#7529 ⟨a, b | aaa=1, abbbba=b⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 1083 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.

19 element center:
- 1, b⁵, b¹⁰, b¹⁵, b²⁰, b²⁵, b³⁰, b³⁵, b⁴⁰, b⁴⁵, b⁵⁰, b⁵⁵, b⁶⁰, b⁶⁵, b⁷⁰, b⁷⁵, b⁸⁰, b⁸⁵, b⁹⁰
1 non-trivial idempotent:
- b⁹⁰
Order of generators:
- a: order 3
- b: index 1, period 90

Histogram:

order 3	2 elements	a, a²
index 1, period 1	1 element	b⁹⁰
index 1, period 2	1 element	b⁴⁵
index 1, period 3	62 elements	ab³, bab², b²ab, b³a, abab⁸, ...
index 1, period 4	30 elements	abab²ab⁶, abab²ab¹⁵, abab²⁰, ab²⁴, bab²³, ...
index 1, period 5	24 elements	abab¹¹, b¹⁸, ab²ab¹⁹, bab²ab¹⁸, (b²a)²b¹⁷, ...
index 1, period 6	62 elements	abab², ab²ab, bab²a, ab²ab¹⁰, bab²ab⁹, ...
index 1, period 9	126 elements	ab²ab⁵, bab²ab⁴, (b²a)²b³, b³(ab²)², abab⁷, ...
index 1, period 10	24 elements	ab⁶, bab⁵, b²ab⁴, b³ab³, b⁴ab², ...
index 1, period 12	60 elements	abab²a, abab⁵, abab²ab³, ab⁹, bab⁸, ...
index 1, period 15	48 elements	b⁶, b¹², ab¹², bab¹¹, b²ab¹⁰, ...
index 1, period 18	126 elements	ab²a, b⁵, abab³, ab⁵, bab⁴, ...
index 1, period 30	48 elements	b³, ab²ab⁴, bab²ab³, b²(ab²)², b(b²a)²b, ...
index 1, period 36	180 elements	aba, ab⁴, bab³, b²ab², b³ab, ...
index 1, period 45	144 elements	ab, ba, b², ab², bab, ...
index 1, period 90	144 elements	b, b⁷, ab⁷, bab⁶, b²ab⁵, ...

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

b⁹¹ ⇒ b
bab⁹⁰ ⇒ ba
b⁵a ⇒ ab⁵
a²b ⇒ b⁴a
ab³a ⇒ bab⁸⁶
ab⁴a ⇒ b
ba² ⇒ ab⁴
(ba)² ⇒ ab²ab⁹
b⁴ab²a ⇒ abab⁸⁶
a³ ⇒ 1
a(b²a)² ⇒ bab²ab⁷⁶

# ab:aaa=1,abbbba=b b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
babbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=ba
bbbbba=abbbbb
aab=bbbba
abbba=babbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
abbbba=b
baa=abbbb
baba=abbabbbbbbbbb
bbbbabba=ababbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb
aaa=1
abbabba=babbabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

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.

4 total

Σ	#	Presentation	Mapping
10	8092	⟨a, b \| aaa=1, bbbb=aba⟩	φ(a) = aa, φ(b) = b
11	22248	⟨a, b \| aaa=1, aabbbb=ba⟩	φ(a) = aa, φ(b) = b
11	22820	⟨a, b \| aaa=1, abbbb=baa⟩	φ(a) = a, φ(b) = b
11	22854	⟨a, b \| aaa=1, bbabb=aba⟩	φ(a) = aa, φ(b) = bb

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Next:	#7531 ⟨a, b \| aaa=1, abbbbb=b⟩