#2972 ⟨a, b | aaabbababba=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 = bcb(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. bbababb=d
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	cd ⇒ 1	[16]
2.	dc ⇒ 1	[19]
3.	ac ⇒ ca	[5]
4.	ad ⇒ da	[20]
5.	a⁴ ⇒ c	[2]
6.	ab²cb ⇒ bcb²a	[42]
7.	abab² ⇒ cb²abda	[23]
8.	a³b²ab ⇒ bab²a³	[29]
9.	a³bcb² ⇒ cb²cbda³	[41]
10.	cb²(ab)² ⇒ bcb(ba)²	[39]
11.	dbcb²ab ⇒ b²(ab)²da³	[27]
12.	cab²(ab)² ⇒ abcb(ba)²	[43]
13.	dabcb²ab ⇒ ab²(ab)²da³	[46]
14.	ca²b²(ab)² ⇒ a²bcb(ba)²	[45]
15.	da²bcb²ab ⇒ a²b²(ab)²da³	[48]
16.	b²cb²ab ⇒ da³	[32]
17.	a³b³cb² ⇒ bab²a³bcbda³	[36]
18.	(cb²ab)² ⇒ bcb²aba²b²a³	[49]
19.	d(bcb²)² ⇒ b²(ab)²da³bcbda³	[53]
20.	cab²abcb²ab ⇒ abcb²aba²b²a³	[50]
21.	da(bcb²)² ⇒ ab²(ab)²da³bcbda³	[55]
22.	ca²b²abcb²ab ⇒ a²bcb²aba²b²a³	[54]
23.	da²(bcb²)² ⇒ a²b²(ab)²da³bcbda³	[56]
24.	b(bcb²)² ⇒ da³bcbda³	[37]
25.	cb²a(bcb²)² ⇒ bcb²aba²b²a³bcbda³	[58]
26.	cab²a(bcb²)² ⇒ abcb²aba²b²a³bcbda³	[59]
27.	ca²b²a(bcb²)² ⇒ a²bcb²aba²b²a³bcbda³	[60]

# ab:aaabbababba=1 cda/b aaaa=c,bbababb=d magic:0
cd=1
dc=1
ac=ca
ad=da
aaaa=c
abbcb=bcbba
ababb=cbbabda
aaabbab=babbaaa
aaabcbb=cbbcbdaaa
cbbabab=bcbbaba
dbcbbab=bbababdaaa
cabbabab=abcbbaba
dabcbbab=abbababdaaa
caabbabab=aabcbbaba
daabcbbab=aabbababdaaa
bbcbbab=daaa
aaabbbcbb=babbaaabcbdaaa
cbbabcbbab=bcbbabaabbaaa
dbcbbbcbb=bbababdaaabcbdaaa
cabbabcbbab=abcbbabaabbaaa
dabcbbbcbb=abbababdaaabcbdaaa
caabbabcbbab=aabcbbabaabbaaa
daabcbbbcbb=aabbababdaaabcbdaaa
bbcbbbcbb=daaabcbdaaa
cbbabcbbbcbb=bcbbabaabbaaabcbdaaa
cabbabcbbbcbb=abcbbabaabbaaabcbdaaa
caabbabcbbbcbb=aabcbbabaabbaaabcbdaaa

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	3122	⟨a, b \| aabbababbaa=1⟩	φ(a) = a, φ(b) = b