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The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The direct product problem
The free product
cancellation
problem
Conclusions
Vipul Naik
October 17, 2006
Outline
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The direct product cancellation problem
The free product
cancellation
problem
Conclusions
The free product cancellation problem
Conclusions
The problem statement
The direct product
problem
Vipul Naik
I
Given: Groups G , H and K such that:
G ×H ∼
=G ×K
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
The direct product
problem
The problem statement
Vipul Naik
I
Given: Groups G , H and K such that:
G ×H ∼
=G ×K
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
I
To Prove:
H∼
=K
The direct product
problem
The problem statement
Vipul Naik
I
Given: Groups G , H and K such that:
G ×H ∼
=G ×K
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
I
To Prove:
H∼
=K
I
Question: What hypotheses on G , H and K can take
us from Given to To Prove?
The direct product
problem
The problem statement
Vipul Naik
I
Given: Groups G , H and K such that:
G ×H ∼
=G ×K
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
I
To Prove:
H∼
=K
I
Question: What hypotheses on G , H and K can take
us from Given to To Prove?That is, when can we
cancel G ?
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The question really has two aspects:
I
What conditions on G ensure that G can be cancelled?
The free product
cancellation
problem
Conclusions
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The question really has two aspects:
I
What conditions on G ensure that G can be cancelled?
I
What conditions on H (and/or K ) ensure that G can
be cancelled?
The free product
cancellation
problem
Conclusions
Monoidal interpretation
The direct product
problem
Vipul Naik
Consider a commutative and associative monoid whose:
I
Elements are isomorphism classes of groups
I
Multiplication map is the direct product operation.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Monoidal interpretation
The direct product
problem
Vipul Naik
Consider a commutative and associative monoid whose:
I
Elements are isomorphism classes of groups
I
Multiplication map is the direct product operation.
The big questions:
I
Which elements of the monoid are cancellative?
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Monoidal interpretation
The direct product
problem
Vipul Naik
Consider a commutative and associative monoid whose:
I
Elements are isomorphism classes of groups
I
Multiplication map is the direct product operation.
The big questions:
I
Which elements of the monoid are cancellative?
I
Which elements of the monoid are cancelductive? That
is, for which elements H is it true that G can always be
cancelled from an expression of the form
G ×H ∼
= G × K?
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Where the cancellation fails
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Let H be any nontrivial group. Take the countable direct
product of H with itself. Call that G .
The free product
cancellation
problem
Conclusions
Where the cancellation fails
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Let H be any nontrivial group. Take the countable direct
product of H with itself. Call that G .
Then:
G ×H ∼
= G × {e}
The free product
cancellation
problem
Conclusions
Where the cancellation fails
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Let H be any nontrivial group. Take the countable direct
product of H with itself. Call that G .
Then:
G ×H ∼
= G × {e}
Even though H is nontrivial.
The free product
cancellation
problem
Conclusions
Positive answer for finite groups
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
We’re going to prove that the collection of isomorphism
classes of finite groups is cancellative.
The free product
cancellation
problem
Conclusions
Positive answer for finite groups
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
We’re going to prove that the collection of isomorphism
classes of finite groups is cancellative.
In other words, if G , H and K are finite groups, we’ll prove
that
G ×H ∼
=K
= G × K =⇒ H ∼
The free product
cancellation
problem
Conclusions
Category theoretic definition of direct product
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Given any group L, the homomorphisms from L to G × H
are in bijection with pairs of homomorphisms from L to G
and L to H.
The free product
cancellation
problem
Conclusions
Category theoretic definition of direct product
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Given any group L, the homomorphisms from L to G × H
are in bijection with pairs of homomorphisms from L to G
and L to H.
This follows from the category theoretic interpretation of
direct product as a universal property with respect to maps
to both the groups.
