Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Twisted group ring isomorphism problem
Leo Margolis and Ofir Schnabel
Spa, Belgium
Groups, Rings and the Yang-Baxter equation
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Group ring isomorphism problem
Denote by Ω the class of finite groups and by Ωn the groups of
order n. For a commutative ring R denote by ∆R an equivalence
relation on Ω which is defined by G ∆R H if and only if RG ∼
= RH.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Group ring isomorphism problem
Denote by Ω the class of finite groups and by Ωn the groups of
order n. For a commutative ring R denote by ∆R an equivalence
relation on Ω which is defined by G ∆R H if and only if RG ∼
= RH.
The group ring isomorphism problem [GRIP]
For a given commutative ring R, determine the equivalence classes
of Ω with respect to the relation ∆R . Answer in particular, for
which groups G ∆R H implies G ∼
= H.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Facts and results
If G , H are abelian groups of the same cardinality then
G ∆C H.
Any finite abelian group is a ∆Q singleton [Perlis and Walker
1950].
G ∆Z H ⇒ G ∆R H, for any commutative ring R.
Any p-group G is a ∆Z singleton [Roggenkamp and Scott
1987].
There exist non-isomorphic groups X , Y such that X ∆Z Y
[Hertweck 2001].
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Twisted group rings
We wish to investigate a “twisted” version of the GRIP.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Twisted group rings
We wish to investigate a “twisted” version of the GRIP.
For a 2-cocycle α ∈ Z 2 (G , R ∗ ) the twisted group ring R α G is the
free R-module with basis {ug }g ∈G such that
ug uh = α(g , h)ugh for all g , h ∈ G
and any ug commutes with the elements of R.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Twisted group rings
We wish to investigate a “twisted” version of the GRIP.
For a 2-cocycle α ∈ Z 2 (G , R ∗ ) the twisted group ring R α G is the
free R-module with basis {ug }g ∈G such that
ug uh = α(g , h)ugh for all g , h ∈ G
and any ug commutes with the elements of R.
The ring structure of R α G depends only on the cohomology class
of α, [α] ∈ H 2 (G , R ∗ ) and not on the particular 2-cocycle.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
refinement of ∆
We are now ready to introduce the relation of interest.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
refinement of ∆
We are now ready to introduce the relation of interest.
Definition
Let R be a commutative ring. For G , H ∈ Ω, G ∼R H if and only
if there exists a group isomorphism
ψ : H 2 (G , R ∗ ) → H 2 (H, R ∗ )
such that for any [α] ∈ H 2 (G , R ∗ ),
R αG ∼
= R ψ(α) H.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
refinement of ∆
We are now ready to introduce the relation of interest.
Definition
Let R be a commutative ring. For G , H ∈ Ω, G ∼R H if and only
if there exists a group isomorphism
ψ : H 2 (G , R ∗ ) → H 2 (H, R ∗ )
such that for any [α] ∈ H 2 (G , R ∗ ),
R αG ∼
= R ψ(α) H.
Notice that ∼R is a refinement of ∆R .
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Twisted group ring isomorphism problem
Main Problem: The twisted group ring isomorphism problem
[TGRIP]
For a given commutative ring R, determine the equivalence classes
of Ω with respect to the relation ∼R . Answer in particular, for
which groups G ∼R H, implies G ∼
= H.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Twisted group ring isomorphism problem
Main Problem: The twisted group ring isomorphism problem
[TGRIP]
For a given commutative ring R, determine the equivalence classes
of Ω with respect to the relation ∼R . Answer in particular, for
which groups G ∼R H, implies G ∼
= H.
We will deal mostly with the case R = C. In this case the second
cohomology group is called the Schur multiplier and it is denoted
by M(G ).
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Twisted group ring isomorphism problem
Main Problem: The twisted group ring isomorphism problem
[TGRIP]
For a given commutative ring R, determine the equivalence classes
of Ω with respect to the relation ∼R . Answer in particular, for
which groups G ∼R H, implies G ∼
= H.
