#645 ⟨a, b | aaabbabba=1⟩

Properties

Sum of relation sides is 9
Infinite non-Abelian group
Element a has infinite order
Inverses of generators:
- a^-1 = da²
- b^-1 = ba⁴ca
- c^-1 = a⁴ca
- d^-1 = a³
Reduced group presentation: ⟨a, b | aaaabbabb⟩
- Isomorphism: φ(a) = a, φ(b) = 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

Auxiliary generators:
1. c = bb
2. d = acac
Reduction order:
- Right-to-left recursive path with deg(a) = deg(d) = 0, a < d; deg(c) = deg(b) = 1, c < b
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

# ab:aaabbabba=1 reversed:ad/cb bb=c,acac=d morph:2/0,4/0
ad=da
daaa=1
cd=dc
caaa=aaac
cac=ddaa
cb=bc
cab=ddaabaaaaca
bb=c

3 unique, 3 total

Σ	#	Presentation	Properties
9	1116	⟨a, b \| abaaba=bbb⟩	Can Inf
11	4262	⟨a, b \| abaabaaba=bb⟩	Can Inf
11	4346	⟨a, b \| abbababba=bb⟩	Can Inf

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

8 total

Σ	#	Presentation	Mapping
9	683	⟨a, b \| aabbabbaa=1⟩	φ(a) = a, φ(b) = b
9	690	⟨a, b \| aabbbbaab=1⟩	φ(a) = b, φ(b) = a
9	709	⟨a, b \| abaabbbba=1⟩	φ(a) = b, φ(b) = a
11	3047	⟨a, b \| aabaabbbaab=1⟩	φ(a) = baaaaca, φ(b) = bab
11	3128	⟨a, b \| aabbabbabba=1⟩	φ(a) = bab, φ(b) = baaaaca
11	3139	⟨a, b \| aabbbaabaab=1⟩	φ(a) = baaaaca, φ(b) = bab
11	3206	⟨a, b \| abaabaabbba=1⟩	φ(a) = baaaaca, φ(b) = bab
11	3221	⟨a, b \| abaabbbaaba=1⟩	φ(a) = baaaaca, φ(b) = bab