#4771 ⟨a, b, c | aa=b, ccccc=1⟩

Properties

Sum of relation sides is 8
Isomorphic to ℤ₅ ∗ ℕ
Infinite cancellative non-commutative monoid
Element a has infinite order
Submonoid of enveloping group: ⟨a, b | aaaaa⟩
- Embedding: φ(a) = b, φ(b) = bb, 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(a) = deg(c) = 0, a < c; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	c⁵ ⇒ 1	[2]
2.	b ⇒ a²	[1]

# abc:aa=b,ccccc=1 ac/b - -
ccccc=1
b=aa

1 unique, 1 total

Σ	#	Presentation	Properties	Description
8	3978	⟨a, b, c \| aba=1, ccccc=1⟩	Grp Inf	ℤ₅ ∗ ℤ

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

1 total

Σ	#	Presentation	Mapping
8	4983	⟨a, b, c \| ab=c, aaaaa=1⟩	φ(a) = c, φ(b) = a, φ(c) = ca