BOGOMOLOV MULTIPLIERS FOR SOME p-GROUPS OF
NILPOTENCY CLASS 2

arXiv:1307.0738v3 [] 18 Jul 2016

IVO M. MICHAILOV
Abstract. The Bogomolov multiplier B0 (G) of a finite group G is defined as the
subgroup of the Schur multiplier consisting of the cohomology classes vanishing after
restriction to all abelian subgroups of G. The triviality of the Bogomolov multiplier
is an obstruction to Noether’s problem. We show that if G is a central product of G1
and G2 , regarding Ki ≤ Z(Gi ), i = 1, 2, and θ : G1 → G2 is a group homomorphism
such that its restriction θ|K1 : K1 → K2 is an isomorphism, then the triviality of
B0 (G1 /K1 ), B0 (G1 ) and B0 (G2 ) implies the triviality of B0 (G). We give a positive
answer to Noether’s problem for all 2-generator p-groups of nilpotency class 2, and for
one series of 4-generator p-groups of nilpotency class 2 (with the usual requirement
for the roots of unity).

1. Introduction
Let K be a field, G a finite group and V a faithful representation of G over K. Then
there is a natural action of G upon the field of rational functions K(V ). The rationality
problem (also known as Noether’s problem when G acts on V by permutations) then
asks whether the field of G-invariant functions K(V )G is rational (i.e., purely transcendental) over K. A question related to the above mentioned is whether K(V )G is
stably rational, that is, whether there exist independent variables x1 , . . . , xr such that
K(V )G (x1 , . . . , xr ) becomes a purely transcendental extension of K. This problem has
close connection with Lüroth’s problem [24] and the inverse Galois problem [23, 25].
Saltman [23] found examples of groups G of order p9 such that C(V )G is not stably
rational over C. His main method was application of the unramified cohomology group
2
2
Hnr
(C(V )G , Q/Z) as an obstruction. Bogomolov [4] proved that Hnr
(C(V )G , Q/Z) is
canonically isomorphic to
\
2
2
B0 (G) =
ker{resA
G : H (G, Q/Z) → H (A, Q/Z)}
A

where A runs over all the bicyclic subgroups of G (a group A is called bicyclic if A
is either a cyclic group or a direct product of two cyclic groups). The group B0 (G)
is a subgroup of the Schur multiplier H 2 (G, Q/Z), and Kunyavskiĭ [17] called it the
Date: February 15, 2018.
2000 Mathematics Subject Classification. primary 14E08, 14M20; secondary 13A50,12F12.
Key words and phrases. Bogomolov multiplier, Noether’s problem, Rationality problem.
This work is partially supported by a project No RD-08-241/12.03.2013 of Shumen University.
1

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IVO M. MICHAILOV

Bogomolov multiplier of G. Thus the vanishing of the Bogomolov multiplier is an
obstruction to Noether’s problem.
Recently, Moravec [19] used a notion of the nonabelian exterior square G ∧ G of a
given group G to obtain a new description of B0 (G). Namely, he proved that B0 (G) is
(non-canonically) isomorphic to the quotient group M(G)/M0 (G), where M(G) is the
kernel of the commutator homomorphism G ∧ G → [G, G], and M0 (G) is the subgroup
of M(G) generated by all x ∧ y such that x, y ∈ G commute. Moravec studied the
functor B0 (G) in [19], and in particular he found the five term exact sequence
N
→ E ab → (E/N)ab → 0,
B0 (E) → B0 (E/N) →
hK(E) ∩ Ni
where E is any group, N a normal subgroup of E and K(E) denotes the set of commutators in E.
If we assume that N is a central subgroup of E, we derive in Section 3 a three term
exact sequence
N ∩ [E, E]
(1.1)
B0 (E) → B0 (E/N) →
.
hK(E) ∩ Ni
Let G be a central product of G1 and G2 , regarding Ki ≤ Z(Gi ), where Z(Gi ) is the
centre of Gi for i = 1, 2. Kang and Kunyavskiĭ [16] raised the question whether the
triviality of B0 (G1 ) and B0 (G2 ) implies the triviality of B0 (G). With the aid of (1.1)
we prove our first main result, stating that if θ : G1 → G2 is a group homomorphism
such that its restriction θ|K1 : K1 → K2 is an isomorphism, then the triviality of
B0 (G1 /K1 ), B0 (G1 ) and B0 (G2 ) implies the triviality of B0 (G).
We prove two applications of this result obtaining the triviality of the Bogomolov
multipliers for the extra-special p-groups and for the 4-generator p-groups that are
central products of metacyclic p-groups. Of course, we will not consider groups that
are direct products of smaller groups, because of the following.
Theorem 1.1. (Kang [15, Theorem 4.1]) Let G and H be finite groups. Then B0 (G ×
H) is isomorphic to B0 (G) × B0 (H). As a corollary, if B0 (G) and B0 (H) are both
trivial, then also is B0 (G × H).
The Bogomolov multipliers for the groups of order pn for n ≤ 6 were calculated
recently in [10, 12, 20, 7]. The reader is referred to the paper [16] for a survey of
groups with trivial Bogomolov multipliers.
Definition 1.1. We say that a group G has the AEC (Abelian Extension of a Cyclic
group) property if G has a normal abelian subgroup H such that the quotient group
G/H is cyclic
Bogomolov [4, Lemma 4.9] proved that if G has the AEC property then B0 (G) = 0.
On the other hand, Noether’s problem for p-groups with the AEC property is still not
solved entirely. However, there are number of partial results that have been obtained
recently. We list two of them.