The free product
cancellation
problem
Conclusions
The direct product
problem
A little working out
Vipul Naik
Let h(L, G ) denote the cardinality of Hom(L, G ) where L
and G are finite groups. Then we have:
G ×H ∼
= G ×K
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
A little working out
The direct product
problem
Vipul Naik
Let h(L, G ) denote the cardinality of Hom(L, G ) where L
and G are finite groups. Then we have:
G ×H ∼
= G ×K
=⇒ h(L, G × H) = h(L, G × K )
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
A little working out
The direct product
problem
Vipul Naik
Let h(L, G ) denote the cardinality of Hom(L, G ) where L
and G are finite groups. Then we have:
G ×H ∼
= G ×K
=⇒ h(L, G × H) = h(L, G × K )
=⇒ h(L, G )h(L, H) = h(L, G )h(L, K )
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
A little working out
The direct product
problem
Vipul Naik
Let h(L, G ) denote the cardinality of Hom(L, G ) where L
and G are finite groups. Then we have:
G ×H ∼
= G ×K
=⇒ h(L, G × H) = h(L, G × K )
=⇒ h(L, G )h(L, H) = h(L, G )h(L, K )
=⇒ h(L, G )(h(L, H) − h(L, K )) = 0
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
The direct product
problem
A little working out
Vipul Naik
Let h(L, G ) denote the cardinality of Hom(L, G ) where L
and G are finite groups. Then we have:
G ×H ∼
= G ×K
=⇒ h(L, G × H) = h(L, G × K )
=⇒ h(L, G )h(L, H) = h(L, G )h(L, K )
=⇒ h(L, G )(h(L, H) − h(L, K )) = 0
=⇒ h(L, H) − h(L, K ) = 0
because h(L, G ) is finite nonzero
=⇒ h(L, H) = h(L, K )
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
So what?
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product
cancellation
problem
We showed that if G , H and K are finite groups and
G ×H ∼
= G × K , then for any finite group L,
h(L, H) = h(L, K )
Conclusions
So what?
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product
cancellation
problem
We showed that if G , H and K are finite groups and
G ×H ∼
= G × K , then for any finite group L,
h(L, H) = h(L, K )
Define the to-Hom statistics of a group H as the map
L 7→ h(L, H). Then from the above discussion, H and K
have the same to-Hom statistics.
Conclusions
to-Hom and injective to-Hom
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Let i(L, H) denote the number of injective homomorphisms
from L to H.
The free product
cancellation
problem
Conclusions
The direct product
problem
to-Hom and injective to-Hom
Vipul Naik
The direct product
cancellation
problem
Let i(L, H) denote the number of injective homomorphisms
from L to H.
Then we have:
h(L, G ) =
X
NEL
i(L/N, G )
(1)
The free product
cancellation
problem
Conclusions
The direct product
problem
to-Hom and injective to-Hom
Vipul Naik
The direct product
cancellation
problem
Let i(L, H) denote the number of injective homomorphisms
from L to H.
Then we have:
h(L, G ) =
X
i(L/N, G )
NEL
Manipulating this equation a bit, we get:
H and K have the same to-Hom statistics =⇒
i(L, H) = i(L, K ) for all finite groups L.
(1)
The free product
cancellation
problem
Conclusions
The direct product
problem
to-Hom and injective to-Hom
Vipul Naik
The direct product
cancellation
problem
Let i(L, H) denote the number of injective homomorphisms
from L to H.
Then we have:
h(L, G ) =
X
i(L/N, G )
NEL
Manipulating this equation a bit, we get:
H and K have the same to-Hom statistics =⇒
i(L, H) = i(L, K ) for all finite groups L.
And then, putting H = L, we are done!
(1)
The free product
cancellation
problem
Conclusions
to-Hom finite groups
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Finiteness of G was not important to the above proof. All
we needed was that for every finite group L, h(L, G ) is finite.
The free product
cancellation
problem
Conclusions
to-Hom finite groups
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Finiteness of G was not important to the above proof. All
we needed was that for every finite group L, h(L, G ) is finite.
A group satisfying that property is termed to-Hom finite.
