#21751 ⟨a, b | aaa=1, abbbbbb=b⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 48 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.

1 element center:
- 1
3 non-trivial idempotents:
- b⁵a, b¹⁰a², b¹⁵
Order of generators:
- a: order 3
- b: index 1, period 15

Histogram:

order 3	2 elements	a, a²
index 1, period 1	3 elements	b⁵a, b¹⁰a², b¹⁵
index 1, period 3	6 elements	b⁵, b⁵a², b¹⁰, b¹⁰a, b¹⁵a, ...
index 1, period 5	12 elements	ba², b²a, b³, b⁴a², b⁶, ...
index 1, period 15	24 elements	b, ba, b², b²a², b³a, ...

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

b¹⁶ ⇒ b
ab ⇒ b¹¹
a³ ⇒ 1

# ab:aaa=1,abbbbbb=b b/a
bbbbbbbbbbbbbbbb=b
ab=bbbbbbbbbbb
aaa=1

Right Cayley graph

Left Cayley graph

Others with same cardinality

4 unique, 17 total

Σ	#	Presentation	Description	Related
10	3771	⟨a, b \| aaaa=bb, abab=1⟩	Finite non-Abelian group with 48 elements	5 iso
11	14993	⟨a, b \| aaa=bb, ababbb=1⟩	Finite non-Abelian group with 48 elements	2 iso
11	16966	⟨a, b \| abab=1, aaabbba=1⟩	Finite non-Abelian group with 48 elements	6 iso
11	17503	⟨a, b \| abab=1, aabbaa=b⟩	Finite non-Abelian group with 48 elements

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	22326	⟨a, b \| aaa=1, bbbbbb=ab⟩	φ(a) = aa, φ(b) = b

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