BOGOMOLOV MULTIPLIERS FOR SOME p-GROUPS OF NILPOTENCY CLASS 2

3

Theorem 1.2. (Kang[13, Theorem 1.5]) Let G be a metacyclic p-group with exponent
pe , and let K be any field such that (i) char K = p, or (ii) char K 6= p and K contains
a primitive pe -th root of unity. Then K(G) is rational over K.
Theorem 1.3. (Michailov[18, Theorem 1.8]) Let G be a group of order pn for n ≥ 2
with an abelian subgroup H of order pn−1 , and let G be of exponent pe . Choose any
α ∈ G such that α generates G/H, i.e., α ∈
/ H, αp ∈ H. Denote H(p) = {h ∈ H :
p
p
h = 1, h ∈
/ H } ∪ {1}, and assume that [H(p), α] ⊂ H(p). Denote by G(i) = [G, G(i−1) ]
the lower central series for i ≥ 1 and G(0) = G. Let the p-th lower central subgroup
G(p) be trivial. Assume that (i) char K = p > 0, or (ii) char K 6= p and K contains a
primitive pe -th root of unity. Then K(G) is rational over K.
In Section 4 of the present article we will give a positive answer to Noether’s problem
for all 2-generator p-groups of nilpotency class 2 (with the usual requirement for the
roots of unity). It seems that almost all known results for Noether’s problem regarding p-groups actually hold only for p-groups with the AEC property and their direct
products. In Section 4 we will also give a positive answer to Noether’s problem for
one series of 4-generator p-groups of nilpotency class 2 that do not posses the AEC
property and that are not direct or central products of smaller groups.
2. Preliminaries and notations
Let G be a group and x, y ∈ G. We define xy = y −1 xy and write [x, y] = x−1 xy =
x−1 y −1 xy for the commutator of x and y. We define the commutators of higher weight
as [x1 , x2 , . . . , xn ] = [[x1 , . . . , xn−1 ], xn ] for x1 , x2 , . . . , xn ∈ G.
Let ϕ be an automorphism of G and Gϕ be an isomorphic copy of G via ϕ : x 7→ xϕ .
We define τ (G) to be the group generated by G and Gϕ , subject to the following
ϕ
relations: [x, y ϕ ]z = [xz , (y z )ϕ ] = [x, y ϕ ]z and [x, xϕ ] = 1 for all x, y, z ∈ G. Obviously,
the groups G and Gϕ can be viewed as subgroups of τ (G). Let [G, Gϕ ] = h[x, y ϕ ] :
x, y ∈ Gi be the commutator subgroup.
Next, denote by G∧G the nonabelian exterior square of G, which is a group generated
by the symbols x ∧ y (x, y ∈ G), satisfying the relations
xy ∧ z = (xy ∧ z y )(y ∧ z),
x ∧ yz = (x ∧ z)(xz ∧ y z ),
x∧x

= 1,

for all x, y, z ∈ G. Let [G, G] be the commutator subgroup of G. Observe that the
commutator map κ : G ∧ G → [G, G], given by x ∧ y 7→ [x, y] is a well-defined group
homomorphism. Let M(G) denote the kernel of κ, and M0 (G) detone the subgroup of
M(G) generated by all x ∧ y such that x, y ∈ G commute. Moravec proved in [19] that
the Bogomolov multiplier B0 (G) is (non-canonically) isomorphic to the quotient group
M(G)/M0 (G).
Notice that the map φ : G ∧ G → [G, Gϕ ] given by x ∧ y 7→ [x, y ϕ ] is actually an
isomorphism of groups (see [5]).

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IVO M. MICHAILOV

Now, let κ∗ = κ · φ−1 be the composite map from [G, Gϕ ] to [G, G], M ∗ (G) = ker κ∗
and M0∗ (G) = φ(M0 (G)). Then B0 (G) is clearly isomorphic to M ∗ (G)/M0∗ (G) by [19].
It is not hard to see that
(
)
Y
Y
M ∗ (G) =
[xi , yiϕ ]εi ∈ [G, Gϕ ] : εi = ±1,
[xi , yi ]εi = 1 ,
finite

and

M0∗ (G) =

(

finite

Y

)

[xi , yiϕ ]εi ∈ [G, Gϕ ] : εi = ±1, [xi , yi ] = 1 .

