#3073 ⟨a, b | aabababbbba=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 = da²
- b^-1 = b³c(ba)²
- 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(a) = 0, c < d < a; deg(b) = 1
Auxiliary generators:
1. aaa=c
2. bababbbb=d
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	cd ⇒ 1	[9]
2.	dc ⇒ 1	[11]
3.	ac ⇒ ca	[5]
4.	ad ⇒ da	[10]
5.	a³ ⇒ c	[2]
6.	cb(ab)² ⇒ bc(ba)²	[38]
7.	dbcbab ⇒ b(ab)²da²	[28]
8.	c(ab)³ ⇒ abc(ba)²	[39]
9.	dabcbab ⇒ (ab)³da²	[36]
10.	ca(ab)³ ⇒ a²bc(ba)²	[40]
11.	da²bcbab ⇒ a(ab)³da²	[41]
12.	db²cbab ⇒ (ba)²b²da²	[29]
13.	ab⁴ ⇒ b³cbda	[34]
14.	dab²cbab ⇒ (ab)³bda²	[42]
15.	a²b³cb ⇒ cb⁴a²	[24]
16.	da²b²cbab ⇒ a(ab)³bda²	[33]
17.	db³cbab ⇒ (ba)²b³da²	[26]
18.	dab³cbab ⇒ (ab)³b²da²	[44]
19.	a(ab)³b² ⇒ b⁴cba	[32]
20.	b⁴cbab ⇒ da²	[25]
21.	dbcb⁴cb ⇒ b(ab)²da²b³a²	[46]
22.	dabcb⁴cb ⇒ (ab)³da²b³a²	[47]
23.	da²bcb⁴cb ⇒ a(ab)³da²b³a²	[48]
24.	db²cb⁴cb ⇒ (ba)²b²da²b³a²	[49]
25.	dab²cb⁴cb ⇒ (ab)³bda²b³a²	[50]
26.	da²b²cb⁴cb ⇒ a(ab)³bda²b³a²	[51]
27.	d(b³cb)² ⇒ (ba)²b³da²b³a²	[52]
28.	da(b³cb)² ⇒ (ab)³b²da²b³a²	[53]
29.	(b⁴c)²b ⇒ da²b³a²	[31]

# ab:aabababbbba=1 cda/b aaa=c,bababbbb=d magic:0
cd=1
dc=1
ac=ca
ad=da
aaa=c
cbabab=bcbaba
dbcbab=bababdaa
cababab=abcbaba
dabcbab=abababdaa
caababab=aabcbaba
daabcbab=aabababdaa
dbbcbab=bababbdaa
abbbb=bbbcbda
dabbcbab=abababbdaa
aabbbcb=cbbbbaa
daabbcbab=aabababbdaa
dbbbcbab=bababbbdaa
dabbbcbab=abababbbdaa
aabababbb=bbbbcba
bbbbcbab=daa
dbcbbbbcb=bababdaabbbaa
dabcbbbbcb=abababdaabbbaa
daabcbbbbcb=aabababdaabbbaa
dbbcbbbbcb=bababbdaabbbaa
dabbcbbbbcb=abababbdaabbbaa
daabbcbbbbcb=aabababbdaabbbaa
dbbbcbbbbcb=bababbbdaabbbaa
dabbbcbbbbcb=abababbbdaabbbaa
bbbbcbbbbcb=daabbbaa

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	3264	⟨a, b \| ababbbbaaab=1⟩	φ(a) = a, φ(b) = b