#4917 ⟨a, b | aab=ba, bbbbb=1⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 160 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.

2 element center:
- 1, a³¹
1 non-trivial idempotent:
- a³¹
Order of generators:
- a: index 1, period 31
- b: order 5
Histogram:

order 5 4 elements b, b², b³, b⁴

index 1, period 1 1 element a³¹

index 1, period 5 124 elements ab, a²b, ab², a³b, a²b², ...

index 1, period 31 30 elements a, a², a³, a⁴, a⁵, ...

order 5	4 elements	b, b², b³, b⁴
index 1, period 1	1 element	a³¹
index 1, period 5	124 elements	ab, a²b, ab², a³b, a²b², ...
index 1, period 31	30 elements	a, a², a³, a⁴, a⁵, ...

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:
- Right-to-left recursive path with deg(a) = 0; deg(b) = 1

a³² ⇒ a
ba ⇒ a²b
b⁵ ⇒ 1

# ab:aab=ba,bbbbb=1 reversed:a/b
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa=a
ba=aab
bbbbb=1

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.

1 total

Σ	#	Presentation	Mapping
11	12060	⟨a, b \| abab=ba, aaaaa=1⟩	φ(a) = bbbb, φ(b) = ab

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