finite

Thus, in order to prove that B0 (G) = 0 for a given group G, it suffices to show that
M ∗ (G) = M0∗ (G). This can be achieved by finding a generating set of M ∗ (G) consisting
solely of elements of M0∗ (G).
The following two Lemmas contain various properties of τ (G) and [G, Gϕ ] that will
be used in our considerations.
Lemma 2.1. ([5]) Let G be a group.
(1) [x, yz] = [x, z][x, y][x, y, z] and [xy, z] = [x, z][x, z, y][y, z] for all x, y, z ∈ G.
(2) If G is nilpotent of class c, then τ (G) is nilpotent of class at most c + 1.
(3) If G is nilpotent of class ≤ 2, then [G, Gϕ ] is abelian.
(4) [x, y ϕ ] = [xϕ , y] for all x, y ∈ G.
(5) [x, y, z ϕ ] = [x, y ϕ , z] = [xϕ , y, z] = [xϕ , y ϕ , z] = [xϕ , y, z ϕ ] = [x, y ϕ , z ϕ ] for all
x, y, z ∈ G.
(6) [[x, y ϕ ], [a, bϕ ]] = [[x, y], [a, b]ϕ ] for all x, y, a, b ∈ G.
(7) [xn , y ϕ ] = [x, y ϕ ]n = [x, (y ϕ )n ] for all integers n and x, y ∈ G with [x, y] = 1.
(8) If [G, G] is nilpotent of class c, then [G, Gϕ ] is nilpotent of class c or c + 1.
Lemma 2.2. ([20, Lemma 3.1]) Let G be a nilpotent group of class ≤ 3. Then
n
n
[x, y n ] = [x, y]n [x, y, y](2 ) [x, y, y, y](3 )
for all x, y ∈ τ (G) and every positive integer n.
3. Central products
Let E be a group, and let N be a central subgroup of E. For G = E/N define
Q
Q
a map η : M ∗ (E) → M ∗ (G) by η( [xi , yiϕ ]mi ) = [xi N, (yi N)ϕ ]mi . Clearly, η is a
homomorphism. Now, define a map ξ : [G, Gϕ ] → E by ξ([x, y ϕ ]) = [e
x, ye], where
x=x
eN, y = yeN. Since N ≤ Z(G), the map ξ does not depend on the choice of x
e and
∗
∗
ye. Moreover, ξ is an epimorphism. Denote N1 = ξ(M (G)) and N0 = ξ(M0 (G)). It is
not hard to see that N1 = N ∩ [E, E] and N0 = h[e
x, ye] ∈ Ni.
One can now easily verify that we have the following exact sequences:
η

ξ

η

ξ

M0∗ (E) −→ M0∗ (G) −→ N0

and

M ∗ (E) −→ M ∗ (G) −→ N1 .

BOGOMOLOV MULTIPLIERS FOR SOME p-GROUPS OF NILPOTENCY CLASS 2

5

Hence we obtain the exact sequence
(3.1)

η∗

ξ∗

B0 (E) −→ B0 (G) −→ N1 /N0 ,

and in particular, the isomorphism B0 (G)/η∗ (B0 (E)) ≃ N1 /N0 .
Next, we are going to develop further the case when G is a central product of
two groups G1 and G2 with a common central subgroup. Let θ : K1 → K2 be an
isomorphism, where K1 ≤ Z(G1 ) and K2 ≤ Z(G2 ), and let E = G1 × G2 . Then
the central product of G1 and G2 is defined as the quotient group G = E/N, where
N = {ab : a ∈ K1 , b ∈ K2 , θ(a) = b−1 } ∈ Z(E).
Theorem 3.1. Let θ : G1 → G2 be a group homomorphism such that its restriction
θ|K1 : K1 → K2 is an isomorphism, where K1 ≤ Z(G1 ) and K2 ≤ Z(G2 ). Let
G be a central product of G1 and G2 , i.e., G = E/N, where E = G1 × G2 and
N = {ab : a ∈ K1 , b ∈ K2 , θ(a) = b−1 }. If B0 (G1 /K1 ) = B0 (G1 ) = B0 (G2 ) = 0 then
B0 (G) = 0.
Proof. Theorem 1.1 implies that B0 (E) ≃ B0 (G1 ) × B0 (G2 ) = 0. Then from the exact
sequence (3.1) it follows that B0 (G) ≃ B0 (G)/η∗ (B0 (E)) ≃ N1 /N0 . Therefore, we need
to show only that N0 = N1 .
Q
ei , yei ∈ E.
Take arbitrary [e
xi , yei ]mi = ab ∈ N1 , where a ∈ K1 , b ∈ K2 , θ(a) = b−1 , x
We have now that x
ei = αi1 αi2 and yei = βi1 βi2 for αij , βij ∈ Gj ; 1 ≤ j ≤ 2. Since
the elements of G1 commute with the elements of G2 , we obtain the formula [e
xi , yei ] =
Q
Q
[αi1 , βi1 ][αi2 , βi2 ]. Hence a = [αi1 , βi1 ]mi and b = [αi2 , βi2 ]mi .
Now, we may apply the exact sequence (3.1) for the central group extension
1 → K1 → G1 → G1 /K1 → 1.

From B0 (G1 /K1 ) = 0 it follows that
B0 (G1 /K1 )
K1 ∩ [G1 , G1 ]
≃
≃ 0,
K(G1 ) ∩ K1
η∗ (B0 (G1 ))
where K(G1 ) is the set of commutators in G1 . Therefore we may assume that [αi1 , βi1 ]
Q
mi
are
in
K
1 for all i. Since θ is a homomorphism, we have that θ(a) = θ( [αi1 , βi1 ] ) =
Q
[θ(αi1 ), θ(βi1 )]mi = b−1 . Note that the commutators [θ(αi1 ), θ(βi1 )] are in K2 for all
i, so their position in the decomposition of b is of no importance. Therefore,
Y
Y
Y
ab =
[e
xi , yei ]mi =
[αi1 , βi1 ]mi
[θ(αi1 ), θ(βi1 )]−mi
Y
Y
=
([αi1 , βi1 ][θ(αi1 ), θ(βi1 )]−1 )mi =
[αi1 θ(βi1 ), βi1 θ(αi1 )]mi .

Since [αi1 θ(βi1 ), βi1 θ(αi1 )] = [αi1 , βi1 ](θ([αi1 , βi1 ]))−1 ∈ N, we obtain finally that ab ∈
N0 .


The first straightforward application of Theorem 3.1 is for the extra-special p-groups.
Recall that a p-group G is extra-special if its center Z is cyclic of order p, and the
quotient G/Z is a non-trivial elementary abelian p-group. Kang and Kunyavskiĭ also
proved in another way the following.

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IVO M. MICHAILOV

Corollary 3.2. ([16, Proposition 2.1]) If G is an extra-special p-group of order p2n+1
(for any n ≥ 1), then B0 (G) = 0.
Proof. Note first that for any n there are two extra-special p-groups of order p2n+1 , and
they are isoclinic. According to a recent result by Moravec [21], the isoclinic groups
have isomorphic Bogomolov multipliers. It is well known that one of these two groups
can be obtained as a central product of n Heisenberg groups H3 of order p3 . We
apply induction by n. For n = 1 we have B0 (H3 ) = 0 (see [3]). Let G be a central
product of the extra-special p-group G2 of order p2n−1 and H3 , where we assume that
B0 (G2 ) = 0. Denote by K1 ∼
= Cp the centers of H3 and G2 , respectively.
= Cp and K2 ∼
It is easy to define a monomorphism θ : H3 → G2 , such that θ(K1 ) = K2 , and since
B0 (H3 /K1 ) = B0 (Cp × Cp ) = 0, we may apply Theorem 3.1.

Another application of Theorem 3.1 we find in the following Corollary, where we
consider the central product of two split metacyclic p-groups.
r

r

a1

Corollary 3.3. Let G = hβ1 , β2 , α1 , α2 : [β1 , α1 ] = β1p , [β2 , α2 ] = β2p , α1p = α2p
b
b
b−1
b−1
β1p = β2p = 1, β1p = β2p i, where a1 , a2 , 1 ≤ r ≤ b − 1, 2 ≤ b. Then B0 (G) = 0.
r

ai

a2

=

b

Proof. For 1 ≤ i ≤ 2 define Gi = hβi , αi : [βi , αi ] = βip , αip = βip = 1i. Without loss
of generality we will assume that a1 ≥ a2 . Define a map θ : G1 → G2 by θ(α1 ) = α2
and θ(β1 ) = β2 . We are going to show that θ is a homomorphism. Indeed, any
element from G1 can be written in the form β1y α1z . We have that θ((β1w α1x )(β1y α1z )) =
b−1
w+y(1+pr )x x+z
w+y(1+pr )x x+z
θ(β1
α1 ) = β2
α2 = θ(β1w α1x )θ(β1y α1z ). Next, note that [βip , αi ] =
b−1
b−1
r+b−1
∈ Z(Gi ). For 1 ≤ i ≤ 2 define Ki = hβip i ∼
= 1, so βip
βip
= Cp . Clearly
the restriction θ|K1 : K1 → K2 is an isomorphism, so G is a central product of the
metacyclic p-groups G1 and G2 . Note that G1 /K1 is also metacyclic. Our result now
follows from Theorems 1.2 and 3.1.

The group G given in the statement of Corollary 3.3 is a 4-generator p-group of
nilpotency class 2 that does not posses the AEC property. In the following section we
will consider another group with the same properties which is not a central product of
smaller groups.
4. Noether’s problem for groups of nilpotency class 2
Let K be any field. A field extension L of K is called rational over K (or K-rational,
for short) if L ≃ K(x1 , . . . , xn ) over K for some integer n, with x1 , . . . , xn algebraically
independent over K. Now let G be a finite group. Let G act on the rational function
field K(x(g) : g ∈ G) by K automorphisms defined by g ·x(h) = x(gh) for any g, h ∈ G.
Denote by K(G) the fixed field K(x(g) : g ∈ G)G . Noether’s problem then asks whether
K(G) is rational over K. This is related to the inverse Galois problem, to the existence
of generic G-Galois extensions over K, and to the existence of versal G-torsors over
K-rational field extensions [25, 22, 8, 33.1, p.86]. Noether’s problem for abelian groups

