#2621 ⟨a, b, c | aba=b, bccb=1⟩

Contents

Properties
Rewriting system
Isomorphic instances

Properties

Sum of relation sides is 8
Infinite non-Abelian group
Element a has infinite order
Inverses of generators:
- a^-1 = babc²
- b^-1 = bc²
- c^-1 = cd
- d^-1 = c²

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 = bb
Reduction order:
- Left-to-right shortlex with a < b < c < d
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	b² ⇒ d	[3]
2.	da ⇒ ad	[5]
3.	db ⇒ bd	[4]
4.	dc ⇒ cd	[12]
5.	aba ⇒ b	[1]
6.	c²a ⇒ ac²	[14]
7.	c²b ⇒ bc²	[6]
8.	c²d ⇒ 1	[9]

# abc:aba=b,bccb=1 abcd bb=d morph:2/0
bb=d
da=ad
db=bd
dc=cd
aba=b
cca=acc
ccb=bcc
ccd=1

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
8	2626	⟨a, b, c \| aba=b, cbbc=1⟩	φ(a) = a, φ(b) = b, φ(c) = c