#631 ⟨a, b | aaabaabba=1⟩

Properties

Sum of relation sides is 9
Infinite non-Abelian group
Element a has infinite order
Inverses of generators:
- a^-1 = dba
- b^-1 = a²d
- c^-1 = d
- d^-1 = c
Reduced group presentation: ⟨a, b | aabbab^-1b^-1⟩
- Isomorphism: φ(a) = b^-1, φ(b) = bba

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

# ab:aaabaabba=1 dc/a/b baa=c,bcaa=d morph:3/0,4/0
dc=1
cd=1
daa=aacc
caa=aadd
baa=c
cb=dbd
ddb=bc
adb=dba

5 unique, 5 total

Σ	#	Presentation	Properties
7	222	⟨a, b \| aabba=bb⟩	Can Inf
8	406	⟨a, b \| ababbba=b⟩	Can Inf
9	1074	⟨a, b \| aababa=bab⟩	Can Inf
11	3737	⟨a, b \| abaabababa=b⟩	Can Inf
11	5142	⟨a, b \| aabaaba=baab⟩	Can Inf

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

5 total

Σ	#	Presentation	Mapping
9	641	⟨a, b \| aaabbaaba=1⟩	φ(a) = dba, φ(b) = aaaab
9	662	⟨a, b \| aabaabbaa=1⟩	φ(a) = a, φ(b) = dbadbaaab
9	676	⟨a, b \| aabbaaaab=1⟩	φ(a) = a, φ(b) = dbadbaaab
9	698	⟨a, b \| abaaaabba=1⟩	φ(a) = dba, φ(b) = aaaab
9	727	⟨a, b \| abbabbbba=1⟩	φ(a) = aaaab, φ(b) = dba