#10273 ⟨a, b | aa=1, ababa=bbb⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 64 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, b⁴, b⁸, (ab²)⁴
1 non-trivial idempotent:
- b⁸
Order of generators:
- a: order 2
- b: index 5, period 8

Histogram:

order 2	1 element	a
index 1, period 1	1 element	b⁸
index 1, period 2	13 elements	abab², ab²ab, bab²a, bab³, b²ab², ...
index 1, period 3	8 elements	bab², b²ab, abab²a, abab³, ab⁵, ...
index 1, period 4	6 elements	(ab²)², (bab)², (b²a)², b⁶, b²(ab²)², ...
index 1, period 6	8 elements	ab(ab²)², abab⁵, a(b²a)²b, ab⁷, ba(b²a)², ...
index 1, period 8	12 elements	b⁵, a(bab)², abab⁴, ab⁶, bab⁵, ...
index 2, period 2	2 elements	b⁴, ab⁴
index 2, period 3	2 elements	(ab)², ab³
index 2, period 8	4 elements	ab², bab, b²a, b³
index 3, period 4	2 elements	b², ab²a
index 4, period 6	2 elements	ab, ba
index 5, period 8	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 shortlex with a < b

a² ⇒ 1
(ba)² ⇒ ab³
b³a ⇒ (ab)²
b²ab³ ⇒ abab²a
bab⁷ ⇒ aba(b²a)²
b¹⁰ ⇒ a(b²a)³
ba(b²a)³ ⇒ a(b²a)³b

# ab:aa=1,ababa=bbb ab
aa=1
baba=abbb
bbba=abab
bbabbb=ababba
babbbbbbb=ababbabba
bbbbbbbbbb=abbabbabba
babbabbabba=abbabbabbab

Right Cayley graph

Left Cayley graph

Others with same cardinality

2 unique, 10 total

Σ	#	Presentation	Description	Related
9	1523	⟨a, b \| aab=ba, bbbb=1⟩	Finite non-commutative monoid with 64 elements	6 iso
10	5442	⟨a, b \| aaaa=1, ababbb=1⟩	Finite non-Abelian group with 64 elements	2 iso

Other isomorphic instances

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

5 total

Σ	#	Presentation	Mapping
10	10294	⟨a, b \| aa=1, abbba=bab⟩	φ(a) = a, φ(b) = b
10	10511	⟨a, b \| aa=1, baba=abbb⟩	φ(a) = a, φ(b) = b
11	27113	⟨a, b \| aa=1, abababa=bb⟩	φ(a) = a, φ(b) = ab
11	27640	⟨a, b \| aa=1, ababab=bba⟩	φ(a) = a, φ(b) = ab
11	28255	⟨a, b \| aa=1, babab=abba⟩	φ(a) = a, φ(b) = ab

Up:	Monoids with two generators and two relations
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Next:	#10279 ⟨a, b \| aa=1, ababb=bab⟩