#2922 ⟨a, b | aaabaabbaba=1⟩

Quick links

Properties
Rewriting system

Properties

Presentation has sum-of-sides 11
Infinite non-Abelian group
Inverses of generators:
- a^-1 = a²d
- b^-1 = aca⁴
- c^-1 = a⁴ba
- d^-1 = a³

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(d) = deg(a) = 0, d < a; deg(b) = deg(c) = 1, b < c
Auxiliary generators:
1. abbab=c
2. abac=d
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	da ⇒ ad	[7]
2.	a³d ⇒ 1	[12]
3.	ca⁴b ⇒ a²d	[21]
4.	bac ⇒ a²d²	[6]
5.	cac ⇒ ab²d	[8]
6.	ca³c ⇒ a²dbab	[22]
7.	ca⁴c ⇒ a³(ba²)²d²	[28]
8.	ca⁵c ⇒ ba⁴bad	[30]
9.	ba²b² ⇒ dca³	[17]
10.	b²ab ⇒ a²dc	[13]
11.	ba⁴ba²b ⇒ ca⁴	[31]
12.	baba⁴b ⇒ aca²	[24]
13.	b²c ⇒ a²dcbab	[14]
14.	ba²bc ⇒ dca⁷(ba²)²d²	[42]
15.	ba²bdc ⇒ dca⁵b²d	[29]
16.	b²adc ⇒ a²dca²b²d	[15]
17.	b²a²c ⇒ a²dcaba⁴bad	[39]
18.	ba⁴ba²c ⇒ ca⁸(ba²)²d²	[38]
19.	(ba²)²dc ⇒ dca³bab	[20]
20.	ba⁴ba²dc ⇒ ca⁶b²d	[36]
21.	baba³c ⇒ aca(ab)²	[26]
22.	ba⁴ba³c ⇒ ca(a⁴b)²ad	[35]
23.	baba⁴c ⇒ aca⁶(ba²)²d²	[44]
24.	baba⁵c ⇒ ac(a³ba)²d	[40]

# ab:aaabaabbaba=1 reversed:da/bc abbab=c,abac=d morph:5/0,4/0
da=ad
aaad=1
caaaab=aad
bac=aadd
cac=abbd
caaac=aadbab
caaaac=aaabaabaadd
caaaaac=baaaabad
baabb=dcaaa
bbab=aadc
baaaabaab=caaaa
babaaaab=acaa
bbc=aadcbab
baabc=dcaaaaaaabaabaadd
baabdc=dcaaaaabbd
bbadc=aadcaabbd
bbaac=aadcabaaaabad
baaaabaac=caaaaaaaabaabaadd
baabaadc=dcaaabab
baaaabaadc=caaaaaabbd
babaaac=acaabab
baaaabaaac=caaaaabaaaabad
babaaaac=acaaaaaabaabaadd
babaaaaac=acaaabaaaabad