (5.7), we have
X
X
1 − a−1 =
(1 − g −1)(1 − r −1 )(1 − f −1 ) +
(1 − g ′−1)(1 − r ′−1 )(1 − f ′−1 ).

Repeating the same process as above, we get
X
X
′
1 − a−1 ≡ −
(1 − W (g −1 , f −1 )f )(1 − r) −
(1 − W (g ′−1 , f ′−1 )f )(1 − r ′ )

mod r2 s + sr2 + r2 t + tr2 . (5.10)

Subtracting 1 − a−1 from 1 − a, by (5.9) and (5.10) we get
X
−1
1 − a2 ≡
(1 − W (f, g)g W (g −1 , f −1 )f )(1 − r)+
(1 − W (f ′, g ′)g

′−1

′

W (g ′−1 , f ′−1 )f )(1 − r ′ )

mod r2 s + sr2 + r2 t + tr2 .

Lemma 5.1 implies that
W (f, g)W (g −1, f −1 )f g , W (f ′, g ′ )W (g ′−1, f ′−1 )f
Hence 1 − a2 ∈ r2 s + sr2 + r2 t + tr2 .

′ g′

∈ [R, R].



NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

17

Remark 5.1. On invoking Theorem 4, [21], it follows that under the hypothesis of the
above theorem, a4 ∈ [R, R, ST ].
One can prove the following result by proceeding in a way similar to that for the proof
of the preceding theorem, and therefore we omit the details.
Theorem 5.3. Let R, S, T be normal subgroups of a free group F with R ⊆ S, T . If
a ∈ D(F, srs + trt), then a2 ∈ D(F, rrs + srr + rrt + trr) and a4 ∈ [R, R, ST ].
The general problem in free group rings, of which the foregoing are special cases, asks
for the identification of normal subgroups D(F, a), where a is a sum of ideals of the form
r1 . . . rn with R1 , . . . , Rn normal subgroups of the given free group F . As a contribution
to this general problem, we present the following two results.
Theorem 5.4. Let R ⊆ S, T be normal subgroups of a free group F .
If a ∈ D(F, rsst + tssr), then
and

a2 ∈ D(F, rsss + sssr + tsss + ssst + ([S, S] − 1)s)

(5.11)

a6 ∈ [S, S, S, RT ].

(5.12)

Proof. While the proof of (5.11) is similar to that of Theorem 5.2, and so we omit it, the
assertion (5.12) follows from (5.11) and the following general result:
If R is a normal subgroup of the free group F and
a ∈ D(F, rrrf + frrr + ([R, R] − 1)r), then a3 ∈ [R, R, R, F ]. (5.13)
To prove (5.13) consider the natural exact sequence
γ3 (R)/γ4 (R)

L3 (Rab )
/

/

rrr
frrr+([R, R]−1)r

SP2 (Rab ) ⊗ Rab

//

rrr
frrr+([R, R]−1)r+(γ3 (R)−1)Z[R]

//

SP3 (Rab ),

where L3 and SP3 are the third Lie and symmetric power functor respectively, and Rab
is the abelianization of R. Applying the homology functor H∗ (F, −) to this sequence, we
get a long exact sequence which connects H1 and H0 , which, in turn, implies that
D(F, rrrf + frrr + ([R, R] − 1)r)
= Coker{H1 (F, SP2 (Rab ) ⊗ Rab ) → H1 (F, SP3 (Rab ))}.
[R, R, R, F ]
The assertion (5.13) follows from the simple fact that, for a free abelian A, the natural
composition
SP3 (A) ֒→ SP2 (A) ⊗ A ։ SP3 (A)
is multiplication by 3.


NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

18

Our concluding result is a generalization of Kanta Gupta’s identification of D(F, rffr)
[8].
Theorem 5.5. If R ⊆ S are normal subgroups of a free group F , then
D(F, rsfr) = D(F, rfsr) = [R ∩ S ′ , R ∩ S ′ , R]γ4 (R).

Proof. Observe that D(F, rsfr) ⊆ D(F, rfr) = γ3 (R). Therefore, by Lemma 2.2,
D(F, rsfr) = D(F, ∆4 (R) + ∆(R)∆(R ∩ S ′ )∆(R)).

Recall from [15] that, for a free group F and its normal subgroup N, if (F/N)ab is
2-torsion-free, then
D(F, fnf + f 4 ) ∼
= L1 SP3 ((F/N)ab ).
[N, N, F ]γ4 (F )
In our situation, the quotient R/R ∩ S ′ ⊆ S/S ′ is torsion-free, therefore, the contribution
from the derived functor L1 SP3 vanishes and the result follows.

Observe that, the condition R ⊆ S in Theorem 5.5 significantly simplifies the identification.
For arbitrary normal subgroups R, S, we conjecture that
D(F, rssr) =

p

[γ3 (R ∩ S), R][γ2 (R ∩ S ′ ), R].

Acknowledgement
The research of the first author is supported by the Russian Science Foundation, grant
N 16-11-10073.
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NARAIN GUPTA’S THREE NORMAL SUBGROUP PROBLEM AND GROUP HOMOLOGY

Roman Mikhailov
St Petersburg Department of Steklov Mathematical Institute
and
Chebyshev Laboratory, St Petersburg State University
14th Line, 29b, Saint Petersburg 199178 Russia
Email: romanvm@mi.ras.ru
Inder Bir S. Passi
Centre for Advanced Study in Mathematics
Panjab University, Sector 14, Chandigarh 160014 India
and
Indian Institute of Science Education and Research, Mohali (Punjab) 140306 India
Email: ibspassi@yahoo.co.in

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