#678 ⟨a, b | aabbaaabb=1⟩

Properties

Sum of relation sides is 9
Infinite non-Abelian group
Element c has infinite order
Inverses of generators:
- a^-1 = a²d
- b^-1 = a²b²d⁴bc⁵
- c^-1 = d
- d^-1 = c
Reduced group presentation: ⟨a, b | aaabbaabb⟩
- 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 = aaa
2. d = bbaabb
Reduction order:
- Left-to-right recursive path with deg(d) = 0; deg(c) = 1; deg(a) = deg(b) = 2, a < b
Certificate: derivations of all rewriting rules from the defining relations.

# ab:aabbaaabb=1 d/c/ab aaa=c,bbaabb=d magic:0
cd=1
dc=1
da=ad
ca=ac
dddddb=bddddd
cb=ddddbccccc
ba=addbcc
dbb=bbd
aaa=c
bbbb=add

2 unique, 3 total

Σ	#	Presentation	Properties	φ
9	860	⟨a, b \| abbaabba=b⟩	Can Inf	1
11	4865	⟨a, b \| abbaabba=bab⟩	Can Inf

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

12 total

Σ	#	Presentation	Mapping
9	680	⟨a, b \| aabbaabba=1⟩	φ(a) = a, φ(b) = b
9	722	⟨a, b \| abbaaabba=1⟩	φ(a) = a, φ(b) = b
9	724	⟨a, b \| abbaabbba=1⟩	φ(a) = b, φ(b) = a
11	3056	⟨a, b \| aababaaabba=1⟩	φ(a) = aad, φ(b) = aaba
11	3101	⟨a, b \| aabbaaababa=1⟩	φ(a) = aad, φ(b) = aaba
11	3107	⟨a, b \| aabbaabaabb=1⟩	φ(a) = aabbddddbccccc, φ(b) = bab
11	3110	⟨a, b \| aabbaabbaab=1⟩	φ(a) = aabbddddbccccc, φ(b) = bab
11	3193	⟨a, b \| abaaabbaaab=1⟩	φ(a) = aad, φ(b) = aaba
11	3216	⟨a, b \| abaabbaabba=1⟩	φ(a) = aabbddddbccccc, φ(b) = bab
11	3278	⟨a, b \| abbaabaabba=1⟩	φ(a) = aabbddddbccccc, φ(b) = bab
11	3282	⟨a, b \| abbaabbabba=1⟩	φ(a) = bab, φ(b) = aabbddddbccccc
11	3300	⟨a, b \| abbbababbba=1⟩	φ(a) = aaba, φ(b) = aad