#417 ⟨a, b, c | abc=b, cba=1⟩

Properties

Sum of relation sides is 7
Infinite non-Abelian group
Element a has infinite order
Inverses of generators:
- a^-1 = cb
- b^-1 = d
- c^-1 = ba
- d^-1 = b
Reduced group presentation: ⟨a, b | aaba^-1b^-1⟩
- Isomorphism: φ(a) = aba^-1, φ(b) = a, 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 = ac
Reduction order:
- Left-to-right recursive path with deg(b) = deg(d) = 0, b < d; deg(a) = deg(c) = 1, a < c
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

# abc:abc=b,cba=1 bd/ac ac=d morph:2/0
bd=1
db=1
da=bad
bba=ab
bc=cbb
dc=cdd
ac=d
cba=1

6 unique, 16 total

Σ	#	Presentation	Properties	φ
7	537	⟨a, b, c \| ba=ac, bb=c⟩	Can Inf	4
7	554	⟨a, b, c \| bb=ac, ca=b⟩	Can Inf	3
7	1041	⟨a, b, c \| ab=c, bca=c⟩	Can Inf	2
8	5567	⟨a, b, c \| ab=c, baca=c⟩	Can Inf
8	5675	⟨a, b, c \| aa=b, aac=ca⟩	Can Inf	1
8	6101	⟨a, b, c \| ab=c, cac=ba⟩	Can Inf

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

10 total

Σ	#	Presentation	Mapping
7	700	⟨a, b, c \| abc=1, acba=1⟩	φ(a) = b, φ(b) = dc, φ(c) = ba
7	722	⟨a, b, c \| abc=1, cbaa=1⟩	φ(a) = b, φ(b) = dc, φ(c) = ba
7	723	⟨a, b, c \| abc=1, cbba=1⟩	φ(a) = bad, φ(b) = b, φ(c) = c
8	1980	⟨a, b, c \| abc=ba, cba=1⟩	φ(a) = bab, φ(b) = c, φ(c) = badcb
8	2795	⟨a, b, c \| abc=b, caba=1⟩	φ(a) = bba, φ(b) = c, φ(c) = dc
8	4062	⟨a, b, c \| abc=1, bacba=1⟩	φ(a) = bba, φ(b) = c, φ(c) = d
8	4097	⟨a, b, c \| abc=1, cacba=1⟩	φ(a) = cb, φ(b) = d, φ(c) = ba
8	4099	⟨a, b, c \| abc=1, cbaba=1⟩	φ(a) = bba, φ(b) = c, φ(c) = d
8	4539	⟨a, b, c \| abc=1, acba=b⟩	φ(a) = dcb, φ(b) = dc, φ(c) = babab
8	4540	⟨a, b, c \| abc=1, acba=c⟩	φ(a) = bab, φ(b) = dcc, φ(c) = ba