#2272 ⟨a, b, c | abac=1, caba=1⟩

Properties

Sum of relation sides is 8
Isomorphic to ℤ ∗ ℤ
Infinite non-Abelian group
Element a has infinite order
Inverses of generators:
- a^-1 = bac
- b^-1 = aca
- c^-1 = aba
Reduced group presentation: ⟨a, b | ⟩
- Isomorphism: φ(a) = a, φ(b) = a^-1b^-1a^-1, φ(c) = b

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

# abc:abac=1,caba=1 abc - -
cab=bac
abac=1
baca=1

2 unique, 7 total

Σ	#	Presentation	Properties	Description	φ
8	4375	⟨a, b, c \| aba=1, aabc=c⟩	Can Inf	ℕ ∗ ℤ	2
8	5674	⟨a, b, c \| aa=b, aac=bc⟩	Can Inf	ℕ ∗ ℕ	3

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

1 total

Σ	#	Presentation	Mapping
8	2292	⟨a, b, c \| abbc=1, bcab=1⟩	φ(a) = ababacbac, φ(b) = a, φ(c) = c