#3071 ⟨a, b | aabababbbaa=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 = da³
- b^-1 = b²c(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. aaaa=c
2. bababbb=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.	a⁴ ⇒ c	[2]
6.	cb(ab)² ⇒ bc(ba)²	[17]
7.	dbcbab ⇒ b(ab)²da³	[23]
8.	ab³ ⇒ b²cbda	[37]
9.	c(ab)³ ⇒ abc(ba)²	[18]
10.	dabcbab ⇒ (ab)³da³	[40]
11.	ca(ab)³ ⇒ a²bc(ba)²	[24]
12.	da²bcbab ⇒ a(ab)³da³	[45]
13.	a³b²cb ⇒ cb³a³	[39]
14.	ca²(ab)³ ⇒ a³bc(ba)²	[42]
15.	da³bcbab ⇒ a²(ab)³da³	[50]
16.	db²cbab ⇒ (ba)²b²da³	[44]
17.	dab²cbab ⇒ (ab)³bda³	[46]
18.	da²b²cbab ⇒ a(ab)³bda³	[51]
19.	a²(ab)³b ⇒ b³cba	[49]
20.	b³cbab ⇒ da³	[31]
21.	dbcb³cb ⇒ b(ab)²da³b²a³	[52]
22.	dabcb³cb ⇒ (ab)³da³b²a³	[53]
23.	da²bcb³cb ⇒ a(ab)³da³b²a³	[55]
24.	da³bcb³cb ⇒ a²(ab)³da³b²a³	[57]
25.	d(b²cb)² ⇒ (ba)²b²da³b²a³	[54]
26.	da(b²cb)² ⇒ (ab)³bda³b²a³	[56]
27.	da²(b²cb)² ⇒ a(ab)³bda³b²a³	[58]
28.	(b³c)²b ⇒ da³b²a³	[48]

# ab:aabababbbaa=1 cda/b aaaa=c,bababbb=d magic:0
cd=1
dc=1
ac=ca
ad=da
aaaa=c
cbabab=bcbaba
dbcbab=bababdaaa
abbb=bbcbda
cababab=abcbaba
dabcbab=abababdaaa
caababab=aabcbaba
daabcbab=aabababdaaa
aaabbcb=cbbbaaa
caaababab=aaabcbaba
daaabcbab=aaabababdaaa
dbbcbab=bababbdaaa
dabbcbab=abababbdaaa
daabbcbab=aabababbdaaa
aaabababb=bbbcba
bbbcbab=daaa
dbcbbbcb=bababdaaabbaaa
dabcbbbcb=abababdaaabbaaa
daabcbbbcb=aabababdaaabbaaa
daaabcbbbcb=aaabababdaaabbaaa
dbbcbbbcb=bababbdaaabbaaa
dabbcbbbcb=abababbdaaabbaaa
daabbcbbbcb=aabababbdaaabbaaa
bbbcbbbcb=daaabbaaa

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	3260	⟨a, b \| ababbbaaaab=1⟩	φ(a) = a, φ(b) = b