#2942 ⟨a, b | aaababbaaba=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 = a³ca²
- c^-1 = a²ba³
- 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. ababb=c
2. caab=d
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	da ⇒ ad	[6]
2.	a³d ⇒ 1	[10]
3.	ca²b ⇒ d	[3]
4.	ba³c ⇒ ad	[18]
5.	cac ⇒ adb²	[7]
6.	ca³c ⇒ a(ba)²d	[19]
7.	ca⁴c ⇒ d(ba²)²	[27]
8.	ca⁵c ⇒ a²dba⁴ba²d	[41]
9.	bab² ⇒ a²dc	[11]
10.	ba⁴bab ⇒ aca²	[24]
11.	b²a²b ⇒ a²cad	[12]
12.	ba²ba⁴b ⇒ a³ca	[33]
13.	ba⁴bc ⇒ aca³b²	[25]
14.	b²ac ⇒ a²ca²db²	[16]
15.	(ba)²c ⇒ a²dca⁴(ba)²d	[21]
16.	baba²c ⇒ a²dc(ba²)²	[37]
17.	ba⁴ba²c ⇒ aca⁶(ba)²d	[38]
18.	baba²dc ⇒ a²dcab²	[13]
19.	b²a⁴c ⇒ a²cad(ba²)²	[42]
20.	(ba⁴)²c ⇒ ac(a⁴b)²a²d	[39]
21.	b²a⁵c ⇒ a²cba⁴ba²d	[43]
22.	ba²ba⁵c ⇒ a³ca⁵(ba)²d	[45]
23.	ba²ba⁶c ⇒ a³ca(ba²)²	[44]
24.	ba²ba⁷c ⇒ a³c(a³ba)²ad	[46]

# ab:aaababbaaba=1 reversed:da/bc ababb=c,caab=d morph:5/0,4/0
da=ad
aaad=1
caab=d
baaac=ad
cac=adbb
caaac=ababad
caaaac=dbaabaa
caaaaac=aadbaaaabaad
babb=aadc
baaaabab=acaa
bbaab=aacad
baabaaaab=aaaca
baaaabc=acaaabb
bbac=aacaadbb
babac=aadcaaaababad
babaac=aadcbaabaa
baaaabaac=acaaaaaababad
babaadc=aadcabb
bbaaaac=aacadbaabaa
baaaabaaaac=acaaaabaaaabaad
bbaaaaac=aacbaaaabaad
baabaaaaac=aaacaaaaababad
baabaaaaaac=aaacabaabaa
baabaaaaaaac=aaacaaabaaaabaad