Examples of to-Hom finite groups: torsion-free groups,
quasicyclic groups.
The free product
cancellation
problem
Conclusions
to-Hom finite groups
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Finiteness of G was not important to the above proof. All
we needed was that for every finite group L, h(L, G ) is finite.
A group satisfying that property is termed to-Hom finite.
Examples of to-Hom finite groups: torsion-free groups,
quasicyclic groups.
Reformulation: if G is to-Hom finite and H and K are finite,
then G × H ∼
= G × K =⇒ H ∼
=K
The free product
cancellation
problem
Conclusions
In greater generality
We used three facts to prove:
G ×H ∼
= G × K =⇒ H ∼
=K
1. For every L, we have:
h(L, G × H) = h(L, G ) × h(L, H)
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
The direct product
problem
In greater generality
We used three facts to prove:
Vipul Naik
G ×H ∼
= G × K =⇒ H ∼
=K
The direct product
cancellation
problem
The free product
cancellation
problem
1. For every L, we have:
h(L, G × H) = h(L, G ) × h(L, H)
2. There is a relationship between h and i, namely:
X
h(L, H) =
i(L/N, H)
NEL
Conclusions
The direct product
problem
In greater generality
We used three facts to prove:
Vipul Naik
G ×H ∼
= G × K =⇒ H ∼
=K
The direct product
cancellation
problem
The free product
cancellation
problem
1. For every L, we have:
h(L, G × H) = h(L, G ) × h(L, H)
2. There is a relationship between h and i, namely:
X
h(L, H) =
i(L/N, H)
NEL
3. If there is an injective homomorphism from H to K and
there is an injective homomorphism from K to H, then
H is isomorphic to K .
Question: which of these is true for structures other than
groups?
Conclusions
Variety of algebras
The direct product
problem
Vipul Naik
I
An algebra is a set along with a collection of n-ary
operations on the set) possibly with different n.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Variety of algebras
The direct product
problem
Vipul Naik
I
I
An algebra is a set along with a collection of n-ary
operations on the set) possibly with different n.
A variety of algebras is described by a collection of
operator symbols, each with an arity, and a collection of
equations between expressions involving those operator
symbols. An algebra belongs to the variety if it has
operations for those operator symbols and the
operations satisfy the equations universally.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Variety of algebras
The direct product
problem
Vipul Naik
I
I
I
An algebra is a set along with a collection of n-ary
operations on the set) possibly with different n.
A variety of algebras is described by a collection of
operator symbols, each with an arity, and a collection of
equations between expressions involving those operator
symbols. An algebra belongs to the variety if it has
operations for those operator symbols and the
operations satisfy the equations universally.
For instance, groups form a variety of algebras.
Monoids form a variety of algebra. Rings with identity
form a variety of algebras. Fields do not form a vareity
of algebras.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
In the generality of varieties
The general version says:
If G , H and K are finite algebras of a variety with 0, and
G ×H ∼
= G × K as algebras of the variety, then H ∼
= K as
algebras of the variety.
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
In the generality of varieties
The general version says:
If G , H and K are finite algebras of a variety with 0, and
G ×H ∼
= G × K as algebras of the variety, then H ∼
= K as
algebras of the variety.
1. For every L, we have:
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
h(L, G × H) = h(L, G ) × h(L, H)
This still holds.
In the generality of varieties
The general version says:
If G , H and K are finite algebras of a variety with 0, and
G ×H ∼
= G × K as algebras of the variety, then H ∼
= K as
algebras of the variety.
1. For every L, we have:
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
h(L, G × H) = h(L, G ) × h(L, H)
This still holds.
2. There is a relationship between h and i, namely:
X
h(L, H) =
i(L/ρ, H)
ρ a congruence on L
In the generality of varieties
The general version says:
If G , H and K are finite algebras of a variety with 0, and
G ×H ∼
= G × K as algebras of the variety, then H ∼
= K as
algebras of the variety.