BOGOMOLOV MULTIPLIERS FOR SOME p-GROUPS OF NILPOTENCY CLASS 2

7

was studied extensively by Swan, Voskresenskii, Endo, Miyata and Lenstra, etc. The
reader is referred to Swan’s paper for a survey of this problem [25].
We list several results which will be used in the sequel.
Theorem 4.1. ([9, Theorem 1]) Let G be a finite group acting on L(x1 , . . . , xm ), the
rational function field of m variables over a field L such that
(i): for any σ ∈ G, σ(L) ⊂ L;
(ii): the restriction of the action of G to L is faithful;
(iii): for any σ ∈ G,

 

σ(x1 )
x1
 ...  = A(σ)  ...  + B(σ)
σ(xm )
xm

where A(σ) ∈ GLm (L) and B(σ) is m × 1 matrix over L. Then there exist
z1 , . . . , zm ∈ L(x1 , . . . , xm ) so that L(x1 , . . . , xm )G = LG (z1 , . . . , zm ) and σ(zi ) =
zi for any σ ∈ G, any 1 ≤ i ≤ m.

Theorem 4.2. ([1, Theorem 3.1]) Let G be a finite group acting on L(x), the rational
function field of one variable over a field L. Assume that, for any σ ∈ G, σ(L) ⊂ L
and σ(x) = aσ x + bσ for any aσ , bσ ∈ L with aσ 6= 0. Then L(x)G = LG (z) for some
z ∈ L[x].
Theorem 4.3. (Kuniyoshi [6, Theorem 1.7]) If charK = p > 0 and G is a finite
p-group, then K(G) is rational over K.
Theorem 4.4. (Kang [13, Theorem 4.1]) Let G be a metacyclic p-group of exponent
m
n
pe , generated by σ and τ such that σ p = τ p = 1, τ −1 στ = σ s for s = ε + apr , where
a = 1 + b · pt for t ∈ N and b ∈ {−1, 0, 1}. Let K be a field, containing a primitive pe -th
root of unity, and let ζ be a primitive pm -th root of unity. Then K(u0 , u1, . . . , upn−1 )G
is rational over K, where G acts on u0 , . . . , upn−1 by
i

σ : ui 7→ ζ s ui ,
τ : u0 7→ u1 7→ · · · 7→ upn −1 7→ u0 .
Lemma 4.5. ([14, 11]) Let hτ i be a cyclic group of order n > 1, acting on L(v1 , . . . , vn−1 ),
the rational function field of n − 1 variables over a field L such that
τ : v1 7→ v2 7→ · · · 7→ vn−1 7→ (v1 · · · vn−1 )−1 7→ v1 .
If L contains a primitive nth root of unity ξ, then L(v1 , . . . , vn−1 ) = L(s1 , . . . , sn−1 )
where τ : si 7→ ξ i si for 1 ≤ i ≤ n − 1.
Now, let us consider 2-generator p-groups of nilpotency class 2. They were recently
classified by Ahmad, Magidin and Morse [2], although we will not use their classification
theorems. Let G be any 2-generator p-group of class 2 of order pn . Then [G, G] is a
central subgroup of G that is isomorphic to Cpc with c ≥ 1, and G/[G, G] is isomorphic
to Cpa × Cpb with n = a + b + c. Without loss of generality assume a ≥ b. Let {α, β}

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IVO M. MICHAILOV
a

b

be a transversal of G/[G, G]. Then αp and β p are elements of [G, G], and we have
a
a
b
b
[αp , β] = [α, β]p = 1 and [α, β p ] = [α, β]p = 1. Hence pc , the order of [G, G], divides
both pa and pb . It follows that 1 ≤ c ≤ b ≤ a. From this analysis we may view any
2-generator group G of order pn and class 2 as a central extension of the form
1 −→ Cpc = h[β, α]i −→ G −→ Cpa × Cpb −→ 1.
Since H = h[β, α], αi is a normal abelian subgroup of G, the group G has the AEC
property. Our verification shows however, that this group does not fit in the conditions
of the known rationality criteria. We give an answer to Noether’s problem for G in the
following.
Theorem 4.6. Let n ≥ 3 and let G be a 2-generator p-group of class 2 of order pn .
Let G be of exponent pe . Assume that K is any field satisfying that either (i) char
K = p > 0, or (ii) char K 6= p and K contains a primitive pe -th root of unity. Then
K(G) is rational over K.
L
Proof. Let V be a K-vector space whose dual space V ∗ is defined as V ∗ = g∈G K ·x(g)
where G acts on V ∗ by h · x(g) = x(hg) for any h, g ∈ G. Thus K(V )G = K(x(g) : g ∈
G)G = K(G). The key idea is to find a faithful G-subspace W of V ∗ and to show that
W G is rational over K. Then by Theorem 4.1 it will follow that K(G) rational over K.
Let γ = [β, α], and let H = hβ, γi. The subspace W is obtained as an induced
b
representation from H. Without loss of generality we may assume that β p = 1, so
H ≃ Cp b × Cp c
Define X1 , X2 ∈ V ∗ by
pc −1

pb −1

X1 =

X

x(β i ), X2 =

X

x(γ i ).

