#1585 ⟨a, b, c | aaa=bc, acb=1⟩

Properties

Sum of relation sides is 8
Infinite non-Abelian group
Element a has infinite order
Inverses of generators:
- a^-1 = d
- b^-1 = cd³
- c^-1 = ba
- d^-1 = a
Reduced group presentation: ⟨a, b | aaabab^-1⟩
- Isomorphism: φ(a) = a, φ(b) = aaab, φ(c) = b^-1

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 = cb
Reduction order:
- Left-to-right recursive path with deg(a) = deg(d) = 0, a < d; deg(b) = 1; deg(c) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

# abc:aaa=bc,acb=1 ad/b/c cb=d morph:2/4
ad=1
da=1
aab=dbd
ddb=aba
cb=d
ac=cddd
dc=caaa
bc=aaa

2 unique, 6 total

Σ	#	Presentation	Properties	φ
8	2830	⟨a, b, c \| aaa=b, acb=c⟩	Can Inf	1
8	2970	⟨a, b, c \| aab=c, aca=b⟩	Can Inf	3

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

7 total

Σ	#	Presentation	Mapping
8	3299	⟨a, b, c \| bb=ac, bcba=1⟩	φ(a) = aaba, φ(b) = a, φ(c) = c
8	5068	⟨a, b, c \| ab=c, baccc=1⟩	φ(a) = ac, φ(b) = ba, φ(c) = a
8	5090	⟨a, b, c \| ab=c, bccac=1⟩	φ(a) = aba, φ(b) = c, φ(c) = a
8	5103	⟨a, b, c \| ab=c, cbacc=1⟩	φ(a) = ac, φ(b) = ba, φ(c) = a
8	6548	⟨a, b, c \| ab=1, bcaccc=1⟩	φ(a) = c, φ(b) = ba, φ(c) = a
8	6566	⟨a, b, c \| ab=1, bcccac=1⟩	φ(a) = ba, φ(b) = c, φ(c) = a
8	6605	⟨a, b, c \| ab=1, cbcacc=1⟩	φ(a) = c, φ(b) = ba, φ(c) = a