#53 ⟨a, b, c | bc=ac, cc=1⟩

Properties

Sum of relation sides is 6
Isomorphic to ℤ₂ ∗ ℕ
Infinite cancellative non-commutative monoid
Element a has infinite order
Submonoid of enveloping group: ⟨a, b | aa⟩
- Embedding: φ(a) = b, φ(b) = b, 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

#	Rule	Proof
1.	b ⇒ a	[3]
2.	c² ⇒ 1	[2]

# abc:bc=ac,cc=1 abc - -
b=a
cc=1

2 unique, 292 total

Σ	#	Presentation	Properties	Description	φ
5	7	⟨a, b, c \| aa=b, cc=1⟩	Can Inf	ℤ₂ ∗ ℕ	65
5	21	⟨a, b, c \| aa=1, bcb=1⟩	Grp Inf	ℤ₂ ∗ ℤ	225

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

8 total

Σ	#	Presentation	Mapping
6	211	⟨a, b, c \| aa=1, aab=c⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	6615	⟨a, b, c \| aa=1, aaaab=c⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	6621	⟨a, b, c \| aa=1, aabaa=c⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	7118	⟨a, b, c \| aa=1, aaab=ac⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	7125	⟨a, b, c \| aa=1, aaba=ca⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	7619	⟨a, b, c \| aa=1, aac=aab⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	7627	⟨a, b, c \| aa=1, aca=aba⟩	φ(a) = c, φ(b) = a, φ(c) = a
8	7633	⟨a, b, c \| aa=1, baa=aac⟩	φ(a) = c, φ(b) = a, φ(c) = a