Groups


Definition

x(yz) = (xy)z, xe = x, xx' = e (Common definition)
x(yz) = (xy)z, ex = x, x'x = e (Another common definition)
x(yz) = (xy)z + for all x there exists a unique x' such that xx'x = x
(w((x'w)'z))((yz)'y) = x [McCune1993] shows by exhaustive computer search that this is the shortest single identity in these operations capable of defining groups.

Examples

Structure

Representation

Decision problems

Identity problem: Solvable
Word problem: Unsolvable [Novikov][Boone] Here is a group presentation with 10 generators and 27 relations which has unsolvable word problem. It is given in D.J. Collins, "A simple presentation of a group with unsolvable word problem," Illinois J. Math 30 (1986), 230-234, based on work of G.S. Céjtin and V.V. Borisov:
   <a,b,c,d,e,p,q,r,t,k;
        p^{10}a = ap, p^{10}b =bp, p^{10}c = cp, p^{10}d = dp, p^{10}e = ep,
        aq^{10} = qa, bq^{10} =qb, cq^{10} = qc, dq^{10} = qd, eq^{10} = qe,
        ra=ar,  rb=br,  rc =cr,  rd=dr,  re=er,
        pacqr = rpcaq,            p^2adq^2r = rp^2daq^2,
        p^3bcq^3r = rp^3cbq^3,    p^4bdq^4r = rp^4dbq^4,
        p^5ceq^5r = rp^5ecaq^5,   p^6deq^6r = rp^6edbq^6,
        p^7cdcq^7r = rp^7cdceq^7, p^8ca^3q^8r = rp^8a^3q^8,
        p^9da^3q^9r = rp^9a^3q^9,
        pt = tp, qt = tq, 
        a^{-3}ta^3k = ka^{-3}ta^3> (This last one is misprinted in Collins)
    

Spectra and growth

Finite spectrum: (numbers of groups of orders <= 100, from J.A. Gallian, "Contemporary Abstract Algebra", p. 290)
         1,    1,   1,   2,   1,   2,   1,   5,   2,   2,
         1,    5,   1,   2,   1,  14,   1,   5,   1,   2,
         2,    2,   1,  15,   2,   2,   5,   4,   1,   4,
         1,   51,   1,   2,   1,  14,   1,   2,   2,  14,
         1,    6,   1,   4,   2,   2,   1,  52,   2,   5,
         1,    5,   1,  15,   2,  13,   2,   2,   1,  13,
         1,    2,   4, 267,   1,   4,   1,   5,   1,   4,
         1,   50,   1,   2,   3,   4,   1,   6    1,  52,
        15,    2,   1,  15,   1,   2,   1,  12,   1,  10,
         1,    4,   2,   2,   1, 230,   1,   5,   2,  16,...
         
Free spectrum:
Growth series: (1+z)/(1-(2r-1)z) for the free group on r generators.

History/Importance

They're very important and have a long history.

References

Too many to give here.

Subsystems

Abelian groups, solvable groups, nilpotent groups