#3229 ⟨a, b | ababaaaabba=1⟩

Quick links

Properties
Rewriting system

Properties

Presentation has sum-of-sides 11
Infinite non-Abelian group
Inverses of generators:
- a^-1 = da³
- b^-1 = ba(ab)²c
- 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(a) = 0, c < a; deg(d) = 1; deg(b) = 2
Auxiliary generators:
1. aaaa=c
2. bbaabab=d
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	ac ⇒ ca	[5]
2.	a⁴ ⇒ c	[2]
3.	dc ⇒ 1	[14]
4.	cd ⇒ 1	[27]
5.	ad ⇒ da	[29]
6.	cb²a²b ⇒ ba(ab)²a³	[34]
7.	abcb² ⇒ ba(ab)²a²	[37]
8.	(ab)²cb ⇒ bcb²a²	[48]
9.	cab²a²b ⇒ (aba)²ba³	[35]
10.	a²b²a²b ⇒ babcbda³	[51]
11.	(a²b)²ab ⇒ cbabcbd	[42]
12.	dba(ab)² ⇒ b²a²bda	[31]
13.	d(aba)²b ⇒ ab²a²bda	[33]
14.	b²a(ab)² ⇒ d	[3]
15.	cb³abcb ⇒ ba²b(aba²)²ba	[58]
16.	cb²cbabcb ⇒ ba((ab)²c)²	[41]
17.	cb²(aba)²b ⇒ ba(ab)²ca³b²da²	[62]
18.	cab³abcb ⇒ aba²b(aba²)²ba	[59]
19.	cab²cbabcb ⇒ aba((ab)²c)²	[55]
20.	cab²(aba)²b ⇒ (aba)²bca³b²da²	[63]
21.	a²b³abcb ⇒ babcbda(a²b)²a	[65]
22.	a²b²cbabcb ⇒ babcb(ab)²c	[54]
23.	a²b²(aba)²b ⇒ babcba³b²da²	[61]
24.	(a²b)²bcb² ⇒ cb(abcbd)²a²	[47]
25.	dba²b²cb² ⇒ b²a²bda²bcbda²	[46]
26.	daba²b²cb² ⇒ ab²a²bda²bcbda²	[57]
27.	b²a²b²cb² ⇒ dabcbda²	[45]
28.	cb²aba²b²cb² ⇒ ba(ab)²ca³b²da³bcbda²	[68]
29.	cab²aba²b²cb² ⇒ (aba)²bca³b²da³bcbda²	[69]
30.	a²b²aba²b²cb² ⇒ babcba³b²da³bcbda²	[67]

# ab:ababaaaabba=1 ca/d/b aaaa=c,bbaabab=d magic:0
ac=ca
aaaa=c
dc=1
cd=1
ad=da
cbbaab=baababaaa
abcbb=baababaa
ababcb=bcbbaa
cabbaab=abaababaaa
aabbaab=babcbdaaa
aabaabab=cbabcbd
dbaabab=bbaabda
dabaabab=abbaabda
bbaabab=d
cbbbabcb=baababaaabaaba
cbbcbabcb=baababcababc
cbbabaabab=baababcaaabbdaa
cabbbabcb=abaababaaabaaba
cabbcbabcb=abaababcababc
cabbabaabab=abaababcaaabbdaa
aabbbabcb=babcbdaaabaaba
aabbcbabcb=babcbababc
aabbabaabab=babcbaaabbdaa
aabaabbcbb=cbabcbdabcbdaa
dbaabbcbb=bbaabdaabcbdaa
dabaabbcbb=abbaabdaabcbdaa
bbaabbcbb=dabcbdaa
cbbabaabbcbb=baababcaaabbdaaabcbdaa
cabbabaabbcbb=abaababcaaabbdaaabcbdaa
aabbabaabbcbb=babcbaaabbdaaabcbdaa