#13293 ⟨a, b | aaaaa=1, ababbb=1⟩

Contents

Properties
Finite structure
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Isomorphic instances

Properties

Sum of relation sides is 11
Finite non-Abelian group with 100 elements
Inverses of generators:
- a^-1 = bab³
- b^-1 = b¹⁹

Finite structure

Elements in the center commute with all other elements.
The order of x is the least n such that xⁿ = 1.

10 element center:
- 1, b², b⁴, b⁶, b⁸, b¹⁰, b¹², b¹⁴, b¹⁶, b¹⁸
Order of generators:
- a: order 5
- b: order 20

Histogram:

order 2	1 element	b¹⁰
order 4	10 elements	b⁵, ab⁷, bab⁶, a²b⁹, ba²b⁸, ...
order 5	24 elements	a, a², b⁴, ab⁴, bab³, ...
order 10	24 elements	b², ab², bab, a²b², ba²b, ...
order 20	40 elements	b, ab, ba, a²b, ba², ...

Right Cayley graph

Left Cayley graph

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b²⁰ ⇒ 1	[13]
2.	b²a ⇒ ab²	[8]
3.	aba ⇒ b¹⁷	[12]
4.	a³ ⇒ ba²b⁷	[10]

# ab:aaaaa=1,ababbb=1 b/a - -
bbbbbbbbbbbbbbbbbbbb=1
bba=abb
aba=bbbbbbbbbbbbbbbbb
aaa=baabbbbbbb

Same cardinality

1 unique, 10 total

Σ	#	Presentation	Properties	φ
9	1434	⟨a, b \| baab=a, bbbb=1⟩	Noc 100	9

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	13296	⟨a, b \| aaaaa=1, abbbab=1⟩	φ(a) = a, φ(b) = b
11	13303	⟨a, b \| aaaaa=1, bababb=1⟩	φ(a) = a, φ(b) = b