#21039 ⟨a, b | ab=aa, bbba=bbb⟩

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Contents

  1. Properties
  2. Finite structure
  3. Cayley table
  4. Right Cayley graph
  5. Left Cayley graph
  6. Rewriting system
  7. Same cardinality

Properties

Finite structure

Elements in the center commute with all other elements.
A left zero element x satisfies xy = x for all y.
An idempotent element x satisfies x2 = x.
The index and period of x is the least m (index) and n (period) such that x(m+n) = xm.

Cayley table

Idempotents are shown in bold.

1aba2bab2a3ba2b2ab3a4ba3b2a2ba4b2a3b2a4
11aba2bab2a3ba2b2ab3a4ba3b2a2ba4b2a3b2a4
aaa2a2a3a3a3a4a4a4a4a4a4a4a4a4a4
bbbab2ba2b2ab3ba3b2a2b3b3ba4b2a3b3b2a4b3b3
a2a2a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4
bababa2ba2ba3ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2b2b2ab3b2a2b3b3b2a3b3b3b3b2a4b3b3b3b3b3
a3a3a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
ba2ba2ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2ab2ab2a2b2a2b2a3b2a3b2a3b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
b3b3b3b3b3b3b3b3b3b3b3b3b3b3b3b3b3
a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4a4
ba3ba3ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2a2b2a2b2a3b2a3b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4ba4
b2a3b2a3b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4
b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4b2a4

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Rewriting system

Format:
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:
Result: 1 
1
#RuleProof
1. ab ⇒ a2 [1]
2. b3a ⇒ b3 [2]
3. b4 ⇒ b3 [4]
4. a5 ⇒ a4 [3]
# ab:ab=aa,bbba=bbb ab - -
ab=aa
bbba=bbb
bbbb=bbb
aaaaa=aaaa

Same cardinality

20 unique, 175 total

Σ#PresentationPropertiesDescriptionφ
8628⟨a, b | bb=aa, abab=1⟩Grp 1658
91331⟨a, b | aaaa=b, bbbb=1⟩Grp Com 16ℤ1667
92051⟨a, b | aab=a, bbbb=b⟩Noc 164
103808⟨a, b | aaab=ba, abab=1⟩Grp 167
104630⟨a, b | aaaa=a, bbbb=a⟩Noc Com 16ℕ(16 = 4)1
104648⟨a, b | aaaa=b, bbbb=a⟩Noc Com 16ℕ(16 = 1)5
106205⟨a, b | aba=b, aaaabb=1⟩Grp 163
1112164⟨a, b | aaaa=aa, bbbb=a⟩Noc Com 16ℕ(16 = 8)
1112194⟨a, b | aaaa=ab, bbbb=a⟩Noc Com 16ℕ(16 = 5)2
1112212⟨a, b | aaaa=bb, bbbb=a⟩Noc Com 16ℕ(16 = 2)
1112306⟨a, b | aaab=bb, abba=a⟩Noc 166
1113251⟨a, b | bab=aab, bbb=aa⟩Noc 16
1113259⟨a, b | bab=aba, bbb=aa⟩Noc 16
1116028⟨a, b | aaa=ab, babb=bb⟩Noc 16
1116032⟨a, b | aaa=ab, bbaa=bb⟩Noc 16
1116060⟨a, b | aaa=bb, abab=aa⟩Noc 16
1116371⟨a, b | aba=aa, bbbb=ab⟩Noc 16
1118811⟨a, b | aaa=b, abbbbb=b⟩Noc Com 16ℕ(16 = 3)2
1120251⟨a, b | aba=b, aaaa=abb⟩Noc 16
1124660⟨a, b | aa=a, ababab=bb⟩Noc 16