#7802 ⟨a, b | aaa=1, abbbb=ba⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 192 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.

4 element center:
- 1, b²¹, b⁴², b⁶³
1 non-trivial idempotent:
- b⁶³
Order of generators:
- a: order 3
- b: index 1, period 63

Histogram:

order 3	2 elements	a, a²
index 1, period 1	1 element	b⁶³
index 1, period 3	44 elements	ab³, a²b³, ab⁶, a²b⁶, ab⁹, ...
index 1, period 7	6 elements	b⁹, b¹⁸, b²⁷, b³⁶, b⁴⁵, ...
index 1, period 9	90 elements	ab, a²b, ab², a²b², ab⁴, ...
index 1, period 21	12 elements	b³, b⁶, b¹², b¹⁵, b²⁴, ...
index 1, period 63	36 elements	b, b², b⁴, b⁵, b⁸, ...

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

b⁶⁴ ⇒ b
ba ⇒ ab⁴
a³ ⇒ 1

# ab:aaa=1,abbbb=ba b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
ba=abbbb
aaa=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	23323	⟨a, b \| aaa=1, bbbb=aaba⟩	φ(a) = a, φ(b) = b

Other anti-isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	21705	⟨a, b \| aaa=1, aabbbba=b⟩	φ(a) = a, φ(b) = b

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