#2940 ⟨a, b | aaabababbba=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 = (ba)²ca³
- b^-1 = b²d
- 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:
- Right-to-left recursive path with deg(c) = deg(d) = deg(b) = 0, c < d < b; deg(a) = 1
Auxiliary generators:
1. bbb=c
2. aaaababa=d
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	dc ⇒ 1	[17]
2.	cd ⇒ 1	[39]
3.	cb ⇒ bc	[5]
4.	db ⇒ bd	[32]
5.	b³ ⇒ c	[2]
6.	a(ba)²c ⇒ (ba)²ca	[41]
7.	abacad ⇒ b²da(ba)²	[42]
8.	(ab)³c ⇒ (ba)²cab	[43]
9.	abacabd ⇒ b²d(ab)³	[44]
10.	(ab)³bc ⇒ (ba)²cab²	[45]
11.	abacab²d ⇒ b²d(ab)³b	[47]
12.	abaca²d ⇒ b²da²(ba)²	[48]
13.	a⁴b ⇒ bdaca³	[52]
14.	abaca²bd ⇒ b²da(ab)³	[49]
15.	aca³b² ⇒ b²a⁴c	[57]
16.	abaca²b²d ⇒ b²da(ab)³b	[58]
17.	abaca³d ⇒ b²da³(ba)²	[59]
18.	abaca³bd ⇒ b²da²(ab)³	[61]
19.	a²(ab)³b ⇒ baca⁴	[38]
20.	abaca⁴ ⇒ b²d	[37]
21.	aca⁴cad ⇒ b²a³b²da(ba)²	[63]
22.	aca⁴cabd ⇒ b²a³b²d(ab)³	[64]
23.	aca⁴cab²d ⇒ b²a³b²d(ab)³b	[65]
24.	aca⁴ca²d ⇒ b²a³b²da²(ba)²	[66]
25.	aca⁴ca²bd ⇒ b²a³b²da(ab)³	[67]
26.	aca⁴ca²b²d ⇒ b²a³b²da(ab)³b	[68]
27.	(aca³)²d ⇒ b²a³b²da³(ba)²	[69]
28.	(aca³)²bd ⇒ b²a³b²da²(ab)³	[70]
29.	a(ca⁴)² ⇒ b²a³b²d	[62]

# ab:aaabababbba=1 reversed:cdb/a bbb=c,aaaababa=d magic:1
dc=1
cd=1
cb=bc
db=bd
bbb=c
ababac=babaca
abacad=bbdababa
abababc=babacab
abacabd=bbdababab
abababbc=babacabb
abacabbd=bbdabababb
abacaad=bbdaababa
aaaab=bdacaaa
abacaabd=bbdaababab
acaaabb=bbaaaac
abacaabbd=bbdaabababb
abacaaad=bbdaaababa
abacaaabd=bbdaaababab
aaabababb=bacaaaa
abacaaaa=bbd
acaaaacad=bbaaabbdababa
acaaaacabd=bbaaabbdababab
acaaaacabbd=bbaaabbdabababb
acaaaacaad=bbaaabbdaababa
acaaaacaabd=bbaaabbdaababab
acaaaacaabbd=bbaaabbdaabababb
acaaaacaaad=bbaaabbdaaababa
acaaaacaaabd=bbaaabbdaaababab
acaaaacaaaa=bbaaabbd

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	3157	⟨a, b \| aabbbbababa=1⟩	φ(a) = b, φ(b) = a