1. For every L, we have:
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
h(L, G × H) = h(L, G ) × h(L, H)
This still holds.
2. There is a relationship between h and i, namely:
X
h(L, H) =
i(L/ρ, H)
ρ a congruence on L
3. If there is an injective homomorphism from H to K and
there is an injective homomorphism from K to H, then
H is isomorphic to K .
Outline
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The direct product cancellation problem
The free product
cancellation
problem
Conclusions
The free product cancellation problem
Conclusions
Meaning of free product
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product of two groups G and H is denoted as
G ∗ H and is defined as the set of formal strings whose
letters are elements from G and elements from H modulo
the product relations in G and in H.
The free product
cancellation
problem
Conclusions
Meaning of free product
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product of two groups G and H is denoted as
G ∗ H and is defined as the set of formal strings whose
letters are elements from G and elements from H modulo
the product relations in G and in H.
The collection of isomorphism classes of all groups forms a
commutative monoid with respect to the free product
operation. Finite groups do not form a submonoid: the free
product of any two nontrivial groups is infinite. However,
finitely generated groups do form a submonoid.
The free product
cancellation
problem
Conclusions
The problem statement
The direct product
problem
Vipul Naik
I
Given: Groups G , H and K such that:
G ∗H ∼
=G ∗K
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
The direct product
problem
The problem statement
Vipul Naik
I
Given: Groups G , H and K such that:
G ∗H ∼
=G ∗K
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
I
To Prove:
H∼
=K
The direct product
problem
The problem statement
Vipul Naik
I
Given: Groups G , H and K such that:
G ∗H ∼
=G ∗K
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
I
To Prove:
H∼
=K
I
Question: What hypotheses on G , H and K can take
us from Given to To Prove?
The direct product
problem
The problem statement
Vipul Naik
I
Given: Groups G , H and K such that:
G ∗H ∼
=G ∗K
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
I
To Prove:
H∼
=K
I
Question: What hypotheses on G , H and K can take
us from Given to To Prove?That is, when can we
cancel G ?
Where the cancellation fails
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Let H be any nontrivial group. Take the countable free
product of H with itself. Call that G .
The free product
cancellation
problem
Conclusions
Where the cancellation fails
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Let H be any nontrivial group. Take the countable free
product of H with itself. Call that G .
Then:
G ∗H ∼
= G ∗ {e}
The free product
cancellation
problem
Conclusions
Where the cancellation fails
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Let H be any nontrivial group. Take the countable free
product of H with itself. Call that G .
Then:
G ∗H ∼
= G ∗ {e}
Even though H is nontrivial.
The free product
cancellation
problem
Conclusions
Products and coproducts
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Given groups (or algebras of a variety) L, G , and H, there is
a natural bijection between the homomorphisms from L to
G × H and pairs of homomorphisms L to G , L to H.
We can in fact define G × H from this.
The free product
cancellation
problem
Conclusions
Products and coproducts
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Given groups (or algebras of a variety) L, G , and H, there is
a natural bijection between the homomorphisms from L to
G × H and pairs of homomorphisms L to G , L to H.
We can in fact define G × H from this.
Is it true that h(G × H, L) = h(G , L)h(H, L)?
No. But if we replace direct product by free product, we get:
h(G ∗ H, L) = h(G , L)h(H, L)
The free product can be defined this way.
