#26637 ⟨a, b | aa=1, abbbbbba=b⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 72 elements

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
The order of x is the least n (if it exists) such that xⁿ = 1.
The index and period of x is the least m (index) and n (period) such that x^(m+n) = x^m.

6 element center:
- 1, b⁷, b¹⁴, b²¹, b²⁸, b³⁵
1 non-trivial idempotent:
- b³⁵
Order of generators:
- a: order 2
- b: index 1, period 35

Histogram:

order 2	1 element	a
index 1, period 1	1 element	b³⁵
index 1, period 2	7 elements	ab⁵, ab¹⁰, ab¹⁵, ab²⁰, ab²⁵, ...
index 1, period 5	4 elements	b⁷, b¹⁴, b²¹, b²⁸
index 1, period 7	6 elements	b⁵, b¹⁰, b¹⁵, b²⁰, b²⁵, ...
index 1, period 10	28 elements	ab, ab², ab³, ab⁴, ab⁶, ...
index 1, period 35	24 elements	b, b², b³, b⁴, b⁶, ...

Complete rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1

b³⁶ ⇒ b
ba ⇒ ab⁶
a² ⇒ 1

# ab:aa=1,abbbbbba=b b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
ba=abbbbbb
aa=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

1 unique, 3 total

Σ	#	Presentation	Description	Related
11	15372	⟨a, b \| aba=bb, aaabbb=1⟩	Finite non-Abelian group with 72 elements	2 iso

Other isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	27166	⟨a, b \| aa=1, abbbbbb=ba⟩	φ(a) = a, φ(b) = b
11	27764	⟨a, b \| aa=1, bbbbbb=aba⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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Next:	#26639 ⟨a, b \| aa=1, abbbbbbb=b⟩