#5728 ⟨a, b | aaaa=1, babbb=a⟩

Contents

1. Properties
2. Finite structure
3. Right Cayley graph
4. Left Cayley graph
5. Rewriting system

Properties

• Sum of relation sides is 10
• Finite non-Abelian group with 320 elements
• Inverses of generators:
  • a^-1 = a³
  • b^-1 = b⁷⁹

Finite structure

• 4 element center:
  • 1, b²⁰, b⁴⁰, b⁶⁰
• Order of generators:
  • a: order 4
  • b: order 80
• Histogram:

  order 2	11 elements	a2, b8a2, b16a2, b24a2, b32a2, ...
  order 4	52 elements	a, a3, b4a, b4a2, b4a3, ...
  order 5	4 elements	b16, b32, b48, b64
  order 8	64 elements	b2a, b2a2, b2a3, b6a, b6a2, ...
  order 10	4 elements	b8, b24, b56, b72
  order 16	128 elements	ba, ba2, ba3, b3a, b3a2, ...
  order 20	8 elements	b4, b12, b28, b36, b44, ...
  order 40	16 elements	b2, b6, b14, b18, b22, ...
  order 80	32 elements	b, b3, b7, b9, b11, ...

Right Cayley graph

Left Cayley graph

Rewriting system

1

# ab:aaaa=1,babbb=a reversed:b/a - -
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=1
ab=bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbba
aaaa=1