The free product
cancellation
problem
Conclusions
The direct product
problem
A little working out
Vipul Naik
Let h(G , L) denote the cardinality of Hom(G , L) where L
and G are finite groups. Then we have:
The direct product
cancellation
problem
The free product
cancellation
problem
G ∗H ∼
= G ∗K
Conclusions
A little working out
The direct product
problem
Vipul Naik
Let h(G , L) denote the cardinality of Hom(G , L) where L
and G are finite groups. Then we have:
The direct product
cancellation
problem
The free product
cancellation
problem
G ∗H ∼
= G ∗K
=⇒ h(G ∗ H, L) = h(G ∗ K , L)
Conclusions
A little working out
The direct product
problem
Vipul Naik
Let h(G , L) denote the cardinality of Hom(G , L) where L
and G are finite groups. Then we have:
The direct product
cancellation
problem
The free product
cancellation
problem
G ∗H ∼
= G ∗K
=⇒ h(G ∗ H, L) = h(G ∗ K , L)
=⇒ h(G , L)h(H, L) = h(G , L)h(K , L)
Conclusions
A little working out
The direct product
problem
Vipul Naik
Let h(G , L) denote the cardinality of Hom(G , L) where L
and G are finite groups. Then we have:
The direct product
cancellation
problem
The free product
cancellation
problem
G ∗H ∼
= G ∗K
=⇒ h(G ∗ H, L) = h(G ∗ K , L)
=⇒ h(G , L)h(H, L) = h(G , L)h(K , L)
=⇒ h(G , L)(h(H, L) − h(K , L)) = 0
Conclusions
The direct product
problem
A little working out
Vipul Naik
Let h(G , L) denote the cardinality of Hom(G , L) where L
and G are finite groups. Then we have:
The direct product
cancellation
problem
The free product
cancellation
problem
G ∗H ∼
= G ∗K
=⇒ h(G ∗ H, L) = h(G ∗ K , L)
=⇒ h(G , L)h(H, L) = h(G , L)h(K , L)
=⇒ h(G , L)(h(H, L) − h(K , L)) = 0
=⇒ h(H, L) − h(K , L) = 0
because h(G , L) is finite nonzero
=⇒ h(H, L) = h(K , L)
Conclusions
So what?
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product
cancellation
problem
We showed that if G , H and K are finite groups and
G ∗H ∼
= G ∗ K , then for any finite group L,
h(H, L) = h(K , L)
Conclusions
So what?
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The free product
cancellation
problem
We showed that if G , H and K are finite groups and
G ∗H ∼
= G ∗ K , then for any finite group L,
h(H, L) = h(K , L)
Define the from-Hom statistics of a group H as the map
L 7→ h(H, L). Then from the above discussion, H and K
have the same from-Hom statistics.
Conclusions
to-Hom and injective to-Hom
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Let s(H, L) denote the number of surjective homomorphisms
from H to H.
The free product
cancellation
problem
Conclusions
The direct product
problem
to-Hom and injective to-Hom
Vipul Naik
The direct product
cancellation
problem
Let s(H, L) denote the number of surjective homomorphisms
from H to H.
Then we have:
h(G , L) =
X
M≤L
s(G , M)
(2)
The free product
cancellation
problem
Conclusions
The direct product
problem
to-Hom and injective to-Hom
Vipul Naik
The direct product
cancellation
problem
Let s(H, L) denote the number of surjective homomorphisms
from H to H.
Then we have:
h(G , L) =
X
s(G , M)
M≤L
Manipulating this equation a bit, we get:
H and K have the same fom-Hom statistics =⇒
s(H, L) = s(K , L) for all finite groups L.
(2)
The free product
cancellation
problem
Conclusions
The direct product
problem
to-Hom and injective to-Hom
Vipul Naik
The direct product
cancellation
problem
Let s(H, L) denote the number of surjective homomorphisms
from H to H.
Then we have:
h(G , L) =
X
s(G , M)
M≤L
Manipulating this equation a bit, we get:
H and K have the same fom-Hom statistics =⇒
s(H, L) = s(K , L) for all finite groups L.
And then, putting H = L, we are done!
(2)
The free product
cancellation
problem
Conclusions
from-Hom finite groups
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Finiteness of G was not important to the above proof. All
we needed was that for every finite group L, h(G , L) is finite.
The free product
cancellation
problem
Conclusions
from-Hom finite groups
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
Finiteness of G was not important to the above proof. All
we needed was that for every finite group L, h(G , L) is finite.
A group satisfying that property is termed from-Hom finite.
