#2987 ⟨a, b | aaabbbababa=1⟩

Quick links

Properties
Rewriting system

Properties

Presentation has sum-of-sides 11
Infinite non-Abelian group
Inverses of generators:
- a^-1 = a³d
- b^-1 = (ab)²cb²
- c^-1 = d
- d^-1 = c

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(c) = deg(d) = deg(a) = 0, c < d < a; deg(b) = 1
Auxiliary generators:
1. aaaa=c
2. bbbabab=d
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	dc ⇒ 1	[11]
2.	cd ⇒ 1	[8]
3.	ca ⇒ ac	[5]
4.	da ⇒ ad	[10]
5.	a⁴ ⇒ c	[2]
6.	b(ab)²c ⇒ (ab)²cb	[19]
7.	babcbd ⇒ a³db(ab)²	[28]
8.	b³a ⇒ adbcb²	[34]
9.	(ba)³c ⇒ (ab)²cba	[24]
10.	babcbad ⇒ a³d(ba)³	[38]
11.	(ba)³ac ⇒ (ab)²cba²	[29]
12.	babcba²d ⇒ a³d(ba)³a	[43]
13.	bcb²a³ ⇒ a³b³c	[37]
14.	(ba)³a²c ⇒ (ab)²cba³	[40]
15.	babcba³d ⇒ a³d(ba)³a²	[48]
16.	babcb²d ⇒ a³db²(ab)²	[42]
17.	babcb²ad ⇒ a³db(ba)³	[45]
18.	babcb²a²d ⇒ a³db(ba)³a	[49]
19.	b(ba)³a² ⇒ abcb³	[47]
20.	babcb³ ⇒ a³d	[23]
21.	bcb³cbd ⇒ a³b²a³db(ab)²	[50]
22.	bcb³cbad ⇒ a³b²a³d(ba)³	[51]
23.	bcb³cba²d ⇒ a³b²a³d(ba)³a	[53]
24.	bcb³cba³d ⇒ a³b²a³d(ba)³a²	[55]
25.	(bcb²)²d ⇒ a³b²a³db²(ab)²	[52]
26.	(bcb²)²ad ⇒ a³b²a³db(ba)³	[54]
27.	(bcb²)²a²d ⇒ a³b²a³db(ba)³a	[56]
28.	b(cb³)² ⇒ a³b²a³d	[46]

# ab:aaabbbababa=1 reversed:cda/b aaaa=c,bbbabab=d magic:0
dc=1
cd=1
ca=ac
da=ad
aaaa=c
bababc=ababcb
babcbd=aaadbabab
bbba=adbcbb
bababac=ababcba
babcbad=aaadbababa
bababaac=ababcbaa
babcbaad=aaadbababaa
bcbbaaa=aaabbbc
bababaaac=ababcbaaa
babcbaaad=aaadbababaaa
babcbbd=aaadbbabab
babcbbad=aaadbbababa
babcbbaad=aaadbbababaa
bbababaaa=abcbbb
babcbbb=aaad
bcbbbcbd=aaabbaaadbabab
bcbbbcbad=aaabbaaadbababa
bcbbbcbaad=aaabbaaadbababaa
bcbbbcbaaad=aaabbaaadbababaaa
bcbbbcbbd=aaabbaaadbbabab
bcbbbcbbad=aaabbaaadbbababa
bcbbbcbbaad=aaabbaaadbbababaa
bcbbbcbbb=aaabbaaad