#3060 ⟨a, b | aababaabbaa=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 = bcbaba²
- 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. babaabb=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.	cbaba²b ⇒ bcbaba²	[16]
7.	dbcbab ⇒ baba²bda²	[38]
8.	ab²cb ⇒ ba²b²a³	[48]
9.	aba²b² ⇒ bcbabda²	[28]
10.	c(ab)²a²b ⇒ abcbaba²	[17]
11.	dabcbab ⇒ (ab)²a²bda²	[39]
12.	a²bcbab ⇒ cb²cbda³	[40]
13.	a(ab)²a²b ⇒ b²cba	[24]
14.	b²cbab ⇒ da²	[23]
15.	cbab³cb ⇒ bcb(aba²)²bda³	[57]
16.	cba(bcb)² ⇒ bcbabca²(ba)²	[35]
17.	cb(ab)²cbab ⇒ bcbabcb²a²	[53]
18.	dbcb²a²b² ⇒ baba²bda²bcbda	[45]
19.	cabab³cb ⇒ abcb(aba²)²bda³	[58]
20.	caba(bcb)² ⇒ abcbabca²(ba)²	[41]
21.	c(ab)³cbab ⇒ (abcb)²ba²	[54]
22.	dabcb²a²b² ⇒ (ab)²a²bda²bcbda	[46]
23.	a²bcb²a²b² ⇒ cb²cbda³bcbda	[47]
24.	a²bab³cb ⇒ b²cb(a²b)²da³	[56]
25.	a²ba(bcb)² ⇒ b(bc)²a(ba)²	[36]
26.	a(ab)³cbab ⇒ b²cba³b²a²	[50]
27.	b²cb²a²b² ⇒ da²bcbda	[44]
28.	cb(ab)²cb²a²b² ⇒ bcbabcb²a²bcbda	[61]
29.	c(ab)³cb²a²b² ⇒ (abcb)²ba²bcbda	[62]
30.	a(ab)³cb²a²b² ⇒ b²cba³b²a²bcbda	[59]

# ab:aababaabbaa=1 cda/b aaaa=c,babaabb=d magic:0
cd=1
dc=1
ac=ca
ad=da
aaaa=c
cbabaab=bcbabaa
dbcbab=babaabdaa
abbcb=baabbaaa
abaabb=bcbabdaa
cababaab=abcbabaa
dabcbab=ababaabdaa
aabcbab=cbbcbdaaa
aababaab=bbcba
bbcbab=daa
cbabbbcb=bcbabaaabaabdaaa
cbabcbbcb=bcbabcaababa
cbababcbab=bcbabcbbaa
dbcbbaabb=babaabdaabcbda
cababbbcb=abcbabaaabaabdaaa
cababcbbcb=abcbabcaababa
cabababcbab=abcbabcbbaa
dabcbbaabb=ababaabdaabcbda
aabcbbaabb=cbbcbdaaabcbda
aababbbcb=bbcbaabaabdaaa
aababcbbcb=bbcbcababa
aabababcbab=bbcbaaabbaa
bbcbbaabb=daabcbda
cbababcbbaabb=bcbabcbbaabcbda
cabababcbbaabb=abcbabcbbaabcbda
aabababcbbaabb=bbcbaaabbaabcbda

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	3214	⟨a, b \| abaabbaaaab=1⟩	φ(a) = a, φ(b) = b