Examples of from-Hom finite groups: finitely generated
groups, simple groups, groups with only finitely many normal
subgroups of finite index.
Reformulation: if G is from-Hom finite and H and K are
finite, then G ∗ H ∼
= G ∗ K =⇒ H ∼
=K
The free product
cancellation
problem
Conclusions
This also generalizes
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The notion of free product makes sense for any variety of
algebras.
And the earlier result generalizes to saying:
If G , H and K are finite algebras of a variety then
G ∗H ∼
=K
= G ∗ K =⇒ H ∼
The free product
cancellation
problem
Conclusions
Outline
The direct product
problem
Vipul Naik
The direct product
cancellation
problem
The direct product cancellation problem
The free product
cancellation
problem
Conclusions
The free product cancellation problem
Conclusions
Crude summary
The direct product
problem
Vipul Naik
When does G × H ∼
= G × K imply that H ∼
= K?
For groups (and also for varieties of algebras):
I
When all of G , H and K are finite.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Crude summary
The direct product
problem
Vipul Naik
When does G × H ∼
= G × K imply that H ∼
= K?
For groups (and also for varieties of algebras):
I
When all of G , H and K are finite.
I
when H and K are finite and G is to-Hom finite. For
instance, G may be torsion-free, or its torsion part may
be finite.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Crude summary
The direct product
problem
Vipul Naik
When does G × H ∼
= G × K imply that H ∼
= K?
For groups (and also for varieties of algebras):
I
When all of G , H and K are finite.
I
when H and K are finite and G is to-Hom finite. For
instance, G may be torsion-free, or its torsion part may
be finite.
∼ K?
When does G ∗ H ∼
= G ∗ K imply that H =
For groups (and also for varieties of algebras):
I
When all of G , H and K are finite.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Crude summary
The direct product
problem
Vipul Naik
When does G × H ∼
= G × K imply that H ∼
= K?
For groups (and also for varieties of algebras):
I
When all of G , H and K are finite.
I
when H and K are finite and G is to-Hom finite. For
instance, G may be torsion-free, or its torsion part may
be finite.
∼ K?
When does G ∗ H ∼
= G ∗ K imply that H =
For groups (and also for varieties of algebras):
I
When all of G , H and K are finite.
I
When H and K are finite and G is from-Hom finite. For
instance, G may be finitely generated, simple, or have
only finitely many finite index normal subgroups.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Sophisticated summary
The direct product
problem
Vipul Naik
I
The monoid of isomorphism classes of finite algebras of
a variety (such as groups) with respect to the direct
product operation, is cancellative.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Sophisticated summary
The direct product
problem
Vipul Naik
I
I
The monoid of isomorphism classes of finite algebras of
a variety (such as groups) with respect to the direct
product operation, is cancellative.
Even better, in the monoid of isomorphism classes of
algebras of a variety, the to-Hom finite members can
always be cancelled when multiplying with the finite
members.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Sophisticated summary
The direct product
problem
Vipul Naik
I
The monoid of isomorphism classes of finite algebras of
a variety (such as groups) with respect to the direct
product operation, is cancellative.
I
Even better, in the monoid of isomorphism classes of
algebras of a variety, the to-Hom finite members can
always be cancelled when multiplying with the finite
members.
I
In the monoid of isomorphism classes of algebras with
the free product as multiplication, the finite elements
cancel with the finite elements.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions
Sophisticated summary
The direct product
problem
Vipul Naik
I
The monoid of isomorphism classes of finite algebras of
a variety (such as groups) with respect to the direct
product operation, is cancellative.
I
Even better, in the monoid of isomorphism classes of
algebras of a variety, the to-Hom finite members can
always be cancelled when multiplying with the finite
members.
I
In the monoid of isomorphism classes of algebras with
the free product as multiplication, the finite elements
cancel with the finite elements.
I
More generally, the from-Hom finite elements cancel
when multiplying with the finite elements.
The direct product
cancellation
problem
The free product
cancellation
problem
Conclusions