Non-trivial group operations

Transcription

1 Non-trivial group operations Marek Żabka Instytut Matematyki Bedlewo, june 2015r. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 1 / 12

2 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 Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 2 / 12

3 Let G = (G, ) be e group. Let us define a new binary operation of the form: We call it a group operation iff x y = x α1 y β1 x α2 y β2... x αn y βn The pair (G, ) is also a group, named G Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 2 / 12

4 Let G = (G, ) be e group. Let us define a new binary operation of the form: We call it a group operation iff x y = x α1 y β1 x α2 y β2... x αn y βn The pair (G, ) is also a group, named G there exists integers γ 1, γ 2,..., γ m, δ 1, δ 2,... δ m such that x y = x γ 1 y δ 1 x γ 2 y δ 2... x γm y δm, Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 2 / 12

5 Let G = (G, ) be e group. Let us define a new binary operation of the form: We call it a group operation iff x y = x α1 y β1 x α2 y β2... x αn y βn The pair (G, ) is also a group, named G there exists integers γ 1, γ 2,..., γ m, δ 1, δ 2,... δ m such that x y = x γ 1 y δ 1 x γ 2 y δ 2... x γm y δm, In each group we have so call trivial group operations: x y = xy and x y = yx Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 2 / 12

6 Problems: To find non trivial group operations Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 3 / 12

7 Problems: To find non trivial group operations To describe all group operations for some groups or class of groups. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 3 / 12

8 Problems: To find non trivial group operations To describe all group operations for some groups or class of groups. To find class of groups such that a given word u(x, y) is a group operation Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 3 / 12

9 Problems: To find non trivial group operations To describe all group operations for some groups or class of groups. To find class of groups such that a given word u(x, y) is a group operation To check if the groups G and G are isomorphic? Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 3 / 12

10 Problems: To find non trivial group operations To describe all group operations for some groups or class of groups. To find class of groups such that a given word u(x, y) is a group operation To check if the groups G and G are isomorphic? To find classes of groups such that for all group operations x y groups G and G are isomorphic? Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 3 / 12

11 Problems: To find non trivial group operations To describe all group operations for some groups or class of groups. To find class of groups such that a given word u(x, y) is a group operation To check if the groups G and G are isomorphic? 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. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 3 / 12

12 Remark There can be two diferent words of F 2, 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 (x n y n ) m Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 4 / 12

13 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 F 2 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 Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 5 / 12

14 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 F 2 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 Proof: Let us set x = 1, y = 1 in x y = x α1 y β1 x α2 y β2... x αn y βn So 1 1 = 1 and eventually 1 = 1 Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 5 / 12

15 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 F 2 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 Proof: Let us set x = 1, y = 1 in x y = x α1 y β1 x α2 y β2... x αn y βn So 1 1 = 1 and eventually 1 = 1 For x = 1 we have x = x α 1+α 2 + +α n and for y = 1: y = y β 1+β 2 + +β n. So x y = x y y (β 1+β 2 + +β n) x (α 1+α 2 + +α n) x α1 y β1 x α2 y β2... x αn y βn }{{} =u(x,y) F 2 Hence, the inverse elements in groups G and G are identical. The identity x µ = x µ is true by induction. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 5 / 12

16 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) 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 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. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 6 / 12

17 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 S n 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 P lonka E. description for Dihedral groups Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 7 / 12

18 Example: Let G be a 2-nilpotent group of finite exponent of derived group G, that is [[x, y], z] = [x, y] n = 1 Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 8 / 12

19 Example: Let G be a 2-nilpotent group of finite exponent of derived group G, that is We have: [[x, y], z] = [x, y] n = 1 Operation x y = xy[x, y] k is associative for all integer k Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 8 / 12

20 Example: Let G be a 2-nilpotent group of finite exponent of derived group G, that is We have: [[x, y], z] = [x, y] n = 1 Operation x y = xy[x, y] k is associative for all integer k 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. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 8 / 12

21 Example: Let G be a 2-nilpotent group of finite exponent of derived group G, that is We have: [[x, y], z] = [x, y] n = 1 Operation x y = xy[x, y] k is associative for all integer k 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. By Hulanicki, Świerczkowski: binary operation x y is a group operation, gcd(exp G, 2k + 1) = 1. Moreover, for groups of finite exponent, the group G and G are isomorphic. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 8 / 12

22 Example: Let G be a 2-nilpotent group of finite exponent of derived group G, that is We have: [[x, y], z] = [x, y] n = 1 Operation x y = xy[x, y] k is associative for all integer k 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. By Hulanicki, Świerczkowski: binary operation x y is a group operation, gcd(exp G, 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 Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 8 / 12

23 The problem of isomorphisps of group G and G is solved for some relatively free groups. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 9 / 12

24 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 G 0 Var G, then groups G and G are isomorphic (and Var G 0 = Var G). Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 9 / 12

25 Definition Group variety R we would call close over group operations if G : G R = G R Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 10 / 12

26 Definition Group variety R we would call close over group operations if Theorem If the group G: G : G R = G R satisfying maximal condition for normal subgrpups and Var G is closed over group operations then groups G and G are isomorphic (and Var G 0 = Var G). Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 10 / 12

27 Examples of varietis closed over group operations: c-nilpotent groups: N c Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 11 / 12

28 Examples of varietis closed over group operations: c-nilpotent groups: N c c-solvable groups: S c Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 11 / 12

29 Examples of varietis closed over group operations: c-nilpotent groups: N c c-solvable groups: S c Bernside groups of exponent k: B k Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 11 / 12

30 Examples of varietis closed over group operations: c-nilpotent groups: N c c-solvable groups: S c Bernside groups of exponent k: B k 2-engel groups: E 2 (identity [x, y, y] = 1) Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 11 / 12

31 Examples of varietis closed over group operations: c-nilpotent groups: N c c-solvable groups: S c Bernside groups of exponent k: B k 2-engel groups: E 2 (identity [x, y, y] = 1) varity width idenity: [[x, y], z m ] = 1 Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 11 / 12

32 Examples of varietis closed over group operations: c-nilpotent groups: N c c-solvable groups: S c Bernside groups of exponent k: B k 2-engel groups: E 2 (identity [x, y, y] = 1) varity width idenity: [[x, y], z m ] = 1 2-nilpotent groups width indentity [x, y] m = 1 (exp G < ) (Hulanicki Swierczkowski) Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 11 / 12

33 Examples of varietis closed over group operations: c-nilpotent groups: N c c-solvable groups: S c Bernside groups of exponent k: B k 2-engel groups: E 2 (identity [x, y, y] = 1) varity width idenity: [[x, y], z m ] = 1 2-nilpotent groups width indentity [x, y] m = 1 (exp G < ) (Hulanicki Swierczkowski) So for 2-nilpotent of finite exponent of derived subgroup G, 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. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 11 / 12

34 Some more questions: to find group operation not of the power form with isomorphic groups G and G. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 12 / 12

35 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. Marek Żabka (Instytut Matematyki) Non-trivial group operations Bedlewo, june 2015r. 12 / 12