#679 ⟨a, b, c | aba=1, cccc=1⟩

Properties

Sum of relation sides is 7
Isomorphic to ℤ₄ ∗ ℤ
Infinite non-Abelian group
Element a has infinite order
Inverses of generators:
- a^-1 = ab
- b^-1 = a²
- c^-1 = c³
Reduced group presentation: ⟨a, b | aaaa⟩
- Isomorphism: φ(a) = b, φ(b) = b^-1b^-1, 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

# abc:aba=1,cccc=1 abc - -
ba=ab
aab=1
cccc=1

2 unique, 10 total

Σ	#	Presentation	Properties	Description	φ
7	774	⟨a, b, c \| aa=b, cccc=1⟩	Can Inf	ℤ₄ ∗ ℕ	8
8	3138	⟨a, b, c \| ac=ab, aaaa=1⟩	Can Inf	ℤ₄ ∗ ℕ

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

11 total

Σ	#	Presentation	Mapping
8	2036	⟨a, b, c \| aaaa=1, abcb=1⟩	φ(a) = abca, φ(b) = ab, φ(c) = cccaa
8	2040	⟨a, b, c \| aaaa=1, bacb=1⟩	φ(a) = c, φ(b) = ab, φ(c) = cccaa
8	2043	⟨a, b, c \| aaaa=1, bbcb=1⟩	φ(a) = c, φ(b) = a, φ(c) = ababab
8	2232	⟨a, b, c \| aabc=1, cbcb=1⟩	φ(a) = ccc, φ(b) = ccab, φ(c) = a
8	2248	⟨a, b, c \| abab=1, bcca=1⟩	φ(a) = ccab, φ(b) = a, φ(c) = ccc
8	2251	⟨a, b, c \| abab=1, cbac=1⟩	φ(a) = abcc, φ(b) = a, φ(c) = ccc
8	2673	⟨a, b, c \| aba=c, cccc=1⟩	φ(a) = a, φ(b) = abcab, φ(c) = c
8	6497	⟨a, b, c \| ab=1, bacccc=1⟩	φ(a) = ab, φ(b) = a, φ(c) = c
8	6568	⟨a, b, c \| ab=1, bcccca=1⟩	φ(a) = ab, φ(b) = a, φ(c) = c
8	6599	⟨a, b, c \| ab=1, cbaccc=1⟩	φ(a) = ab, φ(b) = a, φ(c) = c
8	6612	⟨a, b, c \| ab=1, ccbacc=1⟩	φ(a) = ab, φ(b) = a, φ(c) = c