#17139 ⟨a, b | aaaa=1, abbbbb=b⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 68 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
4 non-trivial idempotents:
- b⁴a, b⁸a², b¹²a³, b¹⁶
Order of generators:
- a: order 4
- b: index 1, period 16

Histogram:

order 2	1 element	a²
order 4	2 elements	a, a³
index 1, period 1	4 elements	b⁴a, b⁸a², b¹²a³, b¹⁶
index 1, period 2	4 elements	b⁴a³, b⁸, b¹²a, b¹⁶a²
index 1, period 4	8 elements	b⁴, b⁴a², b⁸a, b⁸a³, b¹², ...
index 1, period 8	16 elements	b², b²a, b²a², b²a³, b⁶, ...
index 1, period 16	32 elements	b, ba, ba², b³, ba³, ...

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

b¹⁷ ⇒ b
ab ⇒ b¹³
a⁴ ⇒ 1

# ab:aaaa=1,abbbbb=b b/a
bbbbbbbbbbbbbbbbb=b
ab=bbbbbbbbbbbbb
aaaa=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	17686	⟨a, b \| aaaa=1, bbbbb=ab⟩	φ(a) = aaa, φ(b) = b

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