We will deal mostly with the case R = C. In this case the second
cohomology group is called the Schur multiplier and it is denoted
by M(G ).
The twisted group algebra Cα G may be simple. In these cases α is
called nondegenerate and G is called of central type.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
The abelian case
While all the abelian groups of the same cardinality are ∆C
equivalent, the following holds for ∼C .
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
The abelian case
While all the abelian groups of the same cardinality are ∆C
equivalent, the following holds for ∼C .
Lemma
Any abelian group A is a ∼C singleton.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
The abelian case
While all the abelian groups of the same cardinality are ∆C
equivalent, the following holds for ∼C .
Lemma
Any abelian group A is a ∼C singleton.
This follows from the fact that abelian groups of the same
cardinality admits non-isomorphic Schur multipliers.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Necessary conditions for G ∼C H
Lemma
The following conditions are necessary conditions for G ∼C H.
None of them is sufficient and none of them implies the other.
∼ M(H).
A) M(G ) =
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Necessary conditions for G ∼C H
Lemma
The following conditions are necessary conditions for G ∼C H.
None of them is sufficient and none of them implies the other.
∼ M(H).
A) M(G ) =
B) There exists a set bijection φ : M(G ) → M(H) such that
Cα G ∼
= Cφ(α) H for any [α] ∈ M(G ) and in particular
∼
CG = CH.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
The case |G | = p 4
Lemma
Let p be prime and let G and H be groups of order p 4 with the
following properties
1
2
G , H are both not of central type.
M(G ) ∼
= M(H).
CG ∼
= CH.
Then, G ∼C H.
3
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
The case |G | = p 4
Lemma
Let p be prime and let G and H be groups of order p 4 with the
following properties
1
2
G , H are both not of central type.
M(G ) ∼
= M(H).
CG ∼
= CH.
Then, G ∼C H.
3
Using the above and the classification of groups of central type of
order p 4 we partition Ωp4 to ∼C -classes for any prime p. In
particular we show that there exist G 6∼
= H such that G ∼C H.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
The case |G | = p 2 q 2
Theorem
Let G and H be groups of cardinality p 2 q 2 for primes p < q. If
M(G ) ∼
= M(H) and CG ∼
= CH, then G ∼C H.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
The case |G | = p 2 q 2
Theorem
Let G and H be groups of cardinality p 2 q 2 for primes p < q. If
M(G ) ∼
= M(H) and CG ∼
= CH, then G ∼C H.
Using this theorem and a known description of groups of order
p 2 q 2 we are able to partition Ωp2 q2 to ∼C -classes for any primes
p, q.
Example
In the partition of the 21 groups of order 32 192 to ∼C classes there
are 11 singletons, 3 classes containing 2 groups and 1 class
containing 4 groups.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
It turns out that any group of central type of cardinality p 4 , p 2 q 2
or 64 is a ∼C -singleton.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
It turns out that any group of central type of cardinality p 4 , p 2 q 2
or 64 is a ∼C -singleton.
One might conjecture that any group of central type is a
∼C -singleton.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Groups of Central Type
Using the description of groups of central type of order n2 for n
square-free we prove.
Theorem
Let G be a group of central type of order n2 where n is a square
free number. Then in the following cases G is a ∼C -singleton.
1
If |G p | is divisible by at most two primes.
2
If
|G |
|G p ||Z (G )|
is square-free.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Groups of Central Type
Using the description of groups of central type of order n2 for n
square-free we prove.
Theorem
Let G be a group of central type of order n2 where n is a square
free number. Then in the following cases G is a ∼C -singleton.
1
If |G p | is divisible by at most two primes.
2
If
|G |
|G p ||Z (G )|
is square-free.
However there exist non-isomorphic groups of central type of order
n2 where n is a square-free number which are ∼C equivalent.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem
Group ring isomorphism problem
Twisted group ring isomorphism problem
Groups of order p 4 and p 2 q 2
Groups of central type
Thanks for your attention.
Leo Margolis and Ofir Schnabel
Twisted group ring isomorphism problem