Non-trivial group operations

Transkrypt

Marek Żabka
Instytut Matematyki
Bedlewo, june 2015r.
Marek Żabka (Instytut Matematyki)
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Let G = (G, ·) be e group.
Let us define a new binary operation ◦ of the form:
x ◦ y = x α1 · y β1 · x α2 · y β2 · . . . · x αn · y βn
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We call it a group operation iff
The pair (G, ◦) is also a group, named G◦
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there exists integers γ1 , γ2 , . . . , γm , δ1 , δ2 , . . . δm such that
x · y = x◦γ1 ◦ y◦δ1 ◦ x◦γ2 ◦ y◦δ2 ◦ . . . ◦ x◦γm ◦ y◦δm ,
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there exists integers γ1 , γ2 , . . . , γm , δ1 , δ2 , . . . δm such that
x · y = x◦γ1 ◦ y◦δ1 ◦ x◦γ2 ◦ y◦δ2 ◦ . . . ◦ x◦γm ◦ y◦δm ,
In each group we have so call trivial group operations:
x ◦ y = xy and x ◦ y = yx
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Problems:
To find non trivial group operations
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Problems:
To describe all group operations for some groups or class of groups.
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Problems:
To find class of groups such that a given word u(x, y ) is a group
operation
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Problems:
operation
To check if the groups G and G◦ are isomorphic?
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Problems:
operation
To find classes of groups such that for all group operations x ◦ y
groups G and G◦ are isomorphic?
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Problems:
operation
To find classes of groups such that for all group operations x ◦ y
groups G and G◦ are isomorphic?
Definition
The isomorphism between groups G and G◦ we call a weak automorphism
of group G .
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Remark
There can be two diferent words of F2 , which define the same group
operation for a given group.
For example, further we show, that
x · y · [x, y ]k oraz (x n · y n )m
define the same group opration in a group, while
x · y · [x, y ]k 6= (x n · y n )m
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Lat us observe some simple properties of group operations.
The same element of a sat G is a identity element for groups G and
G◦
There exist a word u commutator subgroup F20 of free group, such
that
x ◦ y = x · y · u(x, y )
Groups G and G◦ have the same power of elements: x◦µ = x µ
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G◦
that
x ◦ y = x · y · u(x, y )
Proof:
Let us set x = 1, y = 1 in
So 1 ◦ 1 = 1 and eventually 1◦ = 1
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G◦
that
x ◦ y = x · y · u(x, y )
Proof:
Let us set x = 1, y = 1 in
So 1 ◦ 1 = 1 and eventually 1◦ = 1
For x = 1 we have x = x α1 +α2 +···+αn and for y = 1: y = y β1 +β2 +···+βn . So
x ◦ y = x · y ·y −(β1 +β2 +···+βn ) ·x −(α1 +α2 +···+αn ) x α1 ·y β1 ·x α2 ·y β2 ·. . .·x αn ·y βn
|
{z
}
=u(x,y )∈F20
Hence, the inverse elements in groups G and G◦ are identical. The identity
x◦µ = x µ is true by induction.
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Let us preview some results:
1961 Neumann Hanna – for free groups there are no other associative
operations then a,x,y,xay,yax (Kerész problem).
1962 Hulanicki A. Świerczkowski S. – 2-nilpotent groups
1966 Goetz A. – if squer of every group element is in a center of group, all
group operations are trivial
Moreover, groups G and G◦ have the same subgroups and normal
subgroups.
1968 Street A, some group operations for some nilpotent groups of class 3
and 4, and remarks of group operations in metabelian groups.
Also, more common properties for letices of subgroups and normal
subgroups.
An example that finite groups G and G◦ can be not isomorphic.
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1974 Solecki A. – for groups of finite there exists group operations of the
form x · y = (x m y m )n , where m n = 1 mod exp G
1993 Żabka M., – For finite symmetric groups Sn all group operations have
the form x · y = (x m y m )n ,
1996 Żabka M., – for finite Coxeter and some generalizations and for groups
of order pq also all group operations have the form x · y = (x m y m )n ,
2010 Plonka E. – description for Dihedral groups
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Example:
Let G be a 2-nilpotent group of finite exponent of derived group G 0 , that is
[[x, y ], z] = [x, y ]n = 1
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Example:
[[x, y ], z] = [x, y ]n = 1
We have:
Operation x ◦ y = xy [x, y ]k is associative for all integer k
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Example:
[[x, y ], z] = [x, y ]n = 1
We have:
It is not an operation group for some of the k, for example if
n = 2k + 1 than operation x ◦ y = xy [x, y ]k is commutative, so
cannot be a group operation.
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Example:
[[x, y ], z] = [x, y ]n = 1
We have:
By Hulanicki, Świerczkowski: binary operation x ◦ y is a group
operation, ⇐⇒ gcd(exp G 0 , 2k + 1) = 1. Moreover, for groups of
finite exponent, the group G and G◦ are isomorphic.
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Example:
[[x, y ], z] = [x, y ]n = 1
We have:
By Hulanicki, Świerczkowski: binary operation x ◦ y is a group
operation, ⇐⇒ gcd(exp G 0 , 2k + 1) = 1. Moreover, for groups of
finite exponent, the group G and G◦ are isomorphic.
It is easy to show, that in this situation,
xy [x, y ]k = (x m y m )r , for some m, r
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The problem of isomorphisps of group G and G◦ is solved for some
relatively free groups.
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The problem of isomorphisps of group G and G◦ is solved for some
relatively free groups.
Theorem
If the group G is relatively free, then the same is the group G◦ .
The groups G and G◦ has the same set of free generators.
If the group G :
satisfying maximal condition for normal subgrpups and
G0 ∈ Var G ,
then groups G and G◦ are isomorphic
(and Var G0 = Var G ).
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Definition
Group variety R we would call close over group operations if
∀G : G ∈ R =⇒ G◦ ∈ R
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Definition
Group variety R we would call close over group operations if
∀G : G ∈ R =⇒ G◦ ∈ R
Theorem
If the group G :
satisfying maximal condition for normal subgrpups and
Var G is closed over group operations
then groups G and G◦ are isomorphic
(and Var G0 = Var G ).
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Examples of varietis closed over group operations:
c-nilpotent groups: Nc
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c-solvable groups: Sc
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Bernside groups of exponent k: Bk
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2-engel groups: E2 (identity [x, y , y ] = 1)
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varity width idenity: [[x, y ], z m ] = 1
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2-nilpotent groups width indentity [x, y ]m = 1 (exp G 0 < ∞ )
(Hulanicki Swierczkowski)
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2-nilpotent groups width indentity [x, y ]m = 1 (exp G 0 < ∞ )
(Hulanicki Swierczkowski)
So for 2-nilpotent of finite exponent of derived subgroup G 0 , groups G and
G◦ are isomorphic for finitely generating groups, and of course, the group
operation is not of the power form and the somorphism also is not of
power form.
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Some more questions:
to find group operation not of the power form with isomorphic groups
G and G◦ .
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Some more questions:
to find group operation not of the power form with isomorphic groups
G and G◦ .
to find varietis not closed over group operations.
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Non-trivial group operations

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