# 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

• Groups of small order
• Any abelian group
• Symmetric group S_n of all permutations of n objects, under composition of mappings. They have order n! and presentations
```s_1,..,s_{n-1}; s_i^2 = 1, s_i s_j = s_j s_i if i < j-1
s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}

```
• The dihedral groups D_n, of order 2n, with presentations
• The braid groups B_n, of infinite order, with presentations
```s_1,..,s_{n-1}; s_i s_j = s_j s_i if i < j-1
s_i s_{i+1} s_i = s_{i+1} s_i s_{i+1}

```
• GL(n,F) = invertible n x n matrices over any field F
• SL(n,F) = n x n matrices of determinant 1 over any field F
• SO_n = rotations of Euclidean n-space
• Automatic groups

## 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

A Catalogue of Algebraic Systems / John Pedersen / jfp@math.usf.edu