#2962 ⟨a, b | aaabbaababa=1⟩

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Properties
Rewriting system
Isomorphic instances

Properties

Presentation has sum-of-sides 11
Infinite non-Abelian group
Inverses of generators:
- a^-1 = a³d
- b^-1 = a(ab)²cb
- 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. bbaabab=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.	ba(ab)²c ⇒ a(ab)²cb	[17]
7.	babcbd ⇒ a²dba(ab)²	[38]
8.	bcb²a ⇒ a³b²a²b	[44]
9.	b²a²ba ⇒ a²dbabcb	[28]
10.	ba²(ba)²c ⇒ a(ab)²cba	[19]
11.	babcbad ⇒ a²dba²(ba)²	[39]
12.	babcba² ⇒ a³dbcb²c	[40]
13.	ba(ab)²a² ⇒ abcb²	[23]
14.	babcb² ⇒ a²d	[22]
15.	bcb³abc ⇒ a³db(a²ba)²bcb	[57]
16.	(bcb)²abc ⇒ (ab)²a²cbabcb	[36]
17.	babcb(ab)²c ⇒ a²b²cbabcb	[52]
18.	b²a²b²cbd ⇒ adbcba²dba(ab)²	[46]
19.	bcb³abac ⇒ a³db(a²ba)²bcba	[58]
20.	(bcb)²abac ⇒ (ab)²a²cbabcba	[41]
21.	babc(ba)³c ⇒ a²b(bcba)²	[54]
22.	b²a²b²cbad ⇒ adbcba²dba²(ba)²	[47]
23.	b²a²b²cba² ⇒ adbcba³dbcb²c	[48]
24.	bcb³aba² ⇒ a³db(a²b)²cb²	[55]
25.	(bcb)²aba² ⇒ a(ba)²cbcb²	[37]
26.	babc(ba)³a ⇒ a²b²a³bcb²	[49]
27.	b²a²b²cb² ⇒ adbcba²d	[45]
28.	b²a²b²cb(ab)²c ⇒ adbcba²b²cbabcb	[60]
29.	b²a²b²c(ba)³c ⇒ adbcba²b(bcba)²	[61]
30.	b²a²b²c(ba)³a ⇒ adbcba²b²a³bcb²	[63]

# ab:aaabbaababa=1 reversed:cda/b aaaa=c,bbaabab=d magic:0
dc=1
cd=1
ca=ac
da=ad
aaaa=c
baababc=aababcb
babcbd=aadbaabab
bcbba=aaabbaab
bbaaba=aadbabcb
baababac=aababcba
babcbad=aadbaababa
babcbaa=aaadbcbbc
baababaa=abcbb
babcbb=aad
bcbbbabc=aaadbaabaaababcb
bcbbcbabc=ababaacbabcb
babcbababc=aabbcbabcb
bbaabbcbd=adbcbaadbaabab
bcbbbabac=aaadbaabaaababcba
bcbbcbabac=ababaacbabcba
babcbababac=aabbcbabcba
bbaabbcbad=adbcbaadbaababa
bbaabbcbaa=adbcbaaadbcbbc
bcbbbabaa=aaadbaabaabcbb
bcbbcbabaa=ababacbcbb
babcbababaa=aabbaaabcbb
bbaabbcbb=adbcbaad
bbaabbcbababc=adbcbaabbcbabcb
bbaabbcbababac=adbcbaabbcbabcba
bbaabbcbababaa=adbcbaabbaaabcbb

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

1 total

Σ	#	Presentation	Mapping
11	3289	⟨a, b \| abbababbbba=1⟩	φ(a) = b, φ(b) = a