#2968 ⟨a, b | aaabbabaaba=1⟩

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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. abbab=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 ⇒ dbab	[7]
6.	ca³c ⇒ ab²a²d	[19]
7.	ca⁴c ⇒ a²dba⁴bad	[28]
8.	ca⁵c ⇒ ad(ba²)²	[41]
9.	ba⁴b² ⇒ aca	[22]
10.	b²ab ⇒ a²dc	[11]
11.	baba²b ⇒ a³cad	[12]
12.	ba²ba⁴b ⇒ a²ca²	[32]
13.	b²c ⇒ a²dcbab	[13]
14.	(ba)²c ⇒ a³cadbab	[16]
15.	ba⁴bac ⇒ aca⁵b²a²d	[26]
16.	b²a²c ⇒ a²dca⁴b²a²d	[21]
17.	ba⁴ba²c ⇒ ac(a³ba)²d	[27]
18.	ba⁴ba²dc ⇒ ac(ab)²	[23]
19.	b²a⁴c ⇒ a²dca(ba²)²	[42]
20.	baba⁴c ⇒ a³cba⁴bad	[29]
21.	baba⁵c ⇒ a³ca²d(ba²)²	[43]
22.	ba²ba⁵c ⇒ a²ca⁶b²a²d	[35]
23.	ba²ba⁶c ⇒ a²c(a⁴b)²ad	[36]
24.	ba²ba⁷c ⇒ a²ca³(ba²)²	[37]

# ab:aaabbabaaba=1 reversed:da/bc abbab=c,caab=d morph:5/0,4/0
da=ad
aaad=1
caab=d
baaac=ad
cac=dbab
caaac=abbaad
caaaac=aadbaaaabad
caaaaac=adbaabaa
baaaabb=aca
bbab=aadc
babaab=aaacad
baabaaaab=aacaa
bbc=aadcbab
babac=aaacadbab
baaaabac=acaaaaabbaad
bbaac=aadcaaaabbaad
baaaabaac=acaaabaaaabad
baaaabaadc=acabab
bbaaaac=aadcabaabaa
babaaaac=aaacbaaaabad
babaaaaac=aaacaadbaabaa
baabaaaaac=aacaaaaaabbaad
baabaaaaaac=aacaaaabaaaabad
baabaaaaaaac=aacaaabaabaa