#5442 ⟨a, b | aaaa=1, ababbb=1⟩

Contents

Properties
Finite structure
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Isomorphic instances

Properties

Sum of relation sides is 10
Finite non-Abelian group with 64 elements
Inverses of generators:
- a^-1 = bab³
- b^-1 = b¹⁵

Finite structure

Elements in the center commute with all other elements.
The order of x is the least n such that xⁿ = 1.

8 element center:
- 1, b², b⁴, b⁶, b⁸, b¹⁰, b¹², b¹⁴
Order of generators:
- a: order 4
- b: order 16
Histogram:

order 2 3 elements a², b⁸, a²b⁸

order 4 12 elements a, b⁴, ab⁴, bab³, a²b⁴, ...

order 8 16 elements b², ab², bab, a²b², b⁶, ...

order 16 32 elements b, ab, ba, a²b, b³, ...

Right Cayley graph

Left Cayley graph

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b¹⁶ ⇒ 1	[16]
2.	b²a ⇒ ab²	[8]
3.	aba ⇒ b¹³	[15]
4.	ba² ⇒ a²b⁹	[13]
5.	a³ ⇒ bab³	[3]

# ab:aaaa=1,ababbb=1 b/a - -
bbbbbbbbbbbbbbbb=1
bba=abb
aba=bbbbbbbbbbbbb
baa=aabbbbbbbbb
aaa=babbb

Same cardinality

2 unique, 13 total

Σ	#	Presentation	Properties	φ
9	1523	⟨a, b \| aab=ba, bbbb=1⟩	Noc 64	6
10	10273	⟨a, b \| aa=1, ababa=bbb⟩	Noc 64	5

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

2 total

Σ	#	Presentation	Mapping
10	5445	⟨a, b \| aaaa=1, abbbab=1⟩	φ(a) = a, φ(b) = b
10	5452	⟨a, b \| aaaa=1, bababb=1⟩	φ(a) = a, φ(b) = b