#17503 ⟨a, b | abab=1, aabbaa=b⟩

Contents

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

Properties

Sum of relation sides is 11
Finite non-Abelian group with 48 elements
Inverses of generators:
- a^-1 = a¹¹
- b^-1 = aba

Finite structure

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

4 element center:
- 1, a³, a⁶, a⁹
Order of generators:
- a: order 12
- b: order 6

Histogram:

order 2	7 elements	ab, ba, a⁶, a²ba⁵, aba⁶, ...
order 3	8 elements	a⁴, a²ba⁴, aba⁵, ba⁶, a⁸, ...
order 4	8 elements	a³, a²ba², aba³, ba⁴, a⁹, ...
order 6	8 elements	b, a², a²b, aba, ba², ...
order 12	16 elements	a, a²ba, aba², ba³, a⁵, ...

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(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a¹² ⇒ 1	[20]
2.	a³b ⇒ ba³	[12]
3.	b² ⇒ aba⁷	[19]
4.	bab ⇒ a¹¹	[18]
5.	ba²b ⇒ a²ba²	[13]

# ab:abab=1,aabbaa=b a/b - -
aaaaaaaaaaaa=1
aaab=baaa
bb=abaaaaaaa
bab=aaaaaaaaaaa
baab=aabaa

Same cardinality

4 unique, 18 total

Σ	#	Presentation	Properties	φ
10	3771	⟨a, b \| aaaa=bb, abab=1⟩	Grp 48	5
11	14993	⟨a, b \| aaa=bb, ababbb=1⟩	Grp 48	2
11	16966	⟨a, b \| abab=1, aaabbba=1⟩	Grp 48	6
11	21751	⟨a, b \| aaa=1, abbbbbb=b⟩	Noc 48	1