#15372 ⟨a, b | aba=bb, aaabbb=1⟩

Contents

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

Properties

Sum of relation sides is 11
Finite non-Abelian group with 72 elements
Inverses of generators:
- a^-1 = bab¹⁶
- b^-1 = b¹⁷

Finite structure

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

6 element center:
- 1, b³, b⁶, b⁹, b¹², b¹⁵
Order of generators:
- a: order 18
- b: order 18

Histogram:

order 2	1 element	b⁹
order 3	2 elements	b⁶, b¹²
order 4	6 elements	ab⁴, bab³, b²ab², ab¹³, bab¹², ...
order 6	2 elements	b³, b¹⁵
order 9	24 elements	b², ab², bab, b²a, ab³, ...
order 12	12 elements	ab, ba, ab⁷, bab⁶, b²ab⁵, ...
order 18	24 elements	a, b, b⁵, ab⁵, bab⁴, ...

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.	b¹⁸ ⇒ 1	[10]
2.	b³a ⇒ ab³	[3]
3.	a² ⇒ bab¹³	[11]
4.	aba ⇒ b²	[1]
5.	ab²a ⇒ b²ab⁵	[6]

# ab:aba=bb,aaabbb=1 b/a - -
bbbbbbbbbbbbbbbbbb=1
bbba=abbb
aa=babbbbbbbbbbbbb
aba=bb
abba=bbabbbbb

Same cardinality

1 unique, 3 total

Σ	#	Presentation	Properties	φ
11	26637	⟨a, b \| aa=1, abbbbbba=b⟩	Noc 72	2

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	15378	⟨a, b \| aba=bb, aabbba=1⟩	φ(a) = a, φ(b) = b
11	15392	⟨a, b \| aba=bb, baaabb=1⟩	φ(a) = a, φ(b) = b