i=0

i=0

Note that β · X1 = X1 and γ · X2 = X2 .
Let ζpb ∈ K be a primitive pb -th root of unity, and let ζpc ∈ K be a primitive pc -th
root of unity. Define Y1 , Y2 ∈ V ∗ by
pb −1

pc −1

Y1 =

X
i=0

It follows that

i
ζp−1
c γ

· X1 , Y 2 =

X

i
ζp−1
b β · X2 .

i=0

β : Y1 7→ Y1 , Y2 7→ ζpb Y2 ,
γ : Y1 7→ ζpc Y1 , Y2 7→ Y2 .
Thus K ·Y1 +K ·Y2 is a representation space of the subgroup H. The induced subspase
a
W depends on the relations in G. It is well known that the case αp = β f γ h can easily
a
be reduced to the case αp = 1 (see e.g. [18, Proof of Theorem 1.8, Step 2]).

BOGOMOLOV MULTIPLIERS FOR SOME p-GROUPS OF NILPOTENCY CLASS 2

9

Define xi = αi · Y1 , yi = αi · Y2 for 0 ≤ i ≤ pa − 1. Note that from the relation
[β, α] = γ it follows that βαi = αi βγ i . We have now
β : xi 7→ ζpi c xi , yi 7→ ζpb yi ,
γ : xi 7→ ζpc xi , yi 7→ yi ,
α : x0 7→ x1 7→ · · · 7→ xpa −1 7→ x0 ,
y0 7→ y1 7→ · · · 7→ ypa −1 7→ y0 .
for 0 ≤ i ≤ pa − 1. For 1 ≤ i ≤ pa − 1, define ui = xi /xi−1 and vi = yi /yi−1. Thus
K(xi , yi : 0 ≤ i ≤ pa − 1) = K(x0 , y0 , ui , vi : 1 ≤ i ≤ pa − 1) and for every g ∈ G
g · x0 ∈ K(ui , vi : 1 ≤ i ≤ pa − 1) · x0 , g · y0 ∈ K(ui , vi : 1 ≤ i ≤ pa − 1) · y0 ,
while the subfield K(ui , vi : 1 ≤ i ≤ pa − 1) is invariant by the action of G. Thus
K(xi , yi : 0 ≤ i ≤ pa − 1)G = K(ui , vi : 1 ≤ i ≤ pa − 1)G (u, v) for some u, v such that
α(v) = β(v) = γ(v) = v and α(u) = β(u) = γ(u) = u. We have now
β : ui 7→ ζpc ui , vi 7→ vi ,
(5.1)

γ : ui 7→ ui , vi 7→ vi ,
α : u1 7→ u2 7→ · · · 7→ upa −1 7→ (u1 u2 · · · upa −1 )−1 ,
v1 7→ v2 7→ · · · 7→ vpa −1 7→ (v1 v2 · · · vpa −1 )−1 ,

for 1 ≤ i ≤ pa − 1. From Theorem 4.2 it follows that if K(ui , vi : 1 ≤ i ≤ pa − 1)G is
rational over K, so is K(xi , yi : 0 ≤ i ≤ pa − 1)G over K.
e = hσ, τ : σ p2c = τ pa = 1, τ −1 στ = σ k , k =
Now, consider the metacyclic p-group G
1 + pc i.
P
j
i
a
Define X = 0≤j≤p2c −1 ζp−j
2c x(σ ), Vi = τ X for 0 ≤ i ≤ p − 1. It follows that
i

σ : Vi 7→ ζpk2c Vi ,

τ : V0 7→ V1 7→ · · · 7→ Vpa −1 7→ V0 .
e

Note that K(V0 , V1 , . . . , Vpa −1 )G is rational by Theorem 4.4.
e
Define Ui = Vi /Vi−1 for 1 ≤ i ≤ pa − 1. Then K(V0 , V1 , . . . , Vpa −1 )G = K(U1 , U2 , . . . ,
e
Upa −1 )G (U) where
i

σ : Ui 7→ ζpk2c−k

i−1

Ui , U 7→ U

τ : U1 7→ U2 7→ · · · 7→ Upa −1 7→ (U1 U2 · · · Upa −1 )−1 , U 7→ U.
i

Notice that ζpk2c−k

i−1

(1+pc )i−1 pc

(1+pc )i−1

= ζpc . Compare Formula (5.1) (i.e.,
e on K(Ui :
the actions of β and α on K(ui : 1 ≤ i ≤ p − 1)) with the action of G
a
1 ≤ i ≤ p − 1). They are the same. Hence, according to Theorem 4.4, we get that
e
e
K(u1 , . . . , upa −1 )G (u) ∼
= K(U1 , . . . , Upa −1 )G (U) = K(V0 , V1 , . . . , Vpa−1 )G is rational over
K. Since by Lemma 4.5 we can linearize the action of α on K(vi : 1 ≤ i ≤ pa − 1), we
obtain finally that K(ui , vi : 1 ≤ i ≤ pa − 1)hβ,αi is rational over K.

