#3126 ⟨a, b | aabbabbaaba=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 = a(ac)²a²b
- b^-1 = db
- 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:
- Left-to-right recursive path with deg(c) = deg(d) = deg(b) = 0, c < d < b; deg(a) = 1
Auxiliary generators:
1. bb=c
2. abbaabaaa=d
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	cd ⇒ 1	[11]
2.	dc ⇒ 1	[18]
3.	bc ⇒ cb	[5]
4.	bd ⇒ db	[17]
5.	b² ⇒ c	[2]
6.	ca²ba² ⇒ aba³c	[41]
7.	caca²ba ⇒ (ac)²a²b	[28]
8.	d(ac)²a² ⇒ aca²badb	[33]
9.	daba³ ⇒ a²ba²d	[40]
10.	ba³ca ⇒ (ac)²a²db	[24]
11.	baca²ba ⇒ a³cac	[26]
12.	b(ac)²a² ⇒ ca³cab	[21]
13.	c(ba²)² ⇒ baba³c	[42]
14.	dbaba³ ⇒ (ba²)²d	[36]
15.	da(ac)²a² ⇒ a²ba³b	[44]
16.	ba(ac)²a² ⇒ ca³(ca)²db	[48]
17.	ba²ba³ ⇒ a³(ca)²d	[38]
18.	a³caca² ⇒ db	[19]
19.	ca⁵(ca)² ⇒ a(ba³c)²	[49]
20.	c(a²ca)²ca ⇒ aba³c²aca²b	[46]
21.	caca⁵ca ⇒ (ac)²a²cba²bad	[53]
22.	cac(a²ca)² ⇒ (ac)²a²cbaca²db	[58]
23.	ba⁴ba³ ⇒ (ac)²a²dbaba²d	[37]
24.	baca⁵ca ⇒ a³cac²a²bad	[52]
25.	bac(a²ca)² ⇒ a³(cac)²a²db	[57]
26.	cba⁵(ca)² ⇒ ba(ba³c)²	[50]
27.	cb(a²ca)²ca ⇒ baba³c²aca²b	[47]
28.	ca⁴(aca)²ca² ⇒ a(ba³c)²ca²badb	[63]
29.	c(a²ca)²(ac)²a² ⇒ aba³c²aca²cba²badb	[68]
30.	caca⁶ba³ ⇒ (ac)²a²cba²badaba²d	[61]
31.	caca²ca⁴ba³ ⇒ (ac)²a²cbaca²dbaba²d	[66]
32.	baca⁶ba³ ⇒ a³cac²a²badaba²d	[60]
33.	baca²ca⁴ba³ ⇒ a³(cac)²a²dbaba²d	[65]
34.	cba⁴(aca)²ca² ⇒ ba(ba³c)²ca²badb	[64]
35.	cb(a²ca)²(ac)²a² ⇒ baba³c²aca²cba²badb	[69]

# ab:aabbabbaaba=1 cdb/a bb=c,abbaabaaa=d magic:1
cd=1
dc=1
bc=cb
bd=db
bb=c
caabaa=abaaac
cacaaba=acacaab
dacacaa=acaabadb
dabaaa=aabaad
baaaca=acacaadb
bacaaba=aaacac
bacacaa=caaacab
cbaabaa=babaaac
dbabaaa=baabaad
daacacaa=aabaaab
baacacaa=caaacacadb
baabaaa=aaacacad
aaacacaa=db
caaaaacaca=abaaacbaaac
caacaaacaca=abaaaccacaab
cacaaaaaca=acacaacbaabad
cacaacaaaca=acacaacbacaadb
baaaabaaa=acacaadbabaad
bacaaaaaca=aaacaccaabad
bacaacaaaca=aaacaccacaadb
cbaaaaacaca=babaaacbaaac
cbaacaaacaca=babaaaccacaab
caaaaacaacacaa=abaaacbaaaccaabadb
caacaaacaacacaa=abaaaccacaacbaabadb
cacaaaaaabaaa=acacaacbaabadabaad
cacaacaaaabaaa=acacaacbacaadbabaad
bacaaaaaabaaa=aaacaccaabadabaad
bacaacaaaabaaa=aaacaccacaadbabaad
cbaaaaacaacacaa=babaaacbaaaccaabadb
cbaacaaacaacacaa=babaaaccacaacbaabadb

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	3129	⟨a, b \| aabbabbbaab=1⟩	φ(a) = b, φ(b) = a
11	3196	⟨a, b \| abaaabbabba=1⟩	φ(a) = a, φ(b) = b