#16551 ⟨a, b | aba=bb, bbb=aaa⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 51 elements

Elements

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

5 element center:
- 1, a³, a⁶, a⁹, a¹²
1 non-trivial idempotent:
- a¹²
Order of generators:
- a: index 7, period 6
- b: index 7, period 6

Histogram:

index 1, period 1	1 element	a¹²
index 1, period 2	1 element	a⁹
index 1, period 3	8 elements	a⁸, a²ba⁶, aba⁷, ba⁸, a¹⁰, ...
index 1, period 4	6 elements	a²ba⁵, aba⁶, ba⁷, a²ba⁸, aba⁹, ...
index 1, period 6	8 elements	a²ba³, ba⁵, a⁷, a²ba⁴, aba⁵, ...
index 2, period 1	1 element	a⁶
index 2, period 3	9 elements	a⁴, a²ba, aba², ba³, aba²b, ...
index 2, period 4	4 elements	ba²b, a²ba², aba³, ba⁴
index 2, period 6	2 elements	a⁵, aba⁴
index 3, period 2	1 element	a³
index 3, period 3	1 element	bab
index 4, period 3	4 elements	a², a²b, aba, ba²
index 5, period 4	2 elements	ab, ba
index 7, period 6	2 elements	a, b

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(a) = 0; deg(b) = 1

a¹³ ⇒ a⁷
ba¹¹ ⇒ ba⁵
a²ba⁹ ⇒ a²ba³
a³b ⇒ ba³
b² ⇒ aba
(ba)² ⇒ a³
ba²ba³ ⇒ a²ba⁸
(ab)² ⇒ a³
(aba)² ⇒ ba³

# ab:aba=bb,bbb=aaa a/b
aaaaaaaaaaaaa=aaaaaaa
baaaaaaaaaaa=baaaaa
aabaaaaaaaaa=aabaaa
aaab=baaa
bb=aba
baba=aaa
baabaaa=aabaaaaaaaa
abab=aaa
abaaba=baaa

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#16551 ⟨a, b | aba=bb, bbb=aaa⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph