#6267 ⟨a, b, c | aa=1, bcbbbc=1⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 8
Infinite non-Abelian group
Element ba has infinite order
Inverses of generators:
- a^-1 = a
- b^-1 = d²b
- c^-1 = db³
- d^-1 = db²

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Auxiliary generators:
1. d = bc
Reduction order:
- Right-to-left recursive path with deg(a) = deg(d) = 0, a < d; deg(c) = deg(b) = 1, c < b
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	a² ⇒ 1	[1]
2.	cd² ⇒ d²c	[17]
3.	d⁴b ⇒ cd	[13]
4.	d²bd ⇒ c	[7]
5.	bd² ⇒ d²b	[12]
6.	cdc ⇒ d⁵	[18]
7.	bc ⇒ d	[3]
8.	d²cb ⇒ bdcd	[20]
9.	cdb ⇒ d²	[19]
10.	cbd ⇒ bdc	[11]
11.	d²b² ⇒ 1	[6]
12.	b²d ⇒ db²	[5]
13.	c²b ⇒ (bd)²cd	[21]
14.	cb² ⇒ bd	[8]

# abc:aa=1,bcbbbc=1 reversed:ad/cb bc=d morph:2/0
aa=1
cdd=ddc
ddddb=cd
ddbd=c
bdd=ddb
cdc=ddddd
bc=d
ddcb=bdcd
cdb=dd
cbd=bdc
ddbb=1
bbd=dbb
ccb=bdbdcd
cbb=bd