#14833 ⟨a, b | abba=b, aaaaa=a⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 101 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.

6 element center:
- 1, b³, a⁴, b⁶, b⁹, b¹²
2 non-trivial idempotents:
- a⁴, b¹²
Order of generators:
- a: index 1, period 4
- b: index 1, period 12

Histogram:

index 1, period 1	2 elements	a⁴, b¹²
index 1, period 2	8 elements	a², (ab)², (ba)², b⁶, a²b⁶, ...
index 1, period 3	8 elements	b⁴, a²b⁵, b⁸, a²b⁷, abab⁶, ...
index 1, period 4	34 elements	a, ab, ba, a³, b³, ...
index 1, period 6	8 elements	b², a²b, aba, abab², b(ab)², ...
index 1, period 8	24 elements	ab², bab, b²a, a²ba, a(ab)², ...
index 1, period 12	16 elements	b, a²b², b⁵, a²b⁴, abab³, ...

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

b¹³ ⇒ b
bab¹² ⇒ ba
b³a ⇒ ab³
ab²a ⇒ b
ba² ⇒ abab²
b(ba)² ⇒ a²b
a³b ⇒ b²a
a(ba)² ⇒ bab¹⁰
a⁵ ⇒ a

# ab:abba=b,aaaaa=a b/a
bbbbbbbbbbbbb=b
babbbbbbbbbbbb=ba
bbba=abbb
abba=b
baa=ababb
bbaba=aab
aaab=bba
ababa=babbbbbbbbbb
aaaaa=a

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#14833 ⟨a, b | abba=b, aaaaa=a⟩

Properties

Elements

Complete rewriting system

Right Cayley graph

Left Cayley graph