#27693 ⟨a, b | aa=1, abbbbb=bab⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 82 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.

1 element center:
- 1
2 non-trivial idempotents:
- b¹⁵, ab¹⁵a
Order of generators:
- a: order 2
- b: index 6, period 15

Histogram:

order 2	1 element	a
index 1, period 1	2 elements	b¹⁵, ab¹⁵a
index 1, period 2	10 elements	ab⁶, b⁶a, ab⁹, b⁹a, ab¹², ...
index 1, period 3	4 elements	b¹⁰, ab¹⁰a, b²⁰, ab²⁰a
index 1, period 5	8 elements	b⁶, ab⁶a, b⁹, ab⁹a, b¹², ...
index 1, period 6	20 elements	ab⁷, b⁷a, ab⁸, b⁸a, ab¹⁰, ...
index 1, period 15	16 elements	b⁷, b⁸, ab⁷a, ab⁸a, b¹¹, ...
index 2, period 2	2 elements	ab³, b³a
index 2, period 3	2 elements	b⁵, ab⁵a
index 2, period 5	2 elements	b³, ab³a
index 2, period 6	6 elements	ab², b²a, ab⁴, b⁴a, ab⁵, ...
index 2, period 15	2 elements	b⁴, ab⁴a
index 3, period 6	2 elements	ab, ba
index 3, period 15	2 elements	b², ab²a
index 6, period 15	2 elements	b, aba

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

b²¹ ⇒ b⁶
bab ⇒ ab⁵
a² ⇒ 1

# ab:aa=1,abbbbb=bab b/a
bbbbbbbbbbbbbbbbbbbbb=bbbbbb
bab=abbbbb
aa=1

Right Cayley graph

Left Cayley graph

Other isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	28290	⟨a, b \| aa=1, bbbbb=abab⟩	φ(a) = a, φ(b) = b

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