= ζp2c

= ζpc

a

10

IVO M. MICHAILOV

The next step is to investigate Noether’s problem for groups of nilpotency class 2
that do not have the AEC property. It turns out that the existing methods can not
answer this problem entirely. Indeed, results about Noether’s problem for such groups
are rarely seen in the literature. We managed to discover one series of p-groups for
which we will give a positive answer to Noether’s problem. For any natural number r
define
r

r

(4.1) Gr = hα1 , α2 , β1 , β2 , γ1 , γ2 : [β1 , α1 ] = γ1 = β1p , [β2 , α2 ] = γ2 = β2p , [α1 , α2 ] = γ2 ,
r

r

r

r

γ1p = γ2p = α1p = α2p = 1i.
where all the relations of the form [x, y] = 1 between the generators have been omitted
from the list. Thus γ1 , γ2 ∈ Z(Gr ), i.e., Gr is a p-group of nilpotency class 2 .
By examining the relations in this group we see that it is well defined. It is not hard
to see also that Gr is not a direct or a central product of smaller groups. Moreover, G
does not have the AEC property. First, let us verify that the Bogomolov multiplier of
Gr is trivial.
Theorem 4.7. If G is isomorphic to the group Gr defined by the presentation (4.1),
then B0 (G) = 0.
Proof. The group [G, Gϕ ] is generated modulo M0∗ (G) by [β1 , α1ϕ ], [β2 , α2ϕ ] and [α1 , α2ϕ ].
Every element w ∈ [G, Gϕ ] can be written as w = [β1 , α1ϕ ]m [β2 , α2ϕ ]n [α1 , α2ϕ ]q w̃, where
∗
w̃ ∈ M0∗ (G). This gives w κ = γ1m γ2n+q , therefore w ∈ M ∗ (G) if and only if pr divides
both m and n + q. By Lemma 2.2 we have that
r

1 = [α2ϕ , α1p ]
r

r

p
p
r
= [α2ϕ , α1 ]p [α2ϕ , α1 , α1 ]( 2 ) [α2ϕ , α1 , α1 , α1 ]( 3 )
pr
pr
r
= [αϕ , β ]p [γ −1 , αϕ ]( 2 ) [γ −1 , α , αϕ ]( 3 )

=
and similarly,

2
2
2
ϕ
pr
[α2 , α1 ] ,

r
r
[α1ϕ , β1 ]p , [α2ϕ , β2 ]p

1

1

1

∈ M0∗ (G). It follows that

M ∗ (G) = h[β2 , α2ϕ ][α1 , α2ϕ ]−1 iM0∗ (G).
Note that
[β2 α2 , α2 α1 ] = [β2 , α1 ]α2 [β2 , α2 ]α1 α2 [α2 , α1 ][α2 , α2 ]α1 = [α2 , γ −1 ] = 1,
hence [β2 α2 , (α2 α1 )ϕ ] ∈ M0∗ (G). Expanding the latter using the class restriction and
Lemma 2.1, we get
ϕ

ϕ

[β2 α2 , (α2 α1 )ϕ ] = [β2 , α1ϕ ]α2 [β2 , α2ϕ ]α1 α2 [α2 , α1ϕ ][α2 , α2ϕ ]α1

ϕ

= [β2 , α1ϕ ]α2 [β2 , α2ϕ ][β2 , α2ϕ , α1ϕ α2 ][α2 , α1ϕ ][α2 , α2ϕ ]α1 .
ϕ

Observe that [β2 , α1ϕ ]α2 , [β2 , α2ϕ , α1ϕ α2 ] and [α2 , α2ϕ ]α1 all belong to M0∗ (G). Thus we

conclude that [β2 , α2ϕ ][α1 , α2ϕ ]−1 = [β2 , α2ϕ ][α2 , α1ϕ ] ∈ M0∗ (G), as required.
Finally, we will show that Noether’s problem has a positive answer for Gr .

BOGOMOLOV MULTIPLIERS FOR SOME p-GROUPS OF NILPOTENCY CLASS 2

11

Theorem 4.8. Let G be isomorphic to the group Gr defined by the presentation (4.1).
Assume that K is any field satisfying that either (i) char K = p > 0, or (ii) char K 6= p
and K contains a primitive p2r -th root of unity. Then K(G) is rational over K.
L
Proof. Let V be a K-vector space whose dual space V ∗ is defined as V ∗ = g∈G K ·x(g)
where G acts on V ∗ by h · x(g) = x(hg) for any h, g ∈ G. Thus K(V )G = K(x(g) : g ∈
G)G = K(G).
Consider the subgroups H1 = hβ1 , β2 , α1 i and H2 = hβ1 , β2 , α2 i of G. Note that
H1 = hβ2 i × hβ1 , α1 i ≃ Cp2r × hβ1 , α1 i. Hence we get a linear character of H1 so that
hβ1 , α1 i is in the kernel. Explicitly, we may define an action of H1 on K · X by
β2 (X) = ζp2r X, β1 (X) = X, α1 (X) = X.
Similarly, H2 = hβ1 i × hβ2 , α2 i ≃ Cp2r × hβ1 , α1 i. Hence we get a linear character of H2
so that hβ2 , α2 i is in the kernel. Explicitly, we may define an action of H2 on K · Y by
β1 (Y ) = ζp2r Y, β2 (Y ) = Y, α2 (Y ) = Y.
i
Construct the induced representations of these linear characters by defining
L xi = α1 ·
i
r
Y, yi = α2 · X for 0 ≤ i ≤ p − 1. Thus we get an action of G on ( 0≤i≤pr −1 K ·
L
xi ) ⊕ ( 0≤i≤pr −1 K · yi ). Since βj αji = αji βj γji , α1 α2i = α2i α1 γ2i and α2 α1i = α1i α2 γ2−i for
1 ≤ j ≤ 2, 0 ≤ i ≤ pr − 1, the action of G is given as follows.
r

β1 : xi 7→ ζp1+ip
xi , yi 7→ yi ,
2r
r

β2 : xi 7→ xi , yi 7→ ζp1+ip
yi ,
2r
α1 : x0 7→ x1 7→ · · · 7→ xpr −1 7→ x0 , yi 7→ ζpi r yi ,
α2 : xi 7→ xi , y0 7→ y1 7→ · · · 7→ ypr −1 7→ y0 ,
for 0 ≤ i ≤ pr − 1. For 1 ≤ i ≤ pr − 1, define ui = xi /xi−1 and vi = yi /yi−1 . Thus
K(xi , yi : 0 ≤ i ≤ pr − 1) = K(x0 , y0 , ui , vi : 1 ≤ i ≤ pr − 1) and for every g ∈ G
g · x0 ∈ K(ui , vi : 1 ≤ i ≤ pr − 1) · x0 , g · y0 ∈ K(ui , vi : 1 ≤ i ≤ pr − 1) · y0 ,
while the subfield K(ui , vi : 1 ≤ i ≤ pr − 1) is invariant by the action of G. Thus
K(xi , yi : 0 ≤ i ≤ pr − 1)G = K(ui , vi : 1 ≤ i ≤ pr − 1)G (u, v) for some u, v such that
β1 (v) = β2 (v) = α1 (v) = α2 (v) = v and β1 (u) = β2 (u) = α1 (u) = α2 (u) = u. We have
now
β1 : ui 7→ ζpr ui , vi 7→ vi ,
β2 : ui 7→ ui , vi 7→ ζpr vi ,
α1 : u1 7→ u2 7→ · · · 7→ upr −1 7→ (u1 u2 · · · upr −1 )−1 , vi 7→ ζpr vi ,
α2 : ui 7→ ui , v1 7→ v2 7→ · · · 7→ vpr −1 7→ (v1 v2 · · · vpr −1 )−1 ,
for 1 ≤ i ≤ pr − 1. From Theorem 4.2 it follows that if K(ui , vi : 1 ≤ i ≤ pr − 1)G is
rational over K, so is K(xi , yi : 0 ≤ i ≤ pr − 1)G over K.

12

IVO M. MICHAILOV
r

Define w1 = v1p and wi = vi /vi−1 for 2 ≤ i ≤ pr − 1. Then K(ui , wi : 1 ≤ i ≤
pr − 1) = K(ui , vi : 1 ≤ i ≤ pr − 1)hβ2 i and the action of α2 on wi (1 ≤ i ≤ pr − 1) is
r

α2 : w1 7→ w1 w2p ,
w2 7→ w3 7→ · · · 7→ wpr −1 7→ (w1 w2p
7→ w1 w2p

r −2

w3p

r −3

r −1

w3p

r −2

· · · wp2r −1 )−1 7→

· · · wp2r −2 wpr −1 7→ w2 .

Define z1 = w2 , zi = α2i · w2 for 2 ≤ i ≤ pr − 1. Now the action of α2 is
α2 : z1 7→ z2 7→ · · · 7→ zpr −1 7→ (z1 z2 · · · zpr −1 )−1 .
r

r

Since w1 = (zpr −1 z1p −1 z2p −2 · · · zp2r −2 )−1 , we get that K(w1 , . . . , wpr −1 ) = K(z1 , . . . , zpr −1 ).
From Lemma 4.5 it follows that the action of α2 on K(z1 , . . . , zp2 −1 ) can be linearized.
Moreover, the actions of β1 and α1 on K(z1 , . . . , zp2 −1 ) are trivial. In this way we
reduce the rationality problem of K(ui , vi : 1 ≤ i ≤ pr − 1)G to the rationality of
K(ui : 1 ≤ i ≤ pr − 1)hβ1 ,α1 i over K. It remains to repeat the same argument for β1
and α1 to complete the proof.

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Faculty of Mathematics and Informatics, Shumen University ”Episkop Konstantin
Preslavski”, Universitetska str. 115, 9700 Shumen, Bulgaria
E-mail address: ivo michailov@yahoo.com

