#6222 ⟨a, b, c | aa=1, bacacb=1⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 8
Infinite non-Abelian group
Element b has infinite order
Inverses of generators:
- a^-1 = cdb
- b^-1 = cdbcd
- c^-1 = dbcdb
- d^-1 = bcdbc

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

#	Rule	Proof
1.	cdbcb ⇒ d	[10]
2.	b²d ⇒ db²	[23]
3.	b²cdbc ⇒ db	[20]
4.	a ⇒ cdb	[9]
5.	dbd ⇒ b	[22]
6.	(dbc)² ⇒ 1	[21]
7.	(cdb)² ⇒ 1	[12]
8.	(bcd)² ⇒ 1	[13]
9.	b(cdbc)² ⇒ cdbcd	[17]
10.	dcdbcd ⇒ cdbc	[15]
11.	cdbcd² ⇒ bd(cdbc)²	[25]
12.	(cdbc)²d ⇒ d(cdbc)²	[19]

# abc:aa=1,bacacb=1 bc/da acb=d morph:3/0
cdbcb=d
bbd=dbb
bbcdbc=db
a=cdb
dbd=b
dbcdbc=1
cdbcdb=1
bcdbcd=1
bcdbccdbc=cdbcd
dcdbcd=cdbc
cdbcdd=bdcdbccdbc
cdbccdbcd=dcdbccdbc