#1578 ⟨a, b, c | aaa=bb, cbc=1⟩

Properties

Sum of relation sides is 8
Infinite non-Abelian group
Element aba has infinite order
Inverses of generators:
- a^-1 = a²c⁴
- b^-1 = c²
- c^-1 = cb
Reduced group presentation: ⟨a, b | aaaabbb⟩
- Isomorphism: φ(a) = b, φ(b) = a^-1a^-1, φ(c) = a

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

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

# abc:aaa=bb,cbc=1 cb/a - -
bc=cb
ccb=1
cca=bacccc
bba=abb
aaa=bb

1 unique, 1 total

Σ	#	Presentation	Properties
8	5773	⟨a, b, c \| aa=b, ccc=bb⟩	Can Inf

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

9 total

Σ	#	Presentation	Mapping
8	3755	⟨a, b, c \| aab=1, bbccc=1⟩	φ(a) = cb, φ(b) = cc, φ(c) = a
8	3781	⟨a, b, c \| aab=1, bcccb=1⟩	φ(a) = cb, φ(b) = cc, φ(c) = a
8	3827	⟨a, b, c \| aab=1, cbbcc=1⟩	φ(a) = cb, φ(b) = cc, φ(c) = a
8	3851	⟨a, b, c \| aab=1, ccbbc=1⟩	φ(a) = cb, φ(b) = cc, φ(c) = a
8	3948	⟨a, b, c \| aba=1, bbccc=1⟩	φ(a) = cb, φ(b) = cc, φ(c) = a
8	3959	⟨a, b, c \| aba=1, bcccb=1⟩	φ(a) = cb, φ(b) = cc, φ(c) = a
8	3973	⟨a, b, c \| aba=1, cbbcc=1⟩	φ(a) = cb, φ(b) = cc, φ(c) = a
8	4752	⟨a, b, c \| aa=b, bcccb=1⟩	φ(a) = c, φ(b) = cc, φ(c) = a
8	4766	⟨a, b, c \| aa=b, cbbcc=1⟩	φ(a) = c, φ(b) = cc, φ(c) = a