Commutator Theory for Loops

Commutator Theory for Loops David Stanovský Charles University, Prague, Czech Republic stanovsk@karlin.mff.cuni.cz joint work with Petr Vojtěchovský, University of Denver June 2013 David Stanovský (Prague) Commutators for loops 1 / 11

Feit-Thompson theorem Theorem (Feit-Thompson, 1962) Groups of odd order are solvable. Can be extended? To which class of algebras? (containing groups) What is odd order? What is solvable? David Stanovský (Prague) Commutators for loops 2 / 11

Feit-Thompson theorem Theorem (Feit-Thompson, 1962) Groups of odd order are solvable. Can be extended? To which class of algebras? (containing groups) What is odd order? What is solvable? Theorem (Glauberman 1964/68) Moufang loops of odd order are solvable. Moufang loop = replace associativity by x(z(yz)) = ((xz)y)z solvable = there are N i L such that 1 = N 0 N 1... N k = L and N i+1 /N i are abelian groups. David Stanovský (Prague) Commutators for loops 2 / 11

Loops A loop is an algebra (L,, 1) such that 1x = x1 = x for every x, y there are unique u, v such that xu = y, vx = y For universal algebra purposes: (L,, \, /, 1), where u = x\y, v = y/x. Examples: octonions Moufang loops various other classes of weakly associative loops various combinatorial constructions, e.g., from Steiner triples systems, coordinatization of projective geometries, etc. David Stanovský (Prague) Commutators for loops 3 / 11

Loops A loop is an algebra (L,, 1) such that 1x = x1 = x for every x, y there are unique u, v such that xu = y, vx = y For universal algebra purposes: (L,, \, /, 1), where u = x\y, v = y/x. Examples: octonions Moufang loops various other classes of weakly associative loops various combinatorial constructions, e.g., from Steiner triples systems, coordinatization of projective geometries, etc. Normal subloops congruences = kernels of a homomorphisms = subloops invariant with respect to Inn(L) Inn(L) = Mlt(L) 1, Mlt(L) = L a, R a : a L David Stanovský (Prague) Commutators for loops 3 / 11

Solvability, nilpotence - after R. H. Bruck Bruck s approach (1950 s), by direct analogy to group theory: L is solvable if there are N i L such that 1 = N 0 N 1... N k = L and N i+1 /N i are abelian groups. L is nilpotent if there are N i L such that 1 = N 0 N 1... N k = L and N i+1 /N i Z(L/N i ). Z(L) = {a L : ax = xa, a(xy) = (ax)y, (xa)y = x(ay) x, y L} David Stanovský (Prague) Commutators for loops 4 / 11

Solvability, nilpotence - after R. H. Bruck Bruck s approach (1950 s), by direct analogy to group theory: L is solvable if there are N i L such that 1 = N 0 N 1... N k = L and N i+1 /N i are abelian groups. L is nilpotent if there are N i L such that 1 = N 0 N 1... N k = L and N i+1 /N i Z(L/N i ). Z(L) = {a L : ax = xa, a(xy) = (ax)y, (xa)y = x(ay) x, y L} Alternatively, define L (0) = L (0) = L, L (i+1) = [L (i), L (i) ], L (i+1) = [L (i), L] L solvable iff L (n) = 1 for some n L nilpotent iff L (n) = 1 for some n Need commutator! [N, N] is the smallest M such that N/M is abelian [N, L] is the smallest M such that N/M Z(L/M) [N 1, N 2 ] is??? David Stanovský (Prague) Commutators for loops 4 / 11

Solvability, nilpotence - universal algebra way Commutator theory approach (1970 s): L is solvable if there are α i Con(L) such that 0 L = α 0 α 1... α k = 1 L and α i+1 is abelian over α i. L is nilpotent if there are α i Con(L) such that 0 L = α 0 α 1... α k = 1 L and α i+1 /α i ζ(l/α i ). ζ(l) = the largest ζ such that C(ζ, 1 L ; 0 L ), i.e., [ζ, 1 L ] = 0 L. Alternatively, define α (0) = α (0) = 1 L, α (i+1) = [α (i), α (i) ], α (i+1) = [α (i), 1 L ] L solvable iff α (n) = 1 for some n L nilpotent iff α (n) = 1 for some n We have a commutator! [α, α] is the smallest β such that α/β is abelian [α, 1 L ] is the smallest β such that α/β ζ(l/β) [α, β] is the smallest δ such that C(α, β; δ) David Stanovský (Prague) Commutators for loops 5 / 11

Translating to loops I Good news 1 A loop is abelian if and only if it is an abelian group. 2 The congruence center corresponds to the Bruck s center. Hence, nilpotent loops are really nilpotent loops! David Stanovský (Prague) Commutators for loops 6 / 11

Translating to loops I Good news 1 A loop is abelian if and only if it is an abelian group. 2 The congruence center corresponds to the Bruck s center. Hence, nilpotent loops are really nilpotent loops! Nevertheless, supernilpotence is a stronger property. David Stanovský (Prague) Commutators for loops 6 / 11

Translating to loops II Bad news Abelian congruences normal subloops that are abelian groups N is an abelian group iff [N, N] N = 0, N is abelian in L iff [N, N] L = 0, i.e., [1 N, 1 N ] N = 0 N i.e., [ν, ν] L = 0 L abelian abelian in L!!! Example: L = Z 4 Z 2, redefine (a, 1) + (b, 1) = (a b, 0) 0 1 2 3 0 0 1 2 3 1 1 3 0 2 2 2 0 3 1 3 3 2 1 0 N = Z 4 {0} L N is an abelian group [N, N] L = N, hence N is not abelian in L David Stanovský (Prague) Commutators for loops 7 / 11

Two notions of solvability L is Bruck-solvable if there are N i L such that 1 = N 0 N 1... N k = L and N i+1 /N i are abelian groups (i.e. [N i+1, N i+1 ] Ni+1 N i ) L is congruence-solvable if there are N i L such that 1 = N 0 N 1... N k = L and N i+1 /N i are abelian in L/N i (i.e. [N i+1, N i+1 ] L N i ) The loop from the previous slide is Bruck-solvable NOT congruence-solvable David Stanovský (Prague) Commutators for loops 8 / 11

Commutator in loops L a,b = L 1 ab L al b, R a,b = R 1 ba R ar b, M a,b = M 1 b\a M am b, U a = Ra 1 M a T a = L a R 1 a TotMlt(L) = L a, R a, M a : a L TotInn(L) = TotMlt(L) 1 = L a,b, R a,b, T a, M a,b, U a : a, b L Main Theorem V a variety of loops, Φ a set of words that generates TotInn s in V, then [α, β] = Cg((ϕ u1,...,u n (a), ϕ v1,...,v n (a)) : ϕ Φ, 1 α a, u i β v i ) for every L V, α, β Con(L). Examples: in loops, Φ = {L a,b, R a,b, M a,b, T a, U a } in groups, Φ = {T a } David Stanovský (Prague) Commutators for loops 9 / 11

Commutator in loops L a,b = L 1 ab L al b, R a,b = R 1 ba R ar b, M a,b = M 1 b\a M am b, U a = Ra 1 M a T a = L a R 1 a TotMlt(L) = L a, R a, M a : a L TotInn(L) = TotMlt(L) 1 = L a,b, R a,b, T a, M a,b, U a : a, b L Main Theorem V a variety of loops, Φ a set of words that generates TotInn s in V, then [α, β] = Cg((ϕ u1,...,u n (a), ϕ v1,...,v n (a)) : ϕ Φ, 1 α a, u i β v i ) for every L V, α, β Con(L). Examples: in loops, Φ = {L a,b, R a,b, M a,b, T a, U a } in groups, Φ = {T a } [M, N] L = Ng(ϕ u1,...,u n (a)/ϕ v1,...,v n (a) : ϕ Φ, a M, u i /v i N) David Stanovský (Prague) Commutators for loops 9 / 11

Solvability and nilpotence summarized Mlt(L) nilpotent (trivial) Inn(L) nilpotent (Niemenmaa) L nilpotent (Bruck) Mlt(L) solvable (Vesanen) L Bruck-solvable Where to put L congruence-solvable? David Stanovský (Prague) Commutators for loops 10 / 11

Solvability and nilpotence summarized Mlt(L) nilpotent (trivial) Inn(L) nilpotent (Niemenmaa) L nilpotent (Bruck) Mlt(L) solvable (Vesanen) L Bruck-solvable Where to put L congruence-solvable? We know: stronger Vesanen fails. Problem Let L be a congruence-solvable loop. Is the group Mlt(L) solvable? David Stanovský (Prague) Commutators for loops 10 / 11

Feit-Thompson revisited Theorem (Glauberman 1964/68) Moufang loops of odd order are Bruck-solvable. Problem Are Moufang loops of odd order congruence-solvable? For Moufang loops, we know that abelian abelian in L (in a 16-element loop) is it so that Bruck-solvable iff congruence-solvable? David Stanovský (Prague) Commutators